Three similar temperaments

Most discussions of tunings on this list seem to deal with temperament
families that are related by having the same commas. I'd like to spend just
a few minutes discussing three temperaments that are similar because they
do "the same things" with different tones, and thus need to temper out
different commas.

Naturally, this will all be stuff that probably seems really obvious to
people who have been doing this for a while, but that took time to "click"
with me. I'm writing it in the hope that some newbies will understand some
of the reasons to temper specific commas, and what effects that tempering
can have.

-----
Syntonic Comma, 81/80, | -4 4 -1 >
-----

Meantone is one of the traditional temperaments of Western music, and is
defined by two things:
1. It tempers out the comma 81/80, and
2. It has a period of an octave and a generator of 3/2.
I can say this with some confidence because, for reasons I don't
understand, that information has been encapsulated in the "wedgie" (wedge
product) of the comma, and people who know the math better than I do have
said so.

Tempering out 81/80 means something very important in relation to the
generator: It means that four stacked 3/2s (octave equivalent) = 5/4. In
traditional musical terms, that's the same as saying that four fifths
equals a major third.

That's good if you want 5-limit harmonies in relatively small scale sizes
-- if it took, say, eight fifths to get to a major third, then you'd need
much larger scales to get the gamut of notes needed for 5-limit harmony.
(This is related to "complexity".)

Four stacked 3/2s = (3/2)^4 = 81/16, which is octave-equivalent to 81/64,
which is about 407 cents -- sharper than 5/4, which is 386 cents. You get
the difference by dividing (81/64)/(5/4) = 81/80. In monzo form, that's |
-4 4 -1 >.

This has other implications, too:

1. The first two generator steps takes you to (3/2)*(3/2) = 9/8. The
difference between 5/4 and 9/8 is (5/4)/(9/8) = 10/9. But to get to 5/4,
you're just taking two more generator steps, which is another 9/8. So it's
easy to see that the generation process enforces the rule that, in this
temperament, 9/8 = 10/9. (And what's the difference between 9/8 and 10/9?
Of course: 81/80.) It's also easy to see that 5/4 is made of two equal
steps of 9/8 (or 10/9).

2. 81/64 is sharper than 5/4. We got there by taking four generator steps
of a fifth each. In order to bend the 81/64 down toward 5/4, we need to
flatten the fifths that we're using as a generator. The implication is
that, in meantone, the fifth will always be flatter than a just 3/2 and the
major third will be sharper than or equal to 5/4. (It's possible to do the
math so that these things aren't true, but then you're just making the
tuning worse.)

3. The minor third in this case is 6/5, because 3/2 = (5/4)*(6/5). Since
fifths are flat of 3/2, and major thirds are sharp of 5/4, there's less
space for the minor thirds to squeeze into; they're always going to be
somewhat flat.

4. This temperament has a specific scale structure. It has MOSs at 7 and 12
tones; these MOSs have the familiar Western diatonic modal patterns (Ionian
/ major, Aeolian / minor, Phrygian is a minor with a flat 2nd, Lydian is a
major with a sharp 4th, major sevenths on the I and IV chords, major chord
with a flat seventh on the V chord, etc.).

-----
Archytas' Comma, 64/63, | 6 -2 0 -1 >
-----

Let's do the same thing as we just did with meantone, only instead of a 5/4
major third, let's use a spicy, 435-cent 9/7 major third.

Four 3/2s still equals 81/64, but now our major third is 9/7, which is
significantly sharper than 81/64.

The difference between them is (9/7)/(81/64) = 64/63. Tempering out 64/63
means that four stacked 3/2s once again equals a (9/7) major third.

The major third is once again composed of two equal steps. This time,
though, the steps equate to both 9/8 and 8/7. (8/7)/(9/8) = 64/63.

Because the major third is sharper than 81/64, we have to sharpen the fifth
that we're using as a generator. The major third will always be flatter
than or equal to a just 9/7, and the fifth will always be sharper than a
just 3/2. The minor third is (3/2)/(9/7) = 7/6. The fifth is pretty sharp
and the major third is flat, so the minor third has to be sharp of 7/6.

Finally, this temperament also has the scale structure of meantone: MOSs at
7 and 12 tones with Western diatonic modal patterns. As a result,
everything you're used to -- I-IV-V7 progressions, ii-V-I turnarounds,
minor keys, and so on -- will work, period.

-----
Pentacircle, 896/891, | 7 -4 0 1 -1 >
-----

One more time, but with a major third of 14/11, which is about 418 cents.

Four 3/2s still equals 81/64, which is about 10 cents flat of 14/11. Our
fifths are going to have to be a little sharp to compensate, but not as
sharp as they were with 64/63.

(14/11)/(81/64) = 896/891. Tempering out 896/891 means four stacked 3/2s
equals a (14/11) major third.

The major third is once again composed of two equal steps. This time, the
steps equate to both 9/8 and 112/99. (Yeah, I wouldn't have expected that,
either, but that's where the math leads. It's about 214 cents.)

The major third is slightly sharper than 81/64, so we have to slightly
sharpen the fifth we use as a generator. Our major thirds will be equal to
or flatter than 14/11.

Here's an interesting twist: The minor third isn't a simple ratio. The
difference between 3/2 and 5/4 is 6/5; the difference between 3/2 and 9/7
is 7/6; but the difference between 3/2 and 14/11 is 33/28.

But pentacircle, even though it's an 11-limit comma, can be factored into
two 13-limit superparticular commas: 364/363, which is | 2 -1 0 1 -2 1 >,
and 352/351, which is | 5 -3 0 0 1 -1 >. (Notice that the exponents of 13
cancel out when you add them.)

364/363 is the difference between (14/11)*(13/11) and 3/2. If pentacircle
and 364/363 are both tempered out -- which means, of course, that 352/351
is also tempered out -- then 14/11 is the major third, 13/11 is the minor
third, and a major third stacked on a minor third equals a perfect fifth.

The 14/11 major third will always be a little flat. The fifth will always
be a little sharp. I'm not sure of precisely what the minor third will be
-- someone else may know.

Finally, this temperament -- with pentacircle *and* 364/363 and 352/351
tempered out -- also has the diatonic scale structure of meantone: MOSs at
7 and 12 tones with Western modal patterns.

-----
Conclusion
-----

Though we generally think of temperament families as related by their
commas, I've described one way in which different commas can be used to get
similar scale structures. The scales differ primarily by the types of
intervals being used, which will cause each to have a different tonal
"flavor". In the case of these temperaments, which are similar to meantone,
people who want to experiment with different types of intervals can do so
without losing the distinctive structural characteristics of chord
progressions and scales that they're used to.

I anticipate that other temperament groups -- "clans" and "families" are
taken, perhaps "societies"? -- can be found that enable people to, say,
transfer their expertise from porcupine to other temperaments that have
similar structures, but with different tonal flavors.

And yes, I'm sure plenty of you know this already, but I'm pretty happy to
have figured it out. :)

Regards,
Jake

Thanks - that was very clear!

Chris

On Tue, Oct 9, 2012 at 9:25 PM, Jake Freivald <jdfreivald@...> wrote:

> **
>
>
> Most discussions of tunings on this list seem to deal with temperament
> families that are related by having the same commas. I'd like to spend just
> a few minutes discussing three temperaments that are similar because they
> do "the same things" with different tones, and thus need to temper out
> different commas.
>
> Naturally, this will all be stuff that probably seems really obvious to
> people who have been doing this for a while, but that took time to "click"
> with me. I'm writing it in the hope that some newbies will understand some
> of the reasons to temper specific commas, and what effects that tempering
> can have.
>
> -----
> Syntonic Comma, 81/80, | -4 4 -1 >
> -----
>
> Meantone is one of the traditional temperaments of Western music, and is
> defined by two things:
> 1. It tempers out the comma 81/80, and
> 2. It has a period of an octave and a generator of 3/2.
> I can say this with some confidence because, for reasons I don't
> understand, that information has been encapsulated in the "wedgie" (wedge
> product) of the comma, and people who know the math better than I do have
> said so.
>
> Tempering out 81/80 means something very important in relation to the
> generator: It means that four stacked 3/2s (octave equivalent) = 5/4. In
> traditional musical terms, that's the same as saying that four fifths
> equals a major third.
>
> That's good if you want 5-limit harmonies in relatively small scale sizes
> -- if it took, say, eight fifths to get to a major third, then you'd need
> much larger scales to get the gamut of notes needed for 5-limit harmony.
> (This is related to "complexity".)
>
> Four stacked 3/2s = (3/2)^4 = 81/16, which is octave-equivalent to 81/64,
> which is about 407 cents -- sharper than 5/4, which is 386 cents. You get
> the difference by dividing (81/64)/(5/4) = 81/80. In monzo form, that's |
> -4 4 -1 >.
>
> This has other implications, too:
>
> 1. The first two generator steps takes you to (3/2)*(3/2) = 9/8. The
> difference between 5/4 and 9/8 is (5/4)/(9/8) = 10/9. But to get to 5/4,
> you're just taking two more generator steps, which is another 9/8. So it's
> easy to see that the generation process enforces the rule that, in this
> temperament, 9/8 = 10/9. (And what's the difference between 9/8 and 10/9?
> Of course: 81/80.) It's also easy to see that 5/4 is made of two equal
> steps of 9/8 (or 10/9).
>
> 2. 81/64 is sharper than 5/4. We got there by taking four generator steps
> of a fifth each. In order to bend the 81/64 down toward 5/4, we need to
> flatten the fifths that we're using as a generator. The implication is
> that, in meantone, the fifth will always be flatter than a just 3/2 and the
> major third will be sharper than or equal to 5/4. (It's possible to do the
> math so that these things aren't true, but then you're just making the
> tuning worse.)
>
> 3. The minor third in this case is 6/5, because 3/2 = (5/4)*(6/5). Since
> fifths are flat of 3/2, and major thirds are sharp of 5/4, there's less
> space for the minor thirds to squeeze into; they're always going to be
> somewhat flat.
>
> 4. This temperament has a specific scale structure. It has MOSs at 7 and
> 12 tones; these MOSs have the familiar Western diatonic modal patterns
> (Ionian / major, Aeolian / minor, Phrygian is a minor with a flat 2nd,
> Lydian is a major with a sharp 4th, major sevenths on the I and IV chords,
> major chord with a flat seventh on the V chord, etc.).
>
> -----
> Archytas' Comma, 64/63, | 6 -2 0 -1 >
> -----
>
> Let's do the same thing as we just did with meantone, only instead of a
> 5/4 major third, let's use a spicy, 435-cent 9/7 major third.
>
> Four 3/2s still equals 81/64, but now our major third is 9/7, which is
> significantly sharper than 81/64.
>
> The difference between them is (9/7)/(81/64) = 64/63. Tempering out 64/63
> means that four stacked 3/2s once again equals a (9/7) major third.
>
> The major third is once again composed of two equal steps. This time,
> though, the steps equate to both 9/8 and 8/7. (8/7)/(9/8) = 64/63.
>
> Because the major third is sharper than 81/64, we have to sharpen the
> fifth that we're using as a generator. The major third will always be
> flatter than or equal to a just 9/7, and the fifth will always be sharper
> than a just 3/2. The minor third is (3/2)/(9/7) = 7/6. The fifth is pretty
> sharp and the major third is flat, so the minor third has to be sharp of
> 7/6.
>
> Finally, this temperament also has the scale structure of meantone: MOSs
> at 7 and 12 tones with Western diatonic modal patterns. As a result,
> everything you're used to -- I-IV-V7 progressions, ii-V-I turnarounds,
> minor keys, and so on -- will work, period.
>
> -----
> Pentacircle, 896/891, | 7 -4 0 1 -1 >
> -----
>
> One more time, but with a major third of 14/11, which is about 418 cents.
>
> Four 3/2s still equals 81/64, which is about 10 cents flat of 14/11. Our
> fifths are going to have to be a little sharp to compensate, but not as
> sharp as they were with 64/63.
>
> (14/11)/(81/64) = 896/891. Tempering out 896/891 means four stacked 3/2s
> equals a (14/11) major third.
>
> The major third is once again composed of two equal steps. This time, the
> steps equate to both 9/8 and 112/99. (Yeah, I wouldn't have expected that,
> either, but that's where the math leads. It's about 214 cents.)
>
> The major third is slightly sharper than 81/64, so we have to slightly
> sharpen the fifth we use as a generator. Our major thirds will be equal to
> or flatter than 14/11.
>
> Here's an interesting twist: The minor third isn't a simple ratio. The
> difference between 3/2 and 5/4 is 6/5; the difference between 3/2 and 9/7
> is 7/6; but the difference between 3/2 and 14/11 is 33/28.
>
> But pentacircle, even though it's an 11-limit comma, can be factored into
> two 13-limit superparticular commas: 364/363, which is | 2 -1 0 1 -2 1 >,
> and 352/351, which is | 5 -3 0 0 1 -1 >. (Notice that the exponents of 13
> cancel out when you add them.)
>
> 364/363 is the difference between (14/11)*(13/11) and 3/2. If pentacircle
> and 364/363 are both tempered out -- which means, of course, that 352/351
> is also tempered out -- then 14/11 is the major third, 13/11 is the minor
> third, and a major third stacked on a minor third equals a perfect fifth.
>
> The 14/11 major third will always be a little flat. The fifth will always
> be a little sharp. I'm not sure of precisely what the minor third will be
> -- someone else may know.
>
> Finally, this temperament -- with pentacircle *and* 364/363 and 352/351
> tempered out -- also has the diatonic scale structure of meantone: MOSs at
> 7 and 12 tones with Western modal patterns.
>
> -----
> Conclusion
> -----
>
> Though we generally think of temperament families as related by their
> commas, I've described one way in which different commas can be used to get
> similar scale structures. The scales differ primarily by the types of
> intervals being used, which will cause each to have a different tonal
> "flavor". In the case of these temperaments, which are similar to meantone,
> people who want to experiment with different types of intervals can do so
> without losing the distinctive structural characteristics of chord
> progressions and scales that they're used to.
>
> I anticipate that other temperament groups -- "clans" and "families" are
> taken, perhaps "societies"? -- can be found that enable people to, say,
> transfer their expertise from porcupine to other temperaments that have
> similar structures, but with different tonal flavors.
>
> And yes, I'm sure plenty of you know this already, but I'm pretty happy to
> have figured it out. :)
>
> Regards,
> Jake
>
>
>

Hi Jake,

you can actually continue these patterns 5:4, 9:7, 14:11,... and 6:5, 7:6, 13:11,... to eventually arrive at their noble mediants

http://www.dkeenan.com/Music/NobleMediant.txt

which some speculate are another kind of concordance! That's a way to look at the thirds of 17-tET.

Kalle

--- In tuning@yahoogroups.com, Jake Freivald <jdfreivald@...> wrote:
>
> Most discussions of tunings on this list seem to deal with temperament
> families that are related by having the same commas. I'd like to spend just
> a few minutes discussing three temperaments that are similar because they
> do "the same things" with different tones, and thus need to temper out
> different commas.
>
> Naturally, this will all be stuff that probably seems really obvious to
> people who have been doing this for a while, but that took time to "click"
> with me. I'm writing it in the hope that some newbies will understand some
> of the reasons to temper specific commas, and what effects that tempering
> can have.
>
> -----
> Syntonic Comma, 81/80, | -4 4 -1 >
> -----
>
> Meantone is one of the traditional temperaments of Western music, and is
> defined by two things:
> 1. It tempers out the comma 81/80, and
> 2. It has a period of an octave and a generator of 3/2.
> I can say this with some confidence because, for reasons I don't
> understand, that information has been encapsulated in the "wedgie" (wedge
> product) of the comma, and people who know the math better than I do have
> said so.
>
> Tempering out 81/80 means something very important in relation to the
> generator: It means that four stacked 3/2s (octave equivalent) = 5/4. In
> traditional musical terms, that's the same as saying that four fifths
> equals a major third.
>
> That's good if you want 5-limit harmonies in relatively small scale sizes
> -- if it took, say, eight fifths to get to a major third, then you'd need
> much larger scales to get the gamut of notes needed for 5-limit harmony.
> (This is related to "complexity".)
>
> Four stacked 3/2s = (3/2)^4 = 81/16, which is octave-equivalent to 81/64,
> which is about 407 cents -- sharper than 5/4, which is 386 cents. You get
> the difference by dividing (81/64)/(5/4) = 81/80. In monzo form, that's |
> -4 4 -1 >.
>
> This has other implications, too:
>
> 1. The first two generator steps takes you to (3/2)*(3/2) = 9/8. The
> difference between 5/4 and 9/8 is (5/4)/(9/8) = 10/9. But to get to 5/4,
> you're just taking two more generator steps, which is another 9/8. So it's
> easy to see that the generation process enforces the rule that, in this
> temperament, 9/8 = 10/9. (And what's the difference between 9/8 and 10/9?
> Of course: 81/80.) It's also easy to see that 5/4 is made of two equal
> steps of 9/8 (or 10/9).
>
> 2. 81/64 is sharper than 5/4. We got there by taking four generator steps
> of a fifth each. In order to bend the 81/64 down toward 5/4, we need to
> flatten the fifths that we're using as a generator. The implication is
> that, in meantone, the fifth will always be flatter than a just 3/2 and the
> major third will be sharper than or equal to 5/4. (It's possible to do the
> math so that these things aren't true, but then you're just making the
> tuning worse.)
>
> 3. The minor third in this case is 6/5, because 3/2 = (5/4)*(6/5). Since
> fifths are flat of 3/2, and major thirds are sharp of 5/4, there's less
> space for the minor thirds to squeeze into; they're always going to be
> somewhat flat.
>
> 4. This temperament has a specific scale structure. It has MOSs at 7 and 12
> tones; these MOSs have the familiar Western diatonic modal patterns (Ionian
> / major, Aeolian / minor, Phrygian is a minor with a flat 2nd, Lydian is a
> major with a sharp 4th, major sevenths on the I and IV chords, major chord
> with a flat seventh on the V chord, etc.).
>
> -----
> Archytas' Comma, 64/63, | 6 -2 0 -1 >
> -----
>
> Let's do the same thing as we just did with meantone, only instead of a 5/4
> major third, let's use a spicy, 435-cent 9/7 major third.
>
> Four 3/2s still equals 81/64, but now our major third is 9/7, which is
> significantly sharper than 81/64.
>
> The difference between them is (9/7)/(81/64) = 64/63. Tempering out 64/63
> means that four stacked 3/2s once again equals a (9/7) major third.
>
> The major third is once again composed of two equal steps. This time,
> though, the steps equate to both 9/8 and 8/7. (8/7)/(9/8) = 64/63.
>
> Because the major third is sharper than 81/64, we have to sharpen the fifth
> that we're using as a generator. The major third will always be flatter
> than or equal to a just 9/7, and the fifth will always be sharper than a
> just 3/2. The minor third is (3/2)/(9/7) = 7/6. The fifth is pretty sharp
> and the major third is flat, so the minor third has to be sharp of 7/6.
>
> Finally, this temperament also has the scale structure of meantone: MOSs at
> 7 and 12 tones with Western diatonic modal patterns. As a result,
> everything you're used to -- I-IV-V7 progressions, ii-V-I turnarounds,
> minor keys, and so on -- will work, period.
>
> -----
> Pentacircle, 896/891, | 7 -4 0 1 -1 >
> -----
>
> One more time, but with a major third of 14/11, which is about 418 cents.
>
> Four 3/2s still equals 81/64, which is about 10 cents flat of 14/11. Our
> fifths are going to have to be a little sharp to compensate, but not as
> sharp as they were with 64/63.
>
> (14/11)/(81/64) = 896/891. Tempering out 896/891 means four stacked 3/2s
> equals a (14/11) major third.
>
> The major third is once again composed of two equal steps. This time, the
> steps equate to both 9/8 and 112/99. (Yeah, I wouldn't have expected that,
> either, but that's where the math leads. It's about 214 cents.)
>
> The major third is slightly sharper than 81/64, so we have to slightly
> sharpen the fifth we use as a generator. Our major thirds will be equal to
> or flatter than 14/11.
>
> Here's an interesting twist: The minor third isn't a simple ratio. The
> difference between 3/2 and 5/4 is 6/5; the difference between 3/2 and 9/7
> is 7/6; but the difference between 3/2 and 14/11 is 33/28.
>
> But pentacircle, even though it's an 11-limit comma, can be factored into
> two 13-limit superparticular commas: 364/363, which is | 2 -1 0 1 -2 1 >,
> and 352/351, which is | 5 -3 0 0 1 -1 >. (Notice that the exponents of 13
> cancel out when you add them.)
>
> 364/363 is the difference between (14/11)*(13/11) and 3/2. If pentacircle
> and 364/363 are both tempered out -- which means, of course, that 352/351
> is also tempered out -- then 14/11 is the major third, 13/11 is the minor
> third, and a major third stacked on a minor third equals a perfect fifth.
>
> The 14/11 major third will always be a little flat. The fifth will always
> be a little sharp. I'm not sure of precisely what the minor third will be
> -- someone else may know.
>
> Finally, this temperament -- with pentacircle *and* 364/363 and 352/351
> tempered out -- also has the diatonic scale structure of meantone: MOSs at
> 7 and 12 tones with Western modal patterns.
>
> -----
> Conclusion
> -----
>
> Though we generally think of temperament families as related by their
> commas, I've described one way in which different commas can be used to get
> similar scale structures. The scales differ primarily by the types of
> intervals being used, which will cause each to have a different tonal
> "flavor". In the case of these temperaments, which are similar to meantone,
> people who want to experiment with different types of intervals can do so
> without losing the distinctive structural characteristics of chord
> progressions and scales that they're used to.
>
> I anticipate that other temperament groups -- "clans" and "families" are
> taken, perhaps "societies"? -- can be found that enable people to, say,
> transfer their expertise from porcupine to other temperaments that have
> similar structures, but with different tonal flavors.
>
> And yes, I'm sure plenty of you know this already, but I'm pretty happy to
> have figured it out. :)
>
> Regards,
> Jake
>

Thanks, Kalle, I'll check it out. I've been playing a bit more with 17 EDO
recently, so getting some more perspective will be a good thing.

Regards,
Jake

On Wed, Oct 10, 2012 at 7:50 AM, Kalle Aho <kalleaho@...>wrote:

>
> http://www.dkeenan.com/Music/NobleMediant.txt
>
>

--- In tuning@yahoogroups.com, Jake Freivald <jdfreivald@...> wrote:
> Finally, this temperament -- with pentacircle *and* 364/363 and 352/351
> tempered out -- also has the diatonic scale structure of meantone: MOSs at
> 7 and 12 tones with Western modal patterns.

This temperament has a name... it's "pepperoni" (which I guess is indirectly named after me, haha). http://xenharmonic.wikispaces.com/Chromatic+pairs#Pepperoni

Extensions to the full 13-limit include leapday (29&46), and also suprapyth. Suprapyth also tempers out 64/63 though, so it's like a hybrid of two of your categories.

Which pretty much nobody cares about for practical music making, as you describe below...

> Though we generally think of temperament families as related by their
> commas, I've described one way in which different commas can be used to get
> similar scale structures. The scales differ primarily by the types of
> intervals being used, which will cause each to have a different tonal
> "flavor". In the case of these temperaments, which are similar to meantone,
> people who want to experiment with different types of intervals can do so
> without losing the distinctive structural characteristics of chord
> progressions and scales that they're used to.
>
> I anticipate that other temperament groups -- "clans" and "families" are
> taken, perhaps "societies"? -- can be found that enable people to, say,
> transfer their expertise from porcupine to other temperaments that have
> similar structures, but with different tonal flavors.
>
> And yes, I'm sure plenty of you know this already, but I'm pretty happy to
> have figured it out. :)

Yep. This is why I started http://xenharmonic.wikispaces.com/Map+of+rank-2+temperaments . It's a generalization, but it seems like most musicians don't care about commas and mappings in practice; what they care about is scale structures, so it makes sense to organize these named rank-2 scale systems (I won't even call them "temperaments") by the size of the period and generator. So you can easily look up porcupine on here and see that there's something next to it called "greeley" and something a little farther away called "nusecond".

For rank-1 (equal) temperaments it's a little different, because for octave-containing temperaments each EDO is this unique self-contained system; there's not really a "continuum" like there is for rank-2 temperaments. (If you don't assume octaves, you can have things like http://xenharmonic.wikispaces.com/EDT#EDO-EDT%20correspondence )

For rank-3 and higher, nobody knows WTF they're doing yet.

Keenan

Dumb question

Is melodic minor technically rank 3?

For rank-3 and higher, nobody knows WTF they're doing yet.

Keenan
*

--- In tuning@yahoogroups.com, chrisvaisvil@... wrote:
>
> Dumb question
>
> Is melodic minor technically rank 3?

No, the way most people think of melodic minor is as a MODMOS of the rank-2 temperament, meantone. If it's LsLLLLs where each L is the same and each s is the same, then it must be rank-2.

...well actually the way most people think of melodic minor is as a subset of 12edo (rank-1). But the way most *hip* people think of melodic minor is as a MODMOS of meantone. =)

Keenan

I'm a little confused it seems.

Setting the fact 12 equal is rank one as side and considering just the scale.

First I meant harmonic minor which has half, whole, and augmented 2nds. That is a rank 3 ?
*

-----Original Message-----
From: "Keenan Pepper" <keenanpepper@...>
Sender: tuning@yahoogroups.com
Date: Thu, 11 Oct 2012 18:01:06
To: <tuning@yahoogroups.com>
Reply-To: tuning@yahoogroups.com
Subject: [tuning] Re: Three similar temperaments

--- In tuning@yahoogroups.com, chrisvaisvil@... wrote:
>
> Dumb question
>
> Is melodic minor technically rank 3?

No, the way most people think of melodic minor is as a MODMOS of the rank-2 temperament, meantone. If it's LsLLLLs where each L is the same and each s is the same, then it must be rank-2.

...well actually the way most people think of melodic minor is as a subset of 12edo (rank-1). But the way most *hip* people think of melodic minor is as a MODMOS of meantone. =)

Keenan

--- In tuning@yahoogroups.com, chrisvaisvil@... wrote:
>
> I'm a little confused it seems.
>
> Setting the fact 12 equal is rank one as side and considering just the scale.
>
> First I meant harmonic minor which has half, whole, and augmented 2nds. That is a rank 3 ?

No, harmonic minor is another MODMOS of meantone, which is a rank-2 temperament.

Harmonic minor is msmmsLs, so if s, m, and L were all *incommensurate* intervals (which means basically that they're "random" numbers that have nothing to do with each other), then it would be rank-3. But in meantone harmonic minor, they're not incommensurate; they satisfy the simple equation L + s = 2m. That equation implies that every interval in the scale can be derived from only two generators (one of which is usually called the "period"); it's not necessary to have three different generators. Therefore it's rank-2, not rank-3.

So, to summarize the situation:

A scale that's msmmsLs, where s, m, and L are all incommensurate intervals, is rank-3, but this is not what is usually meant by "harmonic minor".

A scale that's msmmsLs, where L + s = 2m, but s and m are incommensurate, is rank-2. An example of this would be harmonic minor in quarter-comma meantone.

A scale that's msmmsLs, but where s, m, and L are all integer multiples of a single step, is rank-1. For example, if m=2s and L=3s, then this is a subset of 12edo (2 1 2 2 1 3 1) - and that's what most non-XA people think of as "harmonic minor".

If you ask me personally what rank I think "harmonic minor" is, I say rank-2, because it's musically essential that it be a subset of meantone temperament, but it's not essential that it be a subset of an equal temperament.

Keenan

It is for this reason the idea of 'ranks' does do seem fruitful and easily leads to a misconception and to overlook the work under ones nose. In the case of constant structure very rarely when an different limit is introduced is it used in a chain so referring to it as a 'generator' seem inappropriate. The most often time when they are used in a chain it is with the context of a repeated triad. It seems more would be gained in that case looking at expansions off of Chalmers tritriadic scales or Wilsons SOME BASIC PATTERNS UNDERLYING GENUS 12 & 17/ MALLET AND KEYBOARD LAYOUTS <http://anaphoria.com/genus.PDF> . Even when does not these serve as suitable models

What you call rank 3 is in fact the material i have always used the most and some of the most developed going back to xenharmonikon 3. one also has those examples of SOME CONSTANT STRUCTURES PREDICTED BY VIGGO BRUN'S ALGORITHM <http://anaphoria.com/viggo3.PDF> where 2 levels are used.

Closely related to MODMOS but expanded quite a bit further can be witnessed in THE MARWA PERMUTATIONS <http://anaphoria.com/xen9mar.PDF> especially under fig.1e the paper like that in the next issue of xenharmonikon. The idea is shown applied to constant structure which are developments of the simpler MOS structures.

All these papers exist on the most basic of material found onhttp://anaphoria.com/wilsonbasic.html
<http://anaphoria.com/wilsonbasic.html>
[For rank-3 and higher, nobody knows WTF they're doing yet.

Keenan]
--
signature file

/^_,',',',_ //^/Kraig Grady_^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere:
North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

a momentary antenna as i turn to water
this evaporates - an island once again

On Thu, Oct 11, 2012 at 1:28 PM, Keenan Pepper <keenanpepper@...>
wrote:
>
> For rank-3 and higher, nobody knows WTF they're doing yet.
>
> Keenan

In all seriousness, what major initiatives are left to figure out? I
feel like I have a crystal clear picture of how to use rank-3 scales
after everything we've discussed in the past year. This is my patented
three step plan:

1) Pick a d-sized "diatonic" Fokker block
2) Pick a c-sized "chromatic" Fokker block that contains #1
3) Continue to modulate through different metamodes of #2 to
accommodate the harmony that you want to play, making sure to
establish as often as possible the notion that there are d different
"regions" of interval space which shift slightly by comma

#3 applies to rank-2 scales as well, though most people aren't doing
that now, except perhaps subconsciously with meantone[12] when they
play in 19-EDO. I would pay $1,000,000 to hear Gene write a piece that
deliberately does all of those things.

In fact, damn, would someone PLEASE write a piece that does this with
15-note Fokker blocks? I think you'd have to try really hard to make
that sound bad.

-Mike

Give me a scala file with the desired 15 note
Fokker bloke and I'll give it a go.

Xhris
*

-----Original Message-----
From: Mike Battaglia <battaglia01@gmail.com>
Sender: tuning@yahoogroups.com
Date: Mon, 15 Oct 2012 07:33:56
To: <tuning@yahoogroups.com>
Reply-To: tuning@yahoogroups.com
Subject: Re: [tuning] Re: Three similar temperaments

On Thu, Oct 11, 2012 at 1:28 PM, Keenan Pepper <keenanpepper@gmail.com>
wrote:
>
> For rank-3 and higher, nobody knows WTF they're doing yet.
>
> Keenan

In all seriousness, what major initiatives are left to figure out? I
feel like I have a crystal clear picture of how to use rank-3 scales
after everything we've discussed in the past year. This is my patented
three step plan:

1) Pick a d-sized "diatonic" Fokker block
2) Pick a c-sized "chromatic" Fokker block that contains #1
3) Continue to modulate through different metamodes of #2 to
accommodate the harmony that you want to play, making sure to
establish as often as possible the notion that there are d different
"regions" of interval space which shift slightly by comma

#3 applies to rank-2 scales as well, though most people aren't doing
that now, except perhaps subconsciously with meantone[12] when they
play in 19-EDO. I would pay $1,000,000 to hear Gene write a piece that
deliberately does all of those things.

In fact, damn, would someone PLEASE write a piece that does this with
15-note Fokker blocks? I think you'd have to try really hard to make
that sound bad.

-Mike

On Mon, Oct 15, 2012 at 8:27 AM, <chrisvaisvil@...> wrote:
>
> Give me a scala file with the desired 15 note
> Fokker bloke and I'll give it a go.
>
> Xhris

There is no way to do what I'm proposing here by playing a single 15
note Fokker block with notes that never change. That's exactly the
opposite of what I'm proposing here. :)

-Nike

Well then give me the two (or three because I'm unsure) scala files for the system you are proposing.

Also you state unequivocally you've done this. So - got an example?
Can you explain how you did it?

Chris
*

-----Original Message-----
From: Mike Battaglia <battaglia01@...>
Sender: tuning@yahoogroups.com
Date: Mon, 15 Oct 2012 09:40:02
To: <tuning@yahoogroups.com>
Reply-To: tuning@yahoogroups.com
Subject: Re: [tuning] Re: Three similar temperaments

On Mon, Oct 15, 2012 at 8:27 AM, <chrisvaisvil@...> wrote:
>
> Give me a scala file with the desired 15 note
> Fokker bloke and I'll give it a go.
>
> Xhris

There is no way to do what I'm proposing here by playing a single 15
note Fokker block with notes that never change. That's exactly the
opposite of what I'm proposing here. :)

-Nike

On Mon, Oct 15, 2012 at 11:19 AM, <chrisvaisvil@...> wrote:
>
> Well then give me the two (or three because I'm unsure) scala files for
> the system you are proposing.

No... the point is that the Fokker block changes dynamically to match
the harmony that you want.

The approach I'm suggesting here involves dynamically shifting the
mode/dome of the Fokker block you're using in such a way that the
following two goals are simultaneously satisfied:

1) The chromatic scale you're choosing at any moment is in harmony
with the chord you want to play
2) All of your chromatic scales have the same size and remain roughly
quasi-equal (and for theorists, are all epimorphic under the same val)

If you want to actually try this approach, I think it'd be simpler to
start out with rank-2. So why not try using different modes of
porcupine[15] in 22-EDO?

> Also you state unequivocally you've done this. So - got an example?
> Can you explain how you did it?

I've done it with rank-2 in meantone in 19-EDO, and have messed around
a bit with doing it with porcupine in 22-EDO. I don't have anything
recorded yet.

What I'm doing is I'm trying to emphasize the meantone chromatic scale
in 19-EDO. The two ways that people do this are:
1) They pick a 12-note subset of 19-EDO and play only those notes, or
2) They just play all 19 notes like it's one huge chromatic scale

I'm trying to find a middle option between the two, which is

3) Pick a 12-note subset of 19-EDO and emphasize it, and then keep
shifting the 12-note subset to accommodate the chords I want, so I
don't get trapped into playing wolf chords I don't want.

So for instance, say you're in 19-EDO and you're playing the chord
progression ||: Fm | Cmaj/E | Emaj | Cmaj/E :||. If you want to play
meantone chromatic melodies over some chord progression, you're going
to pick a chromatic scale that has Ab over the Fm and G# over the
Emaj, which would have to involve a mode shift of the meantone[12]
background, because otherwise the notes will clash. So your only
option is to shift the mode of meantone[12] that you're using
dynamically, or play some 19-tone enharmonic melody instead.

The thing I'm talking about above is just a variant of that for rank-3
scales. I'm not even sure how it's possible to improvise something
with that approach, given the practical restrictions we all currently
have on playing. To realistically do it, you'll need to pick a
massively large tuning, like a large EDO or giant-sized Fokker block
containing all of the different chromatic scales you could possibly
want, and then modulate around within that.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Thu, Oct 11, 2012 at 1:28 PM, Keenan Pepper <keenanpepper@...>
> wrote:
> >
> > For rank-3 and higher, nobody knows WTF they're doing yet.
> >
> > Keenan
>
> In all seriousness, what major initiatives are left to figure out?

Well, the specific thing I had in mind was that http://xenharmonic.wikispaces.com/Map+of+rank-2+temperaments has a unique place for every rank-2 scale system that maps octaves in some way. No matter what subgroup it is, or what commas it tempers out, or even if it's a temperament at all, if it's rank-2 and contains octaves than it has a place on there.

Am I correct in saying that nobody knows how to make an analogous map for all possible rank-3 scale systems that contain octaves? The problem is not merely that there would have to be two degrees of freedom. The problem is that for rank 3, the generators are not unique at all, even up to octave equivalence.

> 1) Pick a d-sized "diatonic" Fokker block
> 2) Pick a c-sized "chromatic" Fokker block that contains #1
> 3) Continue to modulate through different metamodes of #2 to
> accommodate the harmony that you want to play, making sure to
> establish as often as possible the notion that there are d different
> "regions" of interval space which shift slightly by comma
>
> #3 applies to rank-2 scales as well, though most people aren't doing
> that now, except perhaps subconsciously with meantone[12] when they
> play in 19-EDO. I would pay $1,000,000 to hear Gene write a piece that
> deliberately does all of those things.
>
> In fact, damn, would someone PLEASE write a piece that does this with
> 15-note Fokker blocks? I think you'd have to try really hard to make
> that sound bad.

I think you'd have to try really hard to do it at all, hahaha. It's way over my head, at least. You should do it!

Keenan

--- In tuning@yahoogroups.com, kraiggrady <kraiggrady@...> wrote:
>
> It is for this reason the idea of 'ranks' does do seem fruitful and
> easily leads to a misconception and to overlook the work under ones
> nose. In the case of constant structure very rarely when an different
> limit is introduced is it used in a chain so referring to it as a
> 'generator' seem inappropriate. The most often time when they are used
> in a chain it is with the context of a repeated triad. It seems more
> would be gained in that case looking at expansions off of Chalmers
> tritriadic scales or Wilsons SOME BASIC PATTERNS UNDERLYING GENUS 12 &
> 17/ MALLET AND KEYBOARD LAYOUTS <http://anaphoria.com/genus.PDF> . Even
> when does not these serve as suitable models
>
> What you call rank 3 is in fact the material i have always used the
> most and some of the most developed going back to xenharmonikon 3. one
> also has those examples of SOME CONSTANT STRUCTURES PREDICTED BY VIGGO
> BRUN'S ALGORITHM <http://anaphoria.com/viggo3.PDF> where 2 levels are used.
>
> Closely related to MODMOS but expanded quite a bit further can be
> witnessed in THE MARWA PERMUTATIONS <http://anaphoria.com/xen9mar.PDF>
> especially under fig.1e the paper like that in the next issue of
> xenharmonikon. The idea is shown applied to constant structure which are
> developments of the simpler MOS structures.
>
> All these papers exist on the most basic of material found
> onhttp://anaphoria.com/wilsonbasic.html
> <http://anaphoria.com/wilsonbasic.html>

This is definitely some intense rank-3 stuff written by someone who knew what he was doing. We should definitely go over this in detail to see if one of us has an epiphany reading it and discovers something Erv Wilson already knew about decades ago.

But the question is, can this be used to construct a map similar to http://xenharmonic.wikispaces.com/Map+of+rank-2+temperaments but for rank-3 temperaments? I was under the impression not, but I'd love to be proven wrong!

Keenan

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Mon, Oct 15, 2012 at 8:27 AM, <chrisvaisvil@...> wrote:
> >
> > Give me a scala file with the desired 15 note
> > Fokker bloke and I'll give it a go.
> >
> > Xhris
>
> There is no way to do what I'm proposing here by playing a single 15
> note Fokker block with notes that never change. That's exactly the
> opposite of what I'm proposing here. :)

A good start would be to decide on something rank 3, hopefully not 5-limit JI.

--- In tuning@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:

> But the question is, can this be used to construct a map similar to http://xenharmonic.wikispaces.com/Map+of+rank-2+temperaments but for rank-3 temperaments? I was under the impression not, but I'd love to be proven wrong!

You might start by looking at the Hermite normal form, and seeing what the {2,3,5} square matrix part looks like. Then put the three generators in the form of fractions of an octave for things with the same {2,3,5} matrix.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:

> You might start by looking at the Hermite normal form, and seeing what the {2,3,5} square matrix part looks like. Then put the three generators in the form of fractions of an octave for things with the same {2,3,5} matrix.

Or less elaborately, why not just list the vals? Maybe there's a useful way to stick vals on a 2D grid instead of 1D, which is how you can view the generator thing.

On Oct 15, 2012, at 7:29 PM, genewardsmith <genewardsmith@...>
wrote:

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Mon, Oct 15, 2012 at 8:27 AM, <chrisvaisvil@...> wrote:
> >
> > Give me a scala file with the desired 15 note
> > Fokker bloke and I'll give it a go.
> >
> > Xhris
>
> There is no way to do what I'm proposing here by playing a single 15
> note Fokker block with notes that never change. That's exactly the
> opposite of what I'm proposing here. :)

A good start would be to decide on something rank 3, hopefully not 5-limit
JI.

What's your favorite 15-note rank-3 Fokker block?

-Mike

Hi Mike,

Now I think I sort of see what you are saying and the problem.

I'd think the best place to start would be with the simplest version of
this you can imagine. Perhaps just a 3 or 4 chord progression with a more
complex melody?

Nonetheless it sounds like a heck of a challenge - thus my interest in
trying to pull it off. My original thought this morning was to use 2,
perhaps three keyboards like they way different manuals on an organ are
used except in this case I'm mixing tunings, not timbres.

Chris

On Mon, Oct 15, 2012 at 4:10 PM, Mike Battaglia <battaglia01@...>wrote:

> **
>
>
>
>
> The thing I'm talking about above is just a variant of that for rank-3
> scales. I'm not even sure how it's possible to improvise something
> with that approach, given the practical restrictions we all currently
> have on playing. To realistically do it, you'll need to pick a
> massively large tuning, like a large EDO or giant-sized Fokker block
> containing all of the different chromatic scales you could possibly
> want, and then modulate around within that.
>
> -Mike
>
>
> </tuning/post;_ylc=X3oDMTJwZ3VraWcwBF9TAzk3MzU5NzE0BGdycElkAzcwNjA1BGdycHNwSWQDMTcwNTg5Nzc1MwRtc2dJZAMxMDUwMzgEc2VjA2Z0cgRzbGsDcnBseQRzdGltZQMxMzUwMzMxODcy?act=reply&messageNum=105038>
>
>
>
>
>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> What's your favorite 15-note rank-3 Fokker block?

I don't know about favorites, but this might be a place to start:

! dwarf15marv.scl
Marvelous dwarf: 1/4 kleismic dwarf(<15 24 35|) subset rosatimarv
15
!
115.587047
184.331593
200.054240
315.641287
384.385833
499.972880
584.440073
615.559927
700.027120
815.614167
884.358713
999.945760
1015.668407
1084.412953
1200.000000
! six tetrads/pentads representible by
! [[-1, 0, -1], [0, -1, 0], [0, 0, 1], [0, 1, 2], [0, -1, 1], [0, 0, 2]]
! nine-limit comma pump in this ordering (among others)

You could probably pull it off by first starting with meantone[12] in
19-EDO, and then graduating to porcupine[15] in 22-EDO. That's not
rank-3, but the general idea is still the same anyway.

-Mike

On Mon, Oct 15, 2012 at 9:17 PM, Chris Vaisvil <chrisvaisvil@...>
wrote:
>
> Hi Mike,
>
> Now I think I sort of see what you are saying and the problem.
>
> I'd think the best place to start would be with the simplest version of
> this you can imagine. Perhaps just a 3 or 4 chord progression with a more
> complex melody?
>
> Nonetheless it sounds like a heck of a challenge - thus my interest in
> trying to pull it off. My original thought this morning was to use 2,
> perhaps three keyboards like they way different manuals on an organ are used
> except in this case I'm mixing tunings, not timbres.
>
> Chris

On Mon, Oct 15, 2012 at 11:07 PM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > What's your favorite 15-note rank-3 Fokker block?
>
> I don't know about favorites, but this might be a place to start:
>
> ! dwarf15marv.scl

Some of these notes are so close together I doubt I'd ever be able to
tell them apart in a melodic setting. For instance, 184 and 200 cents
are about 16 cents apart. If we call the note 1/1 "C", then this scale
basically to me sounds like a 12-note scale with two intonational
variants for D, F#, and Bb. The two D's and Bb's are literally just 16
cents apart from one another.

How about the Fokker block where 1029/1024 and 126/125 are unison
vectors, and 256/243 and 49/48 and 55/54 are chromatic vectors?

-Mike

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@> wrote:
>
> > You might start by looking at the Hermite normal form, and seeing what the {2,3,5} square matrix part looks like. Then put the three generators in the form of fractions of an octave for things with the same {2,3,5} matrix.
>
> Or less elaborately, why not just list the vals? Maybe there's a useful way to stick vals on a 2D grid instead of 1D, which is how you can view the generator thing.

Can you explain this more? I don't immediately see how to view it that way. Do you mean each octave fraction is associated with a specific val, or what? How do I get the two vals I need to construct a rank-2 temperament?

Keenan

Hi Keenan~
One could change the cycle of any generator from being only one size with a disjunction to two or three etc. as seen in the Purvi and Marwa papers . this differs in having an alternation of generators instead of the process of simultaneous generators. It leaves great room for choice in the process instead of mathematical determinism.
signature file

/^_,',',',_ //^/Kraig Grady_^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere:
North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

a momentary antenna as i turn to water
this evaporates - an island once again

I have this one already but I don't think I tried it - on the list!

On Mon, Oct 15, 2012 at 11:07 PM, genewardsmith <genewardsmith@...
> wrote:

> **
>
>
>
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > What's your favorite 15-note rank-3 Fokker block?
>
> I don't know about favorites, but this might be a place to start:
>
> ! dwarf15marv.scl
> Marvelous dwarf: 1/4 kleismic dwarf(<15 24 35|) subset rosatimarv
> 15
> !
> 115.587047
> 184.331593
> 200.054240
> 315.641287
> 384.385833
> 499.972880
> 584.440073
> 615.559927
> 700.027120
> 815.614167
> 884.358713
> 999.945760
> 1015.668407
> 1084.412953
> 1200.000000
> ! six tetrads/pentads representible by
> ! [[-1, 0, -1], [0, -1, 0], [0, 0, 1], [0, 1, 2], [0, -1, 1], [0, 0, 2]]
> ! nine-limit comma pump in this ordering (among others)
>
>
>

On Thu, Oct 11, 2012 at 2:43 PM, <chrisvaisvil@...> wrote:
>
> I'm a little confused it seems.
>
> Setting the fact 12 equal is rank one as side and considering just the
> scale.
>
> First I meant harmonic minor which has half, whole, and augmented 2nds.
> That is a rank 3 ?

To give a bit nicer explanation, scales don't actually have "ranks."
It's a slight abuse of notation that we're using here. The things
which actually have ranks are what you'd call lattices, and what
mathematicians would call "groups."

If you look at a neverending chain of fifths, for instance, that's
rank 1. Say you come up with a big 1-dimensional number line to
represent your position on the chain of fifths.

Now let's say you add a second neverending chain of octaves. Now
you're at rank 2. If you want to keep track of your position in this
new tuning system, you'll need two number lines - one to represent
your place on the chain of fifths, and another to represent your place
on the chain of octaves. The obvious way to visualize this is to place
these two number lines perpendicular to one another, giving you an X-Y
plot. Now it's clear you have a 2D lattice to move around in, hence
"rank 2."

Now let's say you add a second neverending chain of major thirds. Now
you're at rank 3. You can add a Z-axis to represent this, and so on.

It's important to note that I'm not talking about 12-EDO major thirds
here, where three add up to an octave. I'm talking about, for
instance, a pure 5/4.

So when you ask, "what rank is the melodic minor scale," there's no
actual answer. The real question is: what's the rank of the tuning
system which I'm imagining is underlying this scale?

- If you're imagining the melodic minor scale as existing as a pattern
of equally-tempered whole and half steps in 12-EDO, then for all
intents and purposes you're thinking of it as part of a rank-1 tuning
system.
- If you're imagining the melodic minor scale as occupying a certain
chunk of the spiral of fifths - like Eb-xx-F-C-G-D-A-xx-B - plus the
ability to shift notes by octave, then you're imagining it as part of
a rank-2 tuning system.
- If you're imagining it as being in 5-limit JI, so that it's
something like 1/1 9/8 6/5 4/3 3/2 8/5 15/8 2/1, then you're imagining
it as part of a rank-3 tuning system.

The important thing is that you can create scales with more than two
sizes of step even in a rank-2 tuning system. The rank refers to the
number of generators presumed to exist "in the background" of the
scale, not the number of steps explicit in the scale itself.

-Mike

Mike

Thank you for that nice explanation. I'm tempted to suggest degrees of freedom in a physics sense might be clearer but I'm sure I'm biased from my chemistry background.

Chris
*

-----Original Message-----
From: Mike Battaglia <battaglia01@gmail.com>
Sender: tuning@yahoogroups.com
Date: Tue, 16 Oct 2012 07:09:18
To: <tuning@yahoogroups.com>
Reply-To: tuning@yahoogroups.com
Subject: Re: [tuning] Re: Three similar temperaments

On Thu, Oct 11, 2012 at 2:43 PM, <chrisvaisvil@gmail.com> wrote:
>
> I'm a little confused it seems.
>
> Setting the fact 12 equal is rank one as side and considering just the
> scale.
>
> First I meant harmonic minor which has half, whole, and augmented 2nds.
> That is a rank 3 ?

To give a bit nicer explanation, scales don't actually have "ranks."
It's a slight abuse of notation that we're using here. The things
which actually have ranks are what you'd call lattices, and what
mathematicians would call "groups."

If you look at a neverending chain of fifths, for instance, that's
rank 1. Say you come up with a big 1-dimensional number line to
represent your position on the chain of fifths.

Now let's say you add a second neverending chain of octaves. Now
you're at rank 2. If you want to keep track of your position in this
new tuning system, you'll need two number lines - one to represent
your place on the chain of fifths, and another to represent your place
on the chain of octaves. The obvious way to visualize this is to place
these two number lines perpendicular to one another, giving you an X-Y
plot. Now it's clear you have a 2D lattice to move around in, hence
"rank 2."

Now let's say you add a second neverending chain of major thirds. Now
you're at rank 3. You can add a Z-axis to represent this, and so on.

It's important to note that I'm not talking about 12-EDO major thirds
here, where three add up to an octave. I'm talking about, for
instance, a pure 5/4.

So when you ask, "what rank is the melodic minor scale," there's no
actual answer. The real question is: what's the rank of the tuning
system which I'm imagining is underlying this scale?

- If you're imagining the melodic minor scale as existing as a pattern
of equally-tempered whole and half steps in 12-EDO, then for all
intents and purposes you're thinking of it as part of a rank-1 tuning
system.
- If you're imagining the melodic minor scale as occupying a certain
chunk of the spiral of fifths - like Eb-xx-F-C-G-D-A-xx-B - plus the
ability to shift notes by octave, then you're imagining it as part of
a rank-2 tuning system.
- If you're imagining it as being in 5-limit JI, so that it's
something like 1/1 9/8 6/5 4/3 3/2 8/5 15/8 2/1, then you're imagining
it as part of a rank-3 tuning system.

The important thing is that you can create scales with more than two
sizes of step even in a rank-2 tuning system. The rank refers to the
number of generators presumed to exist "in the background" of the
scale, not the number of steps explicit in the scale itself.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> The important thing is that you can create scales with more than two
> sizes of step even in a rank-2 tuning system. The rank refers to the
> number of generators presumed to exist "in the background" of the
> scale, not the number of steps explicit in the scale itself.

If you have explicit values for the notes of the scale you don't need to presume anything--the rank is just the rank of the group generated by the scale.

On Tue, Oct 16, 2012 at 8:57 AM, genewardsmith <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > The important thing is that you can create scales with more than two
> > sizes of step even in a rank-2 tuning system. The rank refers to the
> > number of generators presumed to exist "in the background" of the
> > scale, not the number of steps explicit in the scale itself.
>
> If you have explicit values for the notes of the scale you don't need to
> presume anything--the rank is just the rank of the group generated by the
> scale.

In this case, we don't. Chris asked what rank "the melodic minor
scale" was, in general, without specifying a tuning.

-Mike

> Most discussions of tunings on this list seem to deal with
> temperament families that are related by having the same commas.
> I'd like to spend just a few minutes discussing three temperaments
> that are similar because they do "the same things" with different
> tones, and thus need to temper out different commas.

Dear Jake,

Thank you for a description that caught my attention devoted to a
style of temperament that's a real favorite for me: your Pentacircle.
You give a nice and very readable summary, and ask about some fine
points which might be interesting to clarify.

Again, your readable style is something I hope to emulate, and I
warmly welcome your comments or questions as to anything that might
not be so clear.

More humorously, for a number of years I've had this motto: "12-EDO
tempers by about the right amount, but in the less interesting
direction!"

> Pentacircle, 896/891, | 7 -4 0 1 -1 >

This is a name that really attracts me, and I'm curious how you
arrived at it. Does "penta" refer to the fifths tempered gently in the
wide direction?

> Four 3/2s still equals 81/64, which is about 10 cents flat of
> 14/11. Our fifths are going to have to be a little sharp to
> compensate, but not as sharp as they were with 64/63.

A good practical observation I'll add here as a medievalist is that
this kind of temperament fits best in styles where 81/64 would make
sense, or more generally a wide variety of major third rather than 5/4
or something reasonably close. Much medieval European music of the
13th-14th centuries fits, and also lots of Near Eastern music favoring
major thirds around 81/64 or a bit larger.

At least in usual harmonic timbres, I wouldn't associate this style of
tuning with European music of the 15th-19th centuries and similar
genres presuming stable major thirds reasonably close to 5/4. This is
a different world, and an equally beautiful one.

For an excellent introduction to this world by George Secor which
focuses on his 17-tone well-temperament, but is equally relevant to
Pentacircle (which the more remote part of his 17-note system uses,
with pure 14/11 thirds), see:

<http://anaphoria.com/Secor17puzzle.pdf>

For Aaron Johnson's wonderful rendition of _Le Greygnour Bien_ by
Matteo de Perugia in a late 14th-century style, listen to

<http://www.akjmusic.com/audio/greygnour.ogg>

> (14/11)/(81/64) = 896/891. Tempering out 896/891 means four stacked
> 3/2s equals a (14/11) major third.

> The major third is once again composed of two equal steps. This
> time, the steps equate to both 9/8 and 112/99. (Yeah, I wouldn't
> have expected that, either, but that's where the math leads. It's
> about 214 cents.)

Yes, and following the presentation of your article, it's the same
pattern as with meantone for 5/4 (9:8 plus 10:9 in JI), or an Archytan
temperament for 9/7 (9:8 plus 9:7). Here we know that 14/11 is 896:891
or 9.688 cents larger than 81/64 (your rounded 10 cents gives an
excellent sense of the magnitude), so we need 9:8 plus another tone or
major second about ten cents larger -- or 214 cents (204 + 214 = 418).
That's indeed the 112/99.

The charm of this, as you nicely communicate, is that tempering out
896:891 and the associated commas you discuss and we'll get to in a
moment is a _much_ more forgiving problem than dealing with the 81:80
or 64:63. We can have mildly tempered fifths (comparable to 12-EDO and
maybe a bit closer to pure) plus a wonderful world of other intervals.

> The major third is slightly sharper than 81/64, so we have to
> slightly sharpen the fifth we use as a generator. Our major thirds
> will be equal to or flatter than 14/11.

This is generally quite right, but it is possible -- at the cost of
some additional compromise to the fifths -- to temper 14/11 slightly
on the wide side, although that's really the upper edge of the region
we're talking about, which I sometimes have called "the 704-cents
neighborhood."

It happens in a temperament I devised in June of 2000, the "e-based"
temperament because the ratio between the whole-tone and diatonic
semitone happens be equal to Euler's e -- no special acoustical
implications, just a fun way to select a shade of tempering. The fifth
is 704.607 cents, and the major third 418.428 cents (0.92 cents wide
of 14/11).

Nowadays my "sweet spot" would be around 703.711 cents, or almost a
cent less; and over the last decade and a bit more, this e-based
temperament has been the only "Pentacircle" one where the major thirds
were larger than 14/11. However, if we go to 17 or especially 24 or
more notes, there is an attraction here if and when we want a regular
tuning with a single chain of fifths.

That advantage is that 15 fifths up is almost precisely equal to 7/4,
and indeed 704.588 cents gives a just 7/4 and a regular major third at
418.354 cents (0.85 cents wide).

Please understand that my purpose is just to say that such "slightly
wider than 14/11" temperaments are out there; my leaning, much
agreeing with your words, is to have 14/11 just or actually a bit
narrow. But this example may help when we look at what happens with
minor thirds in different shades of Pentacircle temperament.

> Here's an interesting twist: The minor third isn't a simple
> ratio. The difference between 3/2 and 5/4 is 6/5; the difference
> between 3/2 and 9/7 is 7/6; but the difference between 3/2 and
> 14/11 is 33/28.

An important point! And the other side of this is that if we had a
pure fifth and 13/11 pure, then we'd get a major third at 33/26 or
412.745 cents.

> But pentacircle, even though it's an 11-limit comma, can be
> factored into two 13-limit superparticular commas: 364/363, which
> is | 2 -1 0 1 -2 1 >, and 352/351, which is | 5 -3 0 0 1 -1
> (Notice that the exponents of 13 cancel out when you add them.)

> 364/363 is the difference between (14/11)*(13/11) and 3/2. If
> pentacircle and 364/363 are both tempered out -- which means, of
> course, that 352/351 is also tempered out -- then 14/11 is the
> major third, 13/11 is the minor third, and a major third stacked on
> a minor third equals a perfect fifth.

This is a nice presentation of the general situation; and I suspect
that we often tend to think of 14/11 and 13/11 more than 33/26 or
33/28 simply because they are simpler ratios. Also, from a melodic
point of view, the difference between 13/11 and 14/11 is a beautiful
small neutral second of 14/13 (128.298 cents), that shows up in these
temperaments as the chromatic semitone, e.g. Eb-E (with C-Eb around
13/11 and C-E around 14/11).

Incidentally, there is a style of tuning which can demonstrate these
small commas or kleismas or whatever at 352:351 and 364:363 and how
they add up to 896:891; but I'll save that for the end, because it's a
bit of a diversion from our main topic of regular or near-regular
temperament.

> The 14/11 major third will always be a little flat. The fifth will
> always be a little sharp. I'm not sure of precisely what the minor
> third will be -- someone else may know.

Here the answer is that different shadings of Pentacircle will give us
different sizes of major and minor thirds, with the size of the
gently widened fifth also part of the equation.

For example, near the lower end of this Pentacircle region, let's
consider George Secor's HTT or High Tolerance Temperament family of
tunings with a generator of 703.579 cents, which he selected to
optimize lots of different ratios.

Our major third is 414.315 cents. That's rather closer to 33/26 (1.569
cents wide) than 14/11 (3.193 cents narrow) -- as you observe, the
difference between these ratios is 364/363 or 4.763 cents. So Secor's
major third is about a third of the way from 33/26 to 14/11.

His minor third at 289.264 cents is a virtual just 13/11 (289.210
cents). So we could say that his Pentacircle shading is closest to
13/11 and 33/26, but still gives a reasonable approximation for 14/11
also. This temperament, by the way, is almost identical with
(352/351)^(1/3) where 13/11 would be precisely just, and 33/26 wide by
the same amount as the fifth (1.642 cents).

As it happens, my own favorite shading of this temperament is MET-24
("Milder Extended Temperament" -- although not quite so mild as
Secor's!) with the fifths averaging at 703.711 cents. I say
"averaging," because my Yamaha TX-802 at 1024-EDO implements this by
tuning alternating fifths at 703.125 and 704.297 cents, which I
actually prefer; but synths with finer resolutions could simply tune
all the regular fifths at 703.711 cents.

Here the major third is 414.844 cents, or 2.664 cents narrow of 14/11,
and 2.098 cents wide of 33/26, to which we're still a bit closer.
As with Secor's shading, we can think of this tempered third as
representing both ratios.

The minor third in my 1024-EDO version is 288.281 cents or 289.453
cents, the former not quite a cent (0.928 cents) narrow of 13/11, and
the latter wide by 0.234 cents. Here, as with Secor's HTT family, we
have a near-just 13/11 almost a full 364:363 from 33/28 (284.447
cents).

Now let's move up to Keenan Pepper's "Noble Fifth" at 704.096 cents,
defining the temperament he announced in September of 2000. In terms
of a tradeoff between 13/11 and 14/11, this is about as optimized as
they come. The major third at 416.382 cents is about 1.126 cents
narrow of 14/11, and clearly much closer to this than to 33/26.

The minor third likewise is at 287.713 cents, or 1.497 cents narrow of
13/11 -- still within 1.5 cents of just, and 3.266 cents wide of
33/28.

With both 13/11 and 14/11 this close to just, we might expect to get a
near-just 14/13 step -- and we do, at 128.669 cents, just 0.371 cents
wide!

So from one point of view, Pepper's tuning (defined by the ratio of
Phi between the major second and near-14/13 chromatic semitone) is
just about the ideal case of what you're describing: 14/11 and 13/11
each just a bit narrow (no more than 1.5 cents each) and the fifth about
two cents wide (here 2.141 cents).

How about a just 14/11? For that, we need a fifth of 704.377 cents,
which gives us our 14/11 major third at 417.508 cents.

Now if the fifth were just, our 14/11 major third would have a 33/28
minor third. Here, the 33/28 will be wider by the same amount as the
fifth, or (896/891)^(1/4) -- about 2.422 cents, or 286.869 cents.
This is 2.341 cents narrow of 13/11.

So, if 14/11 is pure, we get a minor third almost equally distant
from 13/11 and 33/28, which could reasonably represent either.

Finally, let's quickly consider the e-based tuning at 704.607 cents,
where the major third at 418.428 cents, or 0.92 cents wide of 14/11.

Here the minor third is 286.179 cents, 3.030 cents narrow of 13/11 and
now only 1.732 cents wide of 33/28. So we're leaning now in the
direction of 33/28, but could still consider this third as a
reasonable representation of 13/11 also.

I could get more into the fine mechanism of the commas or kleismas
(896:891, 352:351, and 364:363), but this may be enough to address
your question and its opportunity to explore the general lay of the
land a bit.

> Finally, this temperament -- with pentacircle *and* 364/363 and
> 352/351 tempered out -- also has the diatonic scale structure of
> meantone: MOSs at 7 and 12 tones with Western modal patterns.

This is true, but I'd add that the Pentacircle in 12 notes
additionally has some wonderful resources for Near Eastern music which
we can expand either by using a larger MOS (e.g. 17), and/or by adding
another MOS chain of fifths (e.g. 12, or merrier yet with 17) at a
strategically chosen distance to optimize some septimal and extra
neutral ratios. For example, we add that second 12-note or 17-note
chain at 58.090 cents for Secor's HTT shading, 57.422 cents for
MET-24, or 58.680 cents for Peppermint.

However, let's stick to the 12-MOS. Not only do we have a great
"accentuated Pythagorean" diatonic -- major and minor thirds near
14/11 and 13/11, and diatonic semitones near 22/21.

We also get some small and large neutral intervals! The chromatic
semitone is close to 14/13, and we have a small neutral third
(i.e. the augmented second) somewhere around 63/52 (332.208 cents) or
17/14 (336.013 cents), and large neutral third around 26/21 (369.747
cents) or 21/17 (365.825 cents). There's also a diminished third
(e.g. C#-Eb, or twice the diatonic semitone) which in the milder
shadings like Secor's HTT or my MET-24 is quite close to 11:10
(165.004 cents).

With only 12 notes, we get more of a sampler than a full set of these
intervals, but it's a start! For example, if we want a bright Rast
mode of the kind favored in some parts of Syria, the basic notes of
the mode are there: B-C#-Eb-E-F#-G#-Bb-B. This approximates, for
example, a tetrachord of 9:8-11:10-14:13 (204-165-128 cents), which
interestingly would yield a fourth less than a cent narrow of pure!

From a medieval European perspective, we would have some interesting
augmented and diminished intervals very nice for a colorful
14th-century intonation; from a Near Eastern perspective, we have the
beginnings of the vocabulary of neutral intervals and modes so central
to medieval and later styles.

To enlarge on the European side of this: we get a wonderful diatonic
system and set of usual accidentals for 13th-14th century polyphony
and familiar cadences in three or four voices, _plus_ those neutral
intervals which are great when augmented or diminished intervals do
arise. Especially if we go with something like Secor's HTT or MET-24,
the fifths are generally a bit closer to pure than in 12-EDO, so the
Pythagorean ideal of 4:3 and 3:2 isn't too compromised.

Melodically, Pentacircle as you've termed it generally fits some
tendencies not only medieval Pythagorean intonation in Europe, but of
Persian music, where there's often a tendency to tune fifths a bit
larger than pure, say around 704-707 cents, with regular semitones
often around 80 cents or so, and lots of freedom and variability!

Anyway, I hope this isn't getting into too much detail, but addresses
some of your questions. As a medievalist, I'd also emphasize along
with George Secor (see my link above) that this style of tuning has
its own outlook, with a structure analogous to but with typically
different musical qualities and purposes than meantone with its
5-limit outlook

* * *

----------------------------------------------
Note on 896:891, 352:351, 364:363, 10648:10647 ----------------------------------------------

Now for my promised aside about the 352:351, 364:363, and the 896:891
which they almost equally divide (4.925 + 4.763 = 9.688 cents). By the
way, the tiny inequality of the 352:351 and 364:363 gives us a fourth
schisma of sorts: the 10648/10647 or harmonisma (0.163 cents).

While the Pentacircle involves a gentle tempering of the fifths by
something at or often a bit less than (896/891)^1/4, there is another
approach analogous in a way to unequal 12-note well-temperaments as
variations on 5-limit meantone: basically a "just temperament"!

We can get a just 14/11 with a chain of four fifths like this:
3/2-176/117-3/2-182/121.... Note that 176/117 (706.880 cents) is wide
by a 352:351, while 182:121 is wide by a 364:363. So we're alternating
between pure fifths and "virtually tempered" ones about not quite five
cents wide, or very close to (896/891)^1/2.

You might note that the compromise here of the impure fifths is very
close to that of 1/4-comma meantone (5.38 cents). But the contrast
between the pure and tangibly impure fifths is a bit analogous to
that in a colorful well-temperament like Werckmeister III!

Here's a "virtual temperament" of this kind in 7 notes:

! 44_39-diat1.scl
!
Diatonic involving 352:351 and 364:363
7
!
44/39
14/11
4/3
3/2
22/13
39/22
2/1

If we think of the 1/1 as G, this would be a variation on the medieval
European Mixolydian untransposed. Note we get a 14/11 (G-B) at 896:891
wider than 81/64; but also a 44/39 and 39/22 at 352:351 respectively
larger or smaller than 9/8 and 16/9, and also 22/13 likewise larger
than 27/16.

Manuel Op de Coul's Scala program is great for showing all the
intervals in this diatonic, and the interplay of the 896:891, 352:351,
364:363, and 10648:10647 - all of which show up when we look at the
step sizes and intervals of this basic diatonic!

Here's a similar 12-note chromatic:

! 44_39-12.scl
!
12-note chromatic tuning with 352:351, 364:363 (G=1/1, Eb-G#)
12
!
14/13
44/39
13/11
14/11
4/3
56/39
3/2
11/7
22/13
39/22
21/11
2/1

The interesting point here is that if we're ready to live with having
every other fifth as about as impure as in 1/4-comma meantone (or
actually a bit less), we can have a JI system no more difficult to
navigate than an unequal well-temperament -- although without
circulation, not really a consideration in medieval European or Near
Eastern music. I like this listing of the Scala file on G-G because
there are lots of interesting symmetries, but just make the 1/1 the
4/3 step (Scala command KEY 5), and we have the equivalent of a
14th-century European Halberstadt keyboard with octaves of C-C and a
range of Eb-G#.

! 44_39-12_C.scl
!
44_39-12.scl with C as 1/1 (Eb-G#)
12
!
14/13
9/8
33/28
33/26
117/88
63/44
3/2
21/13
22/13
39/22
21/11
2/1

In practice, the reason I usually go with Pentacircle is simply that
for most purposes I want consistently near-pure fifths -- in MET-24,
either 1.170 or 2.342 cents wide. But the "virtual temperament"
approach gives a nice survey of the just ratios we're approximating,
and also shows, at 22/13-2/1-14/11, how a pure 13/11 and 14/11 add up
to a fifth at 182/121, or a 364:363 wide.

For people who may not have Scala, here's a listing of the first
version (G-G) with the cents:

|
0: 1/1 0.000 unison, perfect prime
1: 14/13 128.298 2/3-tone
2: 44/39 208.835
3: 13/11 289.210 tridecimal minor third
4: 14/11 417.508 undecimal diminished fourth or major third
5: 4/3 498.045 perfect fourth
6: 56/39 626.343
7: 3/2 701.955 perfect fifth
8: 11/7 782.492 undecimal augmented fifth
9: 22/13 910.790 tridecimal major sixth
10: 39/22 991.165
11: 21/11 1119.463 undecimal major seventh
12: 2/1 1200.000 octave

And another with C rather than G as the 1/1:

|
0: 1/1 0.000 unison, perfect prime
1: 14/13 128.298 2/3-tone
2: 9/8 203.910 major whole tone
3: 33/28 284.447 undecimal minor third
4: 33/26 412.745 tridecimal major third
5: 117/88 493.120
6: 63/44 621.418
7: 3/2 701.955 perfect fifth
8: 21/13 830.253
9: 22/13 910.790 tridecimal major sixth
10: 39/22 991.165
11: 21/11 1119.463 undecimal major seventh
12: 2/1 1200.000 octave

Compare these with the way things get "averaged out," and altered a
bit, in a Pentacircle temperament like MET-24 (here a 12-note chain,
C-C with range of Eb-G#):

! met12.scl
!
Milder Extended Temperament, 5ths average 703.711 cents
12
!
126.56250
207.42187
289.45313
414.84375
496.87500
622.26563
704.29688
829.68750
911.71875
992.57813
1119.14063
2/1

With many thanks,

Margo Schulter
mschulter@...

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:

> > Pentacircle, 896/891, | 7 -4 0 1 -1 >
>
> This is a name that really attracts me, and I'm curious how you
> arrived at it. Does "penta" refer to the fifths tempered gently in the
> wide direction?

A pentatonic scale with a chain of four slightly wide fifths plus a slightly wide 11/7 can temper out 896/891 and close at the octave; cents(896/891)/5 = 1.94 cents shows about how much to temper the fifths.

Here is something in this tuning I did tonight. Solo piano - somewhat
impressionistic.

http://chrisvaisvil.com/?p=2776

On Mon, Oct 15, 2012 at 11:07 PM, genewardsmith <genewardsmith@...
> wrote:

> **
>
>
>
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > What's your favorite 15-note rank-3 Fokker block?
>
> I don't know about favorites, but this might be a place to start:
>
> ! dwarf15marv.scl
> Marvelous dwarf: 1/4 kleismic dwarf(<15 24 35|) subset rosatimarv
> 15
> !
> 115.587047
> 184.331593
> 200.054240
> 315.641287
> 384.385833
> 499.972880
> 584.440073
> 615.559927
> 700.027120
> 815.614167
> 884.358713
> 999.945760
> 1015.668407
> 1084.412953
> 1200.000000
> ! six tetrads/pentads representible by
> ! [[-1, 0, -1], [0, -1, 0], [0, 0, 1], [0, 1, 2], [0, -1, 1], [0, 0, 2]]
> ! nine-limit comma pump in this ordering (among others)
>
>
>

The melodic minor appears in Wilson's Marwa paper along with many 7 notes scales with aug. 2nd
--
signature file

/^_,',',',_ //^/Kraig Grady_^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere:
North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

a momentary antenna as i turn to water
this evaporates - an island once again

Jake originally wrote:

>>> Pentacircle, 896/891, | 7 -4 0 1 -1 >

I replied:

>> This is a name that really attracts me, and I'm curious how you
>> arrived at it. Does "penta" refer to the fifths tempered gently in the
>> wide direction?

And Gene responded:

> A pentatonic scale with a chain of four slightly wide fifths plus a
> slightly wide 11/7 can temper out 896/891 and close at the octave;
> cents(896/891)/5 = 1.94 cents shows about how much to temper the
> fifths.

Dear Jake and Gene,

After some pondering, I may have figured out the precise temperament
you're defining, Gene, with a just 22/21 (in a diatonic set) but
suspect that you, Jake, may be hinting at something slightly different
(with a just 14/13) in your original post, which asked a question I'll
try to answer below for both of these temperaments. First, Gene,
here's a possible statement of the problem I think you are posing:

Find the regular or eventone temperament for a 14/11
major third where:

(a) The approximations of 3/2 and 14/11 (or 11/7)
are equally impure; and

(b) The diatonic semitone or limma is a just 22/21.

The answer, of course, is specifically and uniquely 1/5-kleisma. As
with other eventones -- tempering out the 81/80 (meantone) or the
64/63 (Archytan) -- tempering out 1/5 of the applicable comma or
kleisma or whatever gets us a temperament where the fifth and major
third are equally impure, _and_ the diatonic semitone has a just
ratio, e.g. 16/15 (1/5-comma meantone), 28/27 (1/5-comma Archytan), or
here 22/21 (1/5-kleisma whatever).

If I'm correct as to the temperament, and were trying to explain it to
a newbie or anyone else, I'd say that 3/2 and 14/11 are equally
impure, and that 22/21 is just.

Here's a post from 2010 defining this temperament and its generator
with a tempering of (896/891)^(1/5), and showing a realization on a
synthesizer in 1024-EDO:

</tuning/topicId_91499.html#91499>

Jake, in your original post, you asked what the size of the minor
third would be in the kind of temperament you were describing, meant
to optimize both 14/11 and 13/11. If you meant this one defined by a
just 22/21 semitone, the answer is 288.322 cents, or about 0.877 cents
narrow of 13/11.

However, given your emphasis on 14/11 and 13/11 (rather than 3/2 and
14/11), and a temperament called Canton I found from 2011, if I'm
correct, I suspect what you might mean is the temperament where 14/11
and 13/11 are equally impure. Curiously, there's a quick way to
calculate this.

If 14/11 and 13/11 are equally impure, then 14/13 must be just! This
is the chromatic semitone of the tuning, e.g. Eb-E, with C-E a bit
narrow of 14/11 and C-Eb equally narrow of 13/11.

Now a chromatic semitone or apotome is equal to 7 fifths up less four
octaves, so we have a generator of (224/13)^1/7, or 704.042606 cents.
The major third is 416.170 cents, or 1.338 cents narrow of 14/11; and
the minor third is 287.872 cents, also 1.338 cents narrow of 13/11.

This Canton temperament, if indeed the idea is to have 14/11 and 13/11
equally impure and 14/13 just, is almost identical to Keenan Pepper's
Noble Fifth tuning at 704.096 cents (September 5, 2000); the
difference between the fifths is 0.053 cents, or about 1/20 cent.
They are distinct, however, with Pepper having a ratio between major
second and chromatic semitone equal to Phi (a la Kornerup's Golden
Meantone); and Canton, if I'm correct, has a just 14/13.

If this is the correct definition, then I might suggest that whoever
created Canton (you, Jake?) use a file like the following one for the
"smooth" or canonical version of Canton with 14/13 just, clearly
distinguishing it from the Pepper tuning. How about canton_14-13.scl
for a Scala file name in the usual style with all lowercase? The only
reason I haven't done this in the following file is a polite desire to
let you verify that this is really Canton in its ideal form, and then
adopt a file like this if you agree with my suggestion.

! 14_13-12.scl
!
Temperament with just 14/13 apotome, close to Pepper Noble Fifth
12
!
128.29824
208.08521
287.87218
416.17043
495.95739
624.25564
704.04261
832.34085
912.12782
991.91479
1120.21303
2/1

Similarly, what I'd call Pentakleismic (i.e. a temperament of 1/5 of
a kleisma, understood to be 896/891) should be distinguished from
Canton. This can be done easily: Pentakleismic has a pure 22/21, and
Canton a pure 14/13 -- respectively as the diatonic and chromatic
semitones, so they're kin in that way.

In fact, why not Pentakleismic-22/21 and Canton-14/13. That identifies
the unique definition of each tuning, and distinguishes both from
other nearby tunings including Pepper. In Scala, given the lowercase
convention and possible complications with slashes or colons in
filenames, maybe pentakleismic-22_21.scl and canton-14_13.scl.

Peace and love,

Margo

--- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@...> wrote:
>
> Here is something in this tuning I did tonight. Solo piano - somewhat
> impressionistic.

I still like it; I'd better link it. Did you find the small steps in some places to be a problem?

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:

> Find the regular or eventone temperament for a 14/11
> major third where:
>
> (a) The approximations of 3/2 and 14/11 (or 11/7)
> are equally impure; and
>
> (b) The diatonic semitone or limma is a just 22/21.

My strange word for (b) is that 22/21 is an "eigenmonzo". I Had this tuning in mind, but mostly just as a way of suggesting the range of tuning possibilities, which should not deviate from this too much. That is, the sharpness of 3/2 and 11/7 about the same.

> Here's a post from 2010 defining this temperament and its generator
> with a tempering of (896/891)^(1/5), and showing a realization on a
> synthesizer in 1024-EDO:
>
> </tuning/topicId_91499.html#91499>

I can't find a link to a music example in there. Is "ozone" the same as "pentacircle"?

Gosh I'm drawing a blank on that question. I guess that means the answer
has to be no.

I would appreciate you linking it to the correct part of the wiki. Do you
know how to embed the yahoo media player to enable on demand streaming play
back?

On Sun, Oct 21, 2012 at 12:17 PM, genewardsmith <genewardsmith@...
> wrote:

> **
>
>
>
>
> --- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@...> wrote:
> >
> > Here is something in this tuning I did tonight. Solo piano - somewhat
> > impressionistic.
>
> I still like it; I'd better link it. Did you find the small steps in some
> places to be a problem?
>
> _
>

On Sun, Oct 21, 2012 at 1:35 PM, Chris Vaisvil <chrisvaisvil@...> wrote:
>
> I would appreciate you linking it to the correct part of the wiki. Do you know how to embed the yahoo media player to enable on demand streaming play back?

Embed the Yahoo player on the wiki, you mean?

-Mike

On Sun, Oct 21, 2012 at 12:41 PM, genewardsmith
<genewardsmith@...> wrote:
>
> My strange word for (b) is that 22/21 is an "eigenmonzo". I Had this
> tuning in mind, but mostly just as a way of suggesting the range of tuning
> possibilities, which should not deviate from this too much. That is, the
> sharpness of 3/2 and 11/7 about the same.

Eigenmonzos for nonsquare matrices?

Are you first tuning temperament mappings into square projection maps
first when you talk about eigenmonzos?

-Mike

Yes, the subject was the wiki. I was checking to see if Gene knew how. I
have been embedding the player on the wiki pages I've found that need it.

On Sun, Oct 21, 2012 at 1:50 PM, Mike Battaglia <battaglia01@...>wrote:

> **
>
>
> On Sun, Oct 21, 2012 at 1:35 PM, Chris Vaisvil <chrisvaisvil@gmail.com>
> wrote:
> >
> > I would appreciate you linking it to the correct part of the wiki. Do
> you know how to embed the yahoo media player to enable on demand streaming
> play back?
>
> Embed the Yahoo player on the wiki, you mean?
>
> -Mike
>
>
>

Argh, top posting.

I'll add it to the wiki. Can you link me to the script I'm supposed to embed?

-Mike

On Sun, Oct 21, 2012 at 2:16 PM, Chris Vaisvil <chrisvaisvil@...> wrote:
>
> Yes, the subject was the wiki. I was checking to see if Gene knew how. I have been embedding the player on the wiki pages I've found that need it.
>
> On Sun, Oct 21, 2012 at 1:50 PM, Mike Battaglia <battaglia01@...> wrote:
>>
>> Embed the Yahoo player on the wiki, you mean?
>>
>> On Sun, Oct 21, 2012 at 1:35 PM, Chris Vaisvil <chrisvaisvil@...> wrote:
>> >
>> > I would appreciate you linking it to the correct part of the wiki. Do you know how to embed the yahoo media player to enable on demand streaming play back?

I'm on my tablet so links are a pain.

Go to the 17edo page on the wiki and look for the "other" widget.
Or you could Google yahoo media player. I think ymp could work as well.

It you have the time embed it into the template to eliminate the page by
Page embedding I'm doing now.

I suspect it's real purpose in life is to ferret out RIAA and MPAA posts
(it plays videos as well) but who cares. Its a great tool and I use it to
play all of the albums I host as well as all the music on my blog and the
xenharmonic Alliance blog as well.

Chris

On Sunday, October 21, 2012, Mike Battaglia <battaglia01@...> wrote:
>
>
> Argh, top posting.
>
> I'll add it to the wiki. Can you link me to the script I'm supposed to
embed?
>
> -Mike
>
> On Sun, Oct 21, 2012 at 2:16 PM, Chris Vaisvil <chrisvaisvil@...>
wrote:
>>
>> Yes, the subject was the wiki. I was checking to see if Gene knew how. I
have been embedding the player on the wiki pages I've found that need it.
>>
>> On Sun, Oct 21, 2012 at 1:50 PM, Mike Battaglia <battaglia01@...>
wrote:
>>>
>>> Embed the Yahoo player on the wiki, you mean?
>>>
>>> On Sun, Oct 21, 2012 at 1:35 PM, Chris Vaisvil <chrisvaisvil@...>
wrote:
>>> >
>>> > I would appreciate you linking it to the correct part of the wiki. Do
you know how to embed the yahoo media player to enable on demand streaming
play back?
>
>

OK, I did it. Please let me know if there's any trouble.

-Mike

On Sun, Oct 21, 2012 at 3:51 PM, Chris Vaisvil <chrisvaisvil@...> wrote:
>
>
>
> I'm on my tablet so links are a pain.
>
> Go to the 17edo page on the wiki and look for the "other" widget.
> Or you could Google yahoo media player. I think ymp could work as well.
>
> It you have the time embed it into the template to eliminate the page by Page embedding I'm doing now.
>
> I suspect it's real purpose in life is to ferret out RIAA and MPAA posts (it plays videos as well) but who cares. Its a great tool and I use it to play all of the albums I host as well as all the music on my blog and the xenharmonic Alliance blog as well.
>
> Chris
>
>
>
> On Sunday, October 21, 2012, Mike Battaglia <battaglia01@...> wrote:
> >
> >
> > Argh, top posting.
> >
> > I'll add it to the wiki. Can you link me to the script I'm supposed to embed?
> >
> > -Mike
> >
> > On Sun, Oct 21, 2012 at 2:16 PM, Chris Vaisvil <chrisvaisvil@...> wrote:
> >>
> >> Yes, the subject was the wiki. I was checking to see if Gene knew how. I have been embedding the player on the wiki pages I've found that need it.
> >>
> >> On Sun, Oct 21, 2012 at 1:50 PM, Mike Battaglia <battaglia01@...> wrote:
> >>>
> >>> Embed the Yahoo player on the wiki, you mean?
> >>>
> >>> On Sun, Oct 21, 2012 at 1:35 PM, Chris Vaisvil <chrisvaisvil@...> wrote:
> >>> >
> >>> > I would appreciate you linking it to the correct part of the wiki. Do you know how to embed the yahoo media player to enable on demand streaming play back?

--- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@...> wrote:
>
> Gosh I'm drawing a blank on that question. I guess that means the answer
> has to be no.
>
> I would appreciate you linking it to the correct part of the wiki. Do you
> know how to embed the yahoo media player to enable on demand streaming play
> back?

I just provide a direct link to an mp3 file, since a lot of people will have it set up so that plays on demand anyway.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Sun, Oct 21, 2012 at 12:41 PM, genewardsmith
> <genewardsmith@...> wrote:
> >
> > My strange word for (b) is that 22/21 is an "eigenmonzo". I Had this
> > tuning in mind, but mostly just as a way of suggesting the range of tuning
> > possibilities, which should not deviate from this too much. That is, the
> > sharpness of 3/2 and 11/7 about the same.
>
> Eigenmonzos for nonsquare matrices?

There's a square projection matrix for the 2.3.11/7 group with 2 and 22/21 as eigenmonzos and 896/891 as comma. Of course if you liked you could have 91/90, 121/120, 169/168, 441/440 generate your commas, and with the same eigenmonzos get the same tuning, only now you get a 6x6 square projection matrix for 13-limit leapday to go along with it.

> Are you first tuning temperament mappings into square projection maps
> first when you talk about eigenmonzos?
>
> -Mike
>

--- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@...> wrote:
>
> I'm on my tablet so links are a pain.

Personally, I like my plan of a direct link to an mp3 file the best. KISS.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:

>Of course if you liked you could have 91/90, 121/120, 169/168, 441/440 generate your commas, and with the same eigenmonzos get the same tuning, only now you get a 6x6 square projection matrix for 13-limit leapday to go along with it.

I should add here that the eigenmonzo for the minimax tuning is 6/5, not 22/21.

On Sun, Oct 21, 2012 at 5:25 PM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@...> wrote:
> >
> > I'm on my tablet so links are a pain.
>
> Personally, I like my plan of a direct link to an mp3 file the best. KISS.

It's already done. You still just direct link to an mp3 file, but now
there's a media player as well. Chris previously had it set up so that
the 17EDO, 16EDO, etc pages had this feature, but I've added it to the
whole wiki.

For those who don't have Javascript enabled, you won't see the player,
but you can still click on the link as before.

-Mike

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@> wrote:
> >
> > I'm on my tablet so links are a pain.
>
> Personally, I like my plan of a direct link to an mp3 file the best. KISS.

How long, for instance, will it take before this embedded player becomes a problem? The mp3 file is very robust that way.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> For those who don't have Javascript enabled, you won't see the player,
> but you can still click on the link as before.

Ah. Nice solution.

On Sun, Oct 21, 2012 at 5:29 PM, genewardsmith <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> How long, for instance, will it take before this embedded player becomes a
> problem? The mp3 file is very robust that way.

No, you still just direct link to the mp3. The script autodetects mp3
links in the document and creates a little popup player right in your
window at the bottom of the screen, if you have Javascript enabled,
with a playlist of all the mp3's found in the document.

-Mike

On Sun, Oct 21, 2012 at 5:19 PM, genewardsmith <genewardsmith@...>
wrote:
>
> There's a square projection matrix for the 2.3.11/7 group with 2 and 22/21
> as eigenmonzos and 896/891 as comma.

OK, I see. BTW, we may want to take this to tuning math, but Ryan
Avella's been doing lots of interesting work recently with eigenmonzos
for involutory matrices, since they allow you to transform one
temperament into another; I think that's the same thing you did in
your "Music for your Ears" composition.

-Mike

Hi, Gene, and thank you for a conversation which opens a dialogue
on some intriguing theoretical issues, as well as providing an
opportunity to document some tunings and share a link to music
(actually in an earlier tuning list post).

And greetings to you, Mike. I was thinking of sending a private
e-mail to invite your participation in what I hope will be a very
fruitful dialogue on all sides, but maybe this and related
threads can serve the same purpose. As I was going to say
privately, congratulations on your moderation and your service to
the tuning community!

Gene, lest this get lost in our theoretical niceties below,
please let me give the links to music in O3 now, which I'll repeat
near the end of this post also, giving both the original tuning
list post of 3 September 2010 with the link, and the mp3 link
itself:

</tuning/topicId_92309.html#92309>
<http://www.bestII.com/~mscbulter/Prelude_in_Shur_for_Erv_Wilson.mp3>

Our conversation so far:

Margo, on a possible understanding of Pentacircle:

>> Find the regular or eventone temperament for a 14/11
>> major third where:

>> (a) The approximations of 3/2 and 14/11 (or 11/7)
>> are equally impure; and
>>
>> (b) The diatonic semitone or limma is a just 22/21.

Gene:

> My strange word for (b) is that 22/21 is an "eigenmonzo". I Had
> this tuning in mind, but mostly just as a way of suggesting the
> range of tuning possibilities, which should not deviate from this
> too much. That is, the sharpness of 3/2 and 11/7 about the same.

Yes, my intuition about what "eigenmonzo" might mean and a bit of
Googling agree: it is the distinctive rational interval in a
regular temperament; makes sense, since I recall that in German,
_eigen_ means "singular" or the like. So indeed it's 22/21 here,
or 16/15 in 1/5-comma meantone, or 28/27 in 1/5-Archytan comma
(to which the nearer nine fifths of George Secor's 17-WT are very
close).

> Here's a post from 2010 defining this temperament and its generator
> with a tempering of (896/891)^(1/5), and showing a realization on a
> synthesizer in 1024-EDO:
>
> <[59]/tuning/topicId_91499.html#91499>

> I can't find a link to a music example in there. Is "ozone" the same as
> "pentacircle"?

First the links (repeated from above), then your question:

</tuning/topicId_92309.html#92309>
<http://www.bestII.com/~mscbulter/Prelude_in_Shur_for_Erv_Wilson.mp3>

A quick answer would be that O3 and Pentacenter, if the latter is
defined as a pentatonic set, both use generators of 2/1 and
703.893 cents, but for quite different projects, with lots more
landmarks and constraints for the former, including the 22/21
semitone and an 11/10 approximation at around 161-162 cents, that
simply don't pertain to a pentatonic. So the latter might well
have a fuzzier definition.

With a pentatonic, you're not weighing the tradeoff between 11/10
and 14/13 (an issue in a 12-MOS or larger system like O3), for
example. It's just the idea that 3/2 and 14/11 or 11/7 are
comparably impure: where there are no landmarks or constraints,
why not be flexible?

In another post, I will discuss the question of mapping
subregions for these undecimal/tredecimal or 14/11 and 13/11
temperaments, with links to some Scala files suggesting some of
these subregions: Secor's 29-HTT; MET-24; O3; Canton and the
slightly more tempered Pepperment, etc.

But for Pentacenter, all this might be a bit much, rather like
trying to draw exact lines for a meantone pentatonic called
Pentacomma where the 3/2 and 8/5 are about equally narrow.
People trying to map meantone subregions will probably look at an
MOS of 12, 19, or 31, etc.

Peace and love,

Margo

On Sun, 21 Oct 2012, Margo Schulter wrote:

> </tuning/topicId_92309.html#92309>
> <http://www.bestII.com/~mscbulter/Prelude_in_Shur_for_Erv_Wilson.mp3>

The last URL is an obvious typo, and I apologize for not checking
this in my mail program before posting, although the first link
will get you there:

> <http://www.bestII.com/~mschulter/Prelude_in_Shur_for_Erv_Wilson.mp3>

Best,

Margo

On Sun, Oct 21, 2012 at 9:52 PM, Margo Schulter <mschulter@...>
wrote:
>
> And greetings to you, Mike. I was thinking of sending a private
> e-mail to invite your participation in what I hope will be a very
> fruitful dialogue on all sides, but maybe this and related
> threads can serve the same purpose. As I was going to say
> privately, congratulations on your moderation and your service to
> the tuning community!

Thanks Margo, it's good to see you posting again! I note you recently
wrote a post on rank-3 temperaments as well, which I have yet to check
out (though I read the email linking to the longer document, and it
looked like a very well-written and clear exposition on the subject).

I guess that for now, all I have to contribute is that if your goal is
a chain of fifths such that 81/64 is equated with 14/11 and 32/27 is
equated with 13/11, then the 2.3.14/11.13/11 temperament eliminating
896/891 and 352/351 will do the trick. The TOP tuning for this
temperament is <1199.506 1902.738 413.917 289.315| where the values
represent the tunings for 2/1, 3/1, 14/11, and 13/11 respectively.
This tuning assigns to each interval n/d a weight of 1/(n*d), so that
simpler intervals are weighted more strongly, and then minimizes the
maximum weighted tuning error over all intervals in the lattice. The
pure-octave version is <1200.000 1903.522 414.088 289.434|, which
makes the fifths a bit sharper.

The TE tuning, which is similar but which instead minimizes something
like "the average error over all intervals" rather than the "max error
over all intervals," is <1199.371 1902.859 415.207 288.280|, with the
pure octave version being <1200.000 1903.857 415.425 288.431|, so it's
about the same.

But, to my ears, I like the slightly sharp version more, especially
when it gets to around 17-EDO, where the 14/11 is tuned sharp at
423.529 cents. I suspect that this is because there's no real
noticeable aural effect from the interval being very close to a 13/11
or 14/11 ratio, and that what I really like are slightly sharp fifths,
possibly for the same reason that the ear tends to like slightly sharp
octaves. So I'm happy with that. However, I suspect that if I were an
expert listener in 13-limit harmony, I really would care a bit about
the tuning of 14/11 and 13/11, because I'd be used to hearing those
intervals as "pieces" of much larger 13-limit chords, much like we can
hear the dyads in a two part invention as pieces of an implied triadic
or tetradic background.

I've gotten better at "placing" intervals like 11/8 and 13/8 this way,
but 14/11 still sounds like a mixture between 9/7 and 5/4 to me; e.g.
it sounds like it'd go equally well if you turned it into 4:7:9 (with
the 14/11 being like a tempered 9/7) or 4:5:6 (with the 14/11 being
like a tempered 5/4). Is this a bug or a feature? (I say it's both,
depending on the musical circumstance.)

Lastly, I enjoyed your Prelude above - have you tried writing
something in superpyth that uses 4:6:7 triads as the basis for harmony
rather than 4:5:6? I'm not sure if I've asked you this before, but I
think it'd sound really good. I know that 6:7:9 is trendier, but I
think there's some magic lurking behind 4:6:7, mostly because if you
double the lowest note down an octave, e.g. you do C,,-C,-C-G-Bb or
something, it just becomes 1:2:4:6:7. 6:7:9, on the other hand,
becomes 3:6:12:14:18, which is a bit more complex.

Anyway, my two cents,
Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> Lastly, I enjoyed your Prelude above - have you tried writing
> something in superpyth that uses 4:6:7 triads as the basis for harmony
> rather than 4:5:6?

There's a lot of music in that directory, and it would be nice to have a brief listing of the tunings used.

Dear Mike,

Please let me first offer a brief reply thanking
you for your kind words about my rank-3 paper, and explain
that my longer reply which follows focuses not so much
on optimizing 14/11 and 13/11, where you have some really
interesting ideas and tunings I'll address in another post
soon.

But the purpose is gently and persuasively to put on your
RMP radar screen a very important kind of 2.3.7.11.13
subgroup mapping I propose we call parapyth, which George
Secor discovered and used as a one subset of his 29-note
High Tolerance Temperament (29-HTT, now a member of the
larger HTT family) of 1978.

As I discuss in my next post, you alerted in me in asking
about 4:6:7 and 6:7:9 that you were unaware that they occur
in my _Prelude in Shur for Erv Wilson_, and that the O3
temperament I use for that piece has a just 7/4 as one of
its defining intervals!

<http://www.bestII.com/~mschulter/Prelude_in_Shur_for_Erv_Wilson.mp3>

While superpyth tempers out the 64/63, parapyth leaves it
in, and, not surprisingly, is much more accurate for ratios
of 3 and 7 alike. My longer post gives specifics and links
to Scala files, so you can evaluate the accuracy of parapyth
for yourself.

Since there's a lot more going on in 29-HTT like full support
of odd factors of 2-3-5-7-9-11-13-15, you'll see how it might
take someone with a special interest in 2.3.7.11.13 to pick
out this part of his tuning and make it the basis for a 24-note
rank-3 tuning for this subgroup of his 13-limit.

It happened in 2002 with Peppermint, for which in that year
I posted a catalogue of near-just intervals and their
mapping in its "ratio space" -- not necessarily mapping in
the RMP sense, exactly, but how to find different primes
and ratios. There a link to that in my next message.

The basic idea is simple, with lots of tweaking to taste
possible as with meantones -- but Secor's HTT is the classic
example to show the recipe. Tune a linear temperament where
13/11 and thus 22/13 is at or close to just. Now add another
linear chain -- the rank-3 part -- at a spacing of about
91/88, and you get pure 7/4 minor sevenths! The 7/6 will
have the same error as the fifth, around two cents or a
bit less (1.624 cents for HTT).

We also get 11 and 13 near-just as well: 11/8, 13/8, lots
of neutral seconds of different sizes, etc., etc. Note that
we also get 13/7, for example, within each chain, so the
variety of neutral intervals is one advantage by comparison
with Pythagorean or nearby schismic temperaments, as well
as with superpyth, especially if we're seeking accuracy.

What I haven't understood over the years is why RMP doesn't
seem much to recognize this parapyth 2.3.7.11.13 mapping -- as
important a contribution of George Secor, I would say, as
his Miracle temperament published in 1975.

And the answer your questions about 4:6:7 and 6:7:9 give
me is that you simply haven't been aware of parapyth, as
I'm now proposing to call it in view of your superpyth.

Again, my next and longer post has lots of details and
links. And I'm eager to see how RMP might analyze the
commas and the "mapping" in your sense, or approach a TOP
version.

With many thanks,

Margo

> Thanks Margo, it's good to see you posting again! I note you
> recently wrote a post on rank-3 temperaments as well, which I have
> yet to check out (though I read the email linking to the longer
> document, and it looked like a very well-written and clear
> exposition on the subject).

Dear Mike,

Thank you for your ideas on optimizing 14/11 and 13/11, and
especially the TOP solution, which I'll address in another
reply. This is a fascinating topic in itself, but I'd like to use
this reply as an opportunity to clarify a very basic point about
what a 2.3.7.11.13 subgroup temperament like Peppermint (2002),
O3 (2010) or MET-24 (2011) actually does.

Your discussion of the 4:6:7, for example, is a wonderful
opportunity to point out that while it wouldn't occur in a 12-MOS
optimization for 14/11 and 13/11, it is absolutely central to
Peppermint, MET-24, and this rank-3 family generally. For
example, as I'll discuss below, _Prelude in Shur for Erv Wilson_
uses a near-just 4:6:7 as one of the most important sonorities
(this piece uses O3).

As your remarks about 4:6:7 and 6:7:9 -- very interesting in
themselves, as I note below -- bring home to me, you are
evidently unaware that since 2002, a major purpose of rank-3
temperaments around 704 cents is get ratios of both 3 and 7 far
more accurate than in what is now called superpyth. Maybe we
should call this mapping parapyth, but the reality is more
important than the name. Each is wonderful in its own way, but I
would think that the rank-3 structure of 2.3.7.11.13 parapyth
would be of special interest to people interested in commas,
mapping, and accuracy.

In seeking gently and effectively to get these quite accurate
rank-3 system on your RMP radar screen, I'd like to point out
that the 14/11-13/11 optimization tree is part of a larger
2.3.7.11.13 forest, and that since 2002, optimizing the whole
forest has been the main issue for those of us designing rank-3
temperaments in this family.

On the 50th anniversary of a very fateful day in world history,
I'm writing to urge that communication and cooperation prevail.
In fact, I'd love to see RMP comma or matrix analyses of these
quite intricate and accurate and yet structurally simple tunings,
TOP versions, etc.

In short, this post is a friendly bid for diplomatic recognition
of a beautiful family of rank-3 temperaments, and an intonational
detente permitting us to learn from each other.

The spirit of cooperation is really more important than details
of math, and I hope that that, especially, is what comes across.
There follows some documentation in terms of past posts on the
list, Scala files, and music; but simple recognition is the main
point.

* * *

The post on rank-3 temperaments which you mention is one approach
to some of the basics, and I tried to make things easy for
newbies and others who may be new to this area.

Here I'd like to clarify points about the history of this
2.3.7.11.13 family, which goes back to a subset of George Secor's
29-note High Tolerance Temperament or 29-HTT tuning he designed
in 1978 with a 2/1 octave, a fifth of 703.579 cents, and spacing
between the two relevant chains of fifths at 58.090 cents. This
was just one part of his system supporting odd factors of
2-3-5-7-9-11-13-15, and the part that caught my attention in
2001-2002

That was the prototype and model, or _fons et origo_ ("font and
origin"), as I might say to quote the Latin of Tinctoris, for
Peppermint and all the rest of these rank-3 systems.

> I guess that for now, all I have to contribute is that if your
> goal is a chain of fifths such that 81/64 is equated with
> 14/11 and 32/27 is equated with 13/11, then the
> 2.3.14/11.13/11 temperament eliminating 896/891 and 352/351
> will do the trick. The TOP tuning for this temperament is
> <1199.506 1902.738 413.917 289.315| where the values represent
> the tunings for 2/1, 3/1, 14/11, and 13/11 respectively.

An interesting point of relevance to my immediate purpose is that
these sizes of major and minor thirds at 413.917 and 289.315 are
very close to Secor's 29-HTT at 289.264 and 414.315 cents: your
shade gives a virtually just 13/11 (289.210 cents), with 14/11
off by 3.59 cents. As in meantone with 5/4 and 6/5, there will be
lots of tastes as to which shade is most pleasant for what
purpose.

However, with 29-HTT or its 2.3.7.11.13 offshoots beginning with
Peppermint, 14/11 and 13/11 are only a small part of the total
picture.

Here's an article from 2002 about the mapping of primes and
ratios in the "ratio space" of Peppermint (2/1, 704.096, 58.090):

</tuning/topicId_40057.html#40057>
<http://www.bestII.com/~mschulter/peprmint.scl>

A just 7/6 minor third is part of the definition of the 3-D
tempered lattice.

And we get lots of neutral intervals like 11/9, 13/8, 11/6, not
only for maqam music, but for people who want to try 8:11:13 or
Secor's 7:9:11:13, etc. Here's a piece in Peppermint using
neutral steps of around 12/11 and 13/12, and also George Secor's
articles on lots of scales and chords to be found in his 17-tone
well-temperament, and also, yet more accurately, in his 29-HTT
and its 2.3.7.11.13 offshoots like Peppermint:

<http://www.bestII.com/~mschulter/SubArbore.mp3>
<http://www.bestII.com/~mschulter/SubArbore.pdf>
<http://www.anaphoria.com/Secor17puzzle.pdf>

And here's a lattice mapping for MET-24 (the Milder Extended
Temperament) of 2011, which differs from Peppermint mainly in its
milder shade of temperament so that 11/10 and 14/13 are equally
impure, along with Scala files for the 1024-EDO version I use and
the "canonical" version (2/1, 703.723, 57.423):

<http://www.bestII.com/~mschulter/met24-lattice1.jpg>
<http://www.bestII.com/~mschulter/met24.scl
<http://www.bestII.com/~mschulter/met24-canonical.scl>

> Lastly, I enjoyed your Prelude above - have you tried writing
> something in superpyth that uses 4:6:7 triads as the basis for
> harmony rather than 4:5:6?

<http://www.bestII.com/~mschulter/Prelude_in_Shur_for_Erv_Wilson.mp3>
<http://www.bestII.com/~mschulter/O3-reg-24.scl>
<http://www.bestII.com/~mschulter/O3-24.scl>

The short answer is that the Prelude uses 4:6:7 as one of its
main sonorities, but with much less inaccuracy in the fifth than
you'd need for superpyth! In O3, the 7/4 is by definition just
(2/1, 703.893, 57.148), and the fifth off by 1.938 cents (the
same as 14/11). In the 1024-EDO version for this recording, 7/4
is always within a cent of just, at 826 or 827 tuning units
(967.969 or 969.141 cents).

With superpyth we'd need the fifth impure by 5 cents or more, and
the 2-3-7-9 intervals wouldn't all be within 4.5 cents of just
(in O3, the 9/7 and 14/9 don't fare as well as the just 7/4, or
the 7/6 off by the same amount as the fifth, 1.938 cents).

Again, George Secor discovered or possibly rediscovered this
ingenious mapping for a just 7/4 in 1978 (basically a linear
22/13 major sixth plus a spacing between two chains of 91/88,
which in 29-HTT is what we get to within a tenth of a cent!),
and Peppermint and O3 and MET-24 seek to make the most of it
in different shadings and nuances. I'm confident that you could
come up with a TOP version, too.

In the Prelude you'll also hear 13/8, and lots of 13/12 and 14/13
steps -- but, actually, little of 14/11 and 13/11; this piece has
more of a septimal focus, although, of course, in other contexts
I'm constantly using these regular thirds also.

But my purpose is to put this 2.3.7.11.13 mapping squarely on
your radar screen -- a happier metaphor today than on this day 50
years ago -- so that we can have mutual recognition and fruitful
cooperation.

Of course, superpyth has its place, also, and the nearer portion
of George Secor's 17-WT is a great example, see his "17puzzle"
article linked to above. But the rank-3 approach gets both the
3/2 and 7/4 within two cents or so of just, so both mappings are
indeed very useful options, as I learned from George Secor.

> I'm not sure if I've asked you this before, but I think it'd
> sound really good. I know that 6:7:9 is trendier, but I think
> there's some magic lurking behind 4:6:7, mostly because if you
> double the lowest note down an octave, e.g. you do
> C,,-C,-C-G-Bb or something, it just becomes 1:2:4:6:7. 6:7:9,
> on the other hand, becomes 3:6:12:14:18, which is a bit more
> complex.

Personally I wouldn't live without either: I do agree that if the
idea is to have a more conclusive sonority, the "rooted" quality
of 4:6:7 may help. Since I typically treat anything more
complicated than 2;3:4 as inconclusive, it might not be quite so
much of a contrast for me, but still in an interesting
distinction in terms of sheer sound. At any rate, the Prelude
definitely goes for 4:6:7.

Of course, one solution is have your cake and eat it too, as I've
been doing for a decade and more with 12:14:18:21. In this
mapping on two Halberstadts, let's say in MET-24, it would be:

-2.002 -0.234
264.869 968.592
D* A*
7/6 7/4

C G
1/1 3/2
0 703.723
+1.768

What I now understand, from your question, is that you were
simply unaware of this mapping, which more generally I would
expect to see as a 2.3.7.9.11 subgroup temperament, however tweaked
or TOP-adjusted according to your algorithms. I'm eager to hear
an RMP analysis of the commas and mapping, etc.

> Anyway, my two cents,

A very apt expression, since in this mapping the fifth is
tempered by around 2 cents (in O3, the tuning for the Prelude, a
theoretical 1.938 cents), and that's the accuracy for the 7:6 in
either 6:7:9 or 4:6:7!

With many thanks,

Margo Schulter
mschulter@...

Hi Margo,

On Mon, Oct 22, 2012 at 7:39 PM, Margo Schulter <mschulter@...> wrote:
>
> But the purpose is gently and persuasively to put on your
> RMP radar screen a very important kind of 2.3.7.11.13
> subgroup mapping I propose we call parapyth, which George
> Secor discovered and used as a one subset of his 29-note
> High Tolerance Temperament (29-HTT, now a member of the
> larger HTT family) of 1978.

Sure, what commas does it temper out? If it starts off in a rank-5
tuning (2.3.7.11.13) and tempers down to a rank-3 tuning, then there
must be 5-3=2 commas tempered out...

> As I discuss in my next post, you alerted in me in asking
> about 4:6:7 and 6:7:9 that you were unaware that they occur
> in my _Prelude in Shur for Erv Wilson_, and that the O3
> temperament I use for that piece has a just 7/4 as one of
> its defining intervals!

I do hear that you land on 4:6:7 right before the end at around 0:32
before resolving it up to 2:3:4. It was definitely a nice sound. I
guess I'm just curious what'd happen if someone were to attempt the
following:
1) take the Credo from Machaut's Messe de Nostre Dame, which uses
triads in a very tense way, and write a piece in the same style, but
where 4:6:7 takes the place of the major chord and 14:21:27 takes the
place of the minor chord
2) take Gesualdo's Tristis est Anima Mea, which utilizes triadic
harmony in a much more impressionistic sounding way, and figure out
how to create the same effect with 4:6:7 and 14:21:27
3) take anything by Palestrina, which uses triadic harmony in a very
straightforward and tonal way, but make 4:6:7 and 14:21:27 the basis
of the tonality rather than 4:5:6 and 10:12:15

I just wonder if superpyth implies some sort of secret alternate
tonality that exists in the diatonic scale, provided you intone things
slightly differently. It's an idea I'm stealing from your and George
Secor's paper on 17-EDO, where you suggest that 6:7:9 is a great
alternative way to intone the minor chords of the diatonic scale,
rather than meantone's 10:12:15. So I wonder if it's possible to be
even more avant garde with it, and instead of having root-third-fifth
serve as the basis for triadic harmony, if you could use
root-fifth-seventh instead, owing to the strength of the 4:6:7 that's
present in the scale. (It'd probably require a tuning closer to 22-EDO
to pull off, though I'm 17-EDO would still sound very nice.)

> What I haven't understood over the years is why RMP doesn't
> seem much to recognize this parapyth 2.3.7.11.13 mapping -- as
> important a contribution of George Secor, I would say, as
> his Miracle temperament published in 1975.

I haven't heard of it before, but it's probably because the subgroup
temperament exploration is still very new, and the rank-3 temperament
subgroup exploration is even newer (have we even tried any systematic
search of rank-3 subgroup temperaments yet?).

Here's Graham's list of the best 2.3.7.11.13 rank-3 temperaments:
http://x31eq.com/cgi-bin/more.cgi?r=3&limit=2_3_7_11_13&error=5.0

Almost none of them have names, and of the ones which do have names,
it's just "guessing" the name as an extension of something else. So
this is all uncharted territory, and I'd be interested to see if
Parapyth turns up near the top of these.

I'll be back later to go through your next email and the one from
before, gotta do some work first though...

-Mike

Hi, Mike!

You ask lots of good questions, and I'll try to clear up at least
some of the commas tempered out in parapyth, and see if maybe we
can find others -- exactly the kind of inquiry I was hoping for!

>> But the purpose is gently and persuasively to put on your
>> RMP radar screen a very important kind of 2.3.7.11.13
>> subgroup mapping I propose we call parapyth, which George
>> Secor discovered and used as a one subset of his 29-note
>> High Tolerance Temperament (29-HTT, now a member of the
>> larger HTT family) of 1978.

> Sure, what commas does it temper out? If it starts off in a
> rank-5 tuning (2.3.7.11.13) and tempers down to a rank-3
> tuning, then there must be 5-3=2 commas tempered out...

Absolutely true! Let's see if we can account for some of them,
and also note one which I thought might be tempered out by
apparently isn't -- subject to review and correction by you and
others.

The obvious ones are those which also get tempered out in a
rank-2 temperament for 14/11 and 13/11. These are the 896/891,
352/351, 364/363, and 10648/10647 -- and also another very
important related one, the 28672/28431 (14.613 cents).

This last one is equal to 896/861 plus 352/351, and I propose
that it be named the Secorian comma (or superkleisma?), because
George Secor tempered it out in 29-HTT (secor29htt.scl) in 1978
in order get an augmented second or supraminor third at a just
ratio of 63/52 (332.208 cents). In other words, his generator of
703.579 cents is (504/13)^1/9, and this fifth is tempered by
(28672/28431)^1/9.

This comma is the difference, for example, between the
Pythagorean augmented second at 19683/16384 (317.595 cents), and
Secor's 13-based 63/52 -- the difference between a near-6/5
or schismatic minor third and parapyth supraminor or semineutral
third. Likewise Pythagorean gives an apotome at 2187/2048 or
113.685 cents, very close to 16/15 (111.731 cents); add Secor's
comma, and we get 14/13 (128.298 cents), a parapyth supraminor
or small neutral second.

For now, I'll quickly point out that all five primes are already
represented in a rank-2 12-MOS: e.g. 21/13 (3-7-13) and 14/11
(2-7-11).

But to get the harmonic forms 7/4, 11/8, and 13/8, we need that
extra rank-3 generator of a spacing between two 12-MOS, 17-MOS,
29-MOS or whatever chains at around 91/88, or the difference
between our 22/13 major sixth and a 7/4 minor seventh.

In order to give you some sporting chance to read and absorb what
I've posted (and the earlier rank-3 article might help), why
don't I leave for now with the lattice for MET-24, which may give
more clues on the commas tempered out or observed; and another
diagram showing some of the ones which are tempered out as one
looks at some of the ratios in the rank-3 system:

<http://www.bestII.com/~mschulter/met24-lattice1.jpg>
<http://www.bestII.com/~mschulter/met24-tour.jpg>

>> As I discuss in my next post, you alerted in me in asking
>> about 4:6:7 and 6:7:9 that you were unaware that they occur in
>> my _Prelude in Shur for Erv Wilson_, and that the O3
>> temperament I use for that piece has a just 7/4 as one of its
>> defining intervals!

> I do hear that you land on 4:6:7 right before the end at around 0:32
> before resolving it up to 2:3:4. It was definitely a nice sound. I
> guess I'm just curious what'd happen if someone were to attempt the
> following:

> 1) take the Credo from Machaut's Messe de Nostre Dame, which uses
> triads in a very tense way, and write a piece in the same style, but
> where 4:6:7 takes the place of the major chord and 14:21:27 takes the
> place of the minor chord

Putting aside medievalist quibbles about some terminology, I
think that 4:6:7 as a relative concord, or 12:14:18:21, would be
great! The 14:21:27 is a strong discord, and could be very
effective in that role, although my analogies might be a bit
different. At any rate, these two could play a central role:
Machaut himself is said to have treated the minor seventh almost
as an imperfect concord.

> 2) take Gesualdo's Tristis est Anima Mea, which utilizes triadic
> harmony in a much more impressionistic sounding way, and figure out
> how to create the same effect with 4:6:7 and 14:21:27

How about analogous effects -- that is one awesome piece! In
addition to your suggested sonorities, of course, I would also
feel free to use regular thirds, neutral intervals, thirdtone
steps, everything I have -- this project takes nothing less!

> 3) take anything by Palestrina, which uses triadic harmony in
> a very straightforward and tonal way, but make 4:6:7 and
> 14:21:27 the basis of the tonality rather than 4:5:6 and
> 10:12:15

The real quibble I have here is that while 4:6:7 is relatively
concordant, traditionally major sevenths are strong dissonances;
but let's keep the spirit of this exercise by modifying it a tad.
How about 4:6:7 as a relative concord, but 14:21:27 and related
sonorities with 27/14 (or 28/27) as suspensions? After all,
Palestrina does use major sevenths in suspensions! And George
Secor, in his 17-puzzle article, shows how attractive and intense
these suspensions can be!

Getting a sense of classical balance like Palestrina's with these
materials is a challenging -- and therefore great -- idea!

> I just wonder if superpyth implies some sort of secret
> alternate tonality that exists in the diatonic scale, provided
> you intone things slightly differently. It's an idea I'm
> stealing from your and George Secor's paper on 17-EDO, where
> you suggest that 6:7:9 is a great alternative way to intone
> the minor chords of the diatonic scale, rather than meantone's
> 10:12:15. So I wonder if it's possible to be even more avant
> garde with it, and instead of having root-third-fifth serve as
> the basis for triadic harmony, if you could use
> root-fifth-seventh instead, owing to the strength of the 4:6:7
> that's present in the scale. (It'd probably require a tuning
> closer to 22-EDO to pull off, though I'm 17-EDO would still
> sound very nice.)

Well, Secor is talking not about 17-EDO, really, but about 17-WT,
where the nearer parts of the circle are a lot closer to 7.
If you want to avoid the 64:63 complications, that or 22-EDO
could make sense. I'd go with 17-WT, a really beautiful tuning!

But rank-3 parapyth does about a third as much tempering of the
fifth, and can get 7/6, 7/4, and 9/7 all within about 2.141 cents
of just if we use Peppermint's solution of making 7/6 pure, with
7/4 and 9/7 with the same impurity as the fifth.

>> What I haven't understood over the years is why RMP doesn't
>> seem much to recognize this parapyth 2.3.7.11.13 mapping -- as
>> important a contribution of George Secor, I would say, as his
>> Miracle temperament published in 1975.

> I haven't heard of it before, but it's probably because the
> subgroup temperament exploration is still very new, and the
> rank-3 temperament subgroup exploration is even newer (have we
> even tried any systematic search of rank-3 subgroup
> temperaments yet?).

This is an opportunity for very productive cooperation between
those of us who have been exploring these tunings over the last
decade from a non-RMP perspective, and RMP people like yourself
who would like to delve into them and maybe bring new insights.

> Here's Graham's list of the best 2.3.7.11.13 rank-3 temperaments:
> [59]http://x31eq.com/cgi-bin/more.cgi?r=3&limit=2_3_7_11_13&error=5.0

> Almost none of them have names, and of the ones which do have names,
> it's just "guessing" the name as an extension of something else. So
> this is all uncharted territory, and I'd be interested to see if
> Parapyth turns up near the top of these.

The quick story is that the first couple I've looked at in the
pure octave versions look like variations on O3 or Peppermint
where someone decided to add irregularity -- something where
George Secor, a master of irregular schemes, might be the best
person to judge. Apart from the irregular features, these are
definitely parapyth, coming out when regularized as things like
(2/1, 704.115, 58.115) or (2/1, 703.856, 58.338). These are
almost identical to Peppermint and O3 respectively.

So I'd say that the intonational territory or parapyth itself has
been quite well charted, but the irregularities introduced by RMP
methods are of interest -- a bit like the modified meantone
systems of the 17th and 18th centuries, where regular meantones
were quite familiar but the variations were mostly new.

> I'll be back later to go through your next email and the one from
> before, gotta do some work first though...

Great, and I'll look at more of those RMP tunings you've directed
me to very helpfully. And you've been very generous responding at
this length!

With many thanks,

Margo

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:

> The obvious ones are those which also get tempered out in a
> rank-2 temperament for 14/11 and 13/11. These are the 896/891,
> 352/351, 364/363, and 10648/10647 -- and also another very
> important related one, the 28672/28431 (14.613 cents).

Using the wonders of mathematics, we can show that this is just the same as tempering out 352/351 and 364/363 in 2.3.7.11.13, which is rank 3. Of course, nothing stops you from, for example, adding 325/324 to the mix and now you have 5s.

> This last one is equal to 896/861 plus 352/351, and I propose
> that it be named the Secorian comma (or superkleisma?),

I've added it to the Xenwiki comma list.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:

> Using the wonders of mathematics, we can show that this is just the same as tempering out 352/351 and 364/363 in 2.3.7.11.13, which is rank 3. Of course, nothing stops you from, for example, adding 325/324 to the mix and now you have 5s.

A thought you might like better is that adding 169/168 to the list gives a no 5s version of leapday temperament. The POTE tuning would be a fifth of 704.745 cents. This is probably something you've already done under some name or other.

On Tue, Oct 23, 2012 at 6:16 AM, Margo Schulter <mschulter@...> wrote:
>
> For now, I'll quickly point out that all five primes are already
> represented in a rank-2 12-MOS: e.g. 21/13 (3-7-13) and 14/11
> (2-7-11).
//snip
> In order to give you some sporting chance to read and absorb what
> I've posted (and the earlier rank-3 article might help), why
> don't I leave for now with the lattice for MET-24, which may give
> more clues on the commas tempered out or observed; and another
> diagram showing some of the ones which are tempered out as one
> looks at some of the ratios in the rank-3 system:

So I've finally caught up on everything, and one thing I notice right
away is that your interval C-G#* is 890 cents, only about 5.4 cents
sharp of a 5/3. If we want to call it 5/3, then we've now expanded to
a full 13-limit rank-3 temperament, which also tempers out 196/195.

This temperament apparently already has a name and it's called "Pele"
temperament: http://x31eq.com/cgi-bin/rt.cgi?ets=58_41_87&limit=13.
The POTE optimal tuning for this has the fifths at 703.414 cents and
spacing at 57.468 cents, which leads to 14/11 set to 413.657 cents,
13/11 set to 289.756 cents, 5/4 set to 388.196, and 6/5 set to 315.218
cents.

As for how you get to these new 5-limit intervals, let's say that C^
is one of your "spacing" commas sharp of C, and then that Cv is a
spacing comma flat of it. So 5/3 would be C-G#^, 6/5 would be C-Fbv,
5/4 would be C-D#^, and 8/5 would be C-Bbbv.

If you don't want to deal with the 5-limit, then parapyth as you
outlined it has a POTE tuning with fifths at 703.857 cents and spacing
at 58.338 cents, so all of this is very similar.

> > 2) take Gesualdo's Tristis est Anima Mea, which utilizes triadic
> > harmony in a much more impressionistic sounding way, and figure out
> > how to create the same effect with 4:6:7 and 14:21:27
>
> How about analogous effects -- that is one awesome piece! In
> addition to your suggested sonorities, of course, I would also
> feel free to use regular thirds, neutral intervals, thirdtone
> steps, everything I have -- this project takes nothing less!

There's also 6:9:11 too, which might be a nice counterpart to 4:6:7.
Maqamic and porcupine have the two both mapping to a type of
root-fifth-seventh, so that might be a place to start. There's also
mavila, for those who don't mind really flat fifths.

> > 3) take anything by Palestrina, which uses triadic harmony in
> > a very straightforward and tonal way, but make 4:6:7 and
> > 14:21:27 the basis of the tonality rather than 4:5:6 and
> > 10:12:15
>
> The real quibble I have here is that while 4:6:7 is relatively
> concordant, traditionally major sevenths are strong dissonances;
> but let's keep the spirit of this exercise by modifying it a tad.
> How about 4:6:7 as a relative concord, but 14:21:27 and related
> sonorities with 27/14 (or 28/27) as suspensions? After all,
> Palestrina does use major sevenths in suspensions! And George
> Secor, in his 17-puzzle article, shows how attractive and intense
> these suspensions can be!

Yes, that's probably right; and 14:21:27 is even stronger than 8:12:15
because the outer dyad starts to (to my ears) get into pseudo-octave
territory. But I think it would be a nice effect to somehow make that
chord sound "stable", despite its very harsh intonation; I think that
if you could pull that off, then you could execute various maneuvers
which sound like they're going to resolve to 4:6:7 and then sneak that
chord in instead, which would probably be a very strong emotional
effect.

> Getting a sense of classical balance like Palestrina's with these
> materials is a challenging -- and therefore great -- idea!

I guess I really just wish I had some sort of inkling about how to
generalize the usual voice-leading rules and rules of tonality to
other tunings like this (especially things like porcupine). Do you (or
does anyone) have any idea at all how to start? Or is it basically up
to us to tinker around for a bit until we piece together what the
different resources are?

> This is an opportunity for very productive cooperation between
> those of us who have been exploring these tunings over the last
> decade from a non-RMP perspective, and RMP people like yourself
> who would like to delve into them and maybe bring new insights.

I was talking about this in freenode #xenharmonic last night too. So I
guess my first question is, what are the things you're naming,
specifically? When we name temperaments, each individual thing that we
call a "temperament" which we assign a name to is a different way to
"map JI" onto some set of generators - or, equivalently, to map those
generators back to JI. To be more specific, these are what we call
"abstract temperaments," which we usually just call temperaments.

This, mathematically, is the same as tempering out a set of commas, so
you might also say that we name temperaments after the commas that
they temper out. Of course, just saying what commas are being tempered
out isn't all that helpful, which is why the first thing that we do
once we have the set of vanishing commas is to mathematically "unpack"
that by relating it back to some mapping from JI to a set of generator
chains. There are efficient mathematical ways to do this, but the
average person on here might not know what those are - so they just
see us talking about vanishing commas and wonder why anyone would care
so much about comma pumps. But the goal is to assign every interval in
the temperament some set of JI intervals that temper down to it.

But the key thing is, we classify all tunings of each temperament as
an instance of that temperament - so quarter-comma meantone,
third-comma meantone, POTE meantone, etc are all still part of
meantone temperament. So when you name things like "peppermint" and
"O3" and so on, are you naming temperaments in this sense - e.g.
mappings of JI intervals onto generator chains which admit a range of
valid tunings? Or are you naming specific tunings of these
temperaments? Or both?

Thanks,
Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> So I've finally caught up on everything, and one thing I notice right
> away is that your interval C-G#* is 890 cents, only about 5.4 cents
> sharp of a 5/3. If we want to call it 5/3, then we've now expanded to
> a full 13-limit rank-3 temperament, which also tempers out 196/195.
>
> This temperament apparently already has a name and it's called "Pele"
> temperament: http://x31eq.com/cgi-bin/rt.cgi?ets=58_41_87&limit=13.
> The POTE optimal tuning for this has the fifths at 703.414 cents and
> spacing at 57.468 cents, which leads to 14/11 set to 413.657 cents,
> 13/11 set to 289.756 cents, 5/4 set to 388.196, and 6/5 set to 315.218
> cents.
>
> As for how you get to these new 5-limit intervals, let's say that C^
> is one of your "spacing" commas sharp of C, and then that Cv is a
> spacing comma flat of it. So 5/3 would be C-G#^, 6/5 would be C-Fbv,
> 5/4 would be C-D#^, and 8/5 would be C-Bbbv.
>
> If you don't want to deal with the 5-limit, then parapyth as you
> outlined it has a POTE tuning with fifths at 703.857 cents and spacing
> at 58.338 cents, so all of this is very similar.

Here are some triple Fokker blocks you might be interested in. (A triple Fokker block is a special kind of Fokker block that's a Fokker block in three different senses, which implies that every generic interval comes in no more than three specific sizes. They are the closest things we have to moment of symmetry scales in rank 3.)

! parapyth12.scl
!
A triple Fokker block of the 2.3.7.11.13 temperament called "parapyth" (TOP tuning)
12
!
58.23604
206.95866
265.19471
413.91733
472.15337
554.50965
703.23227
761.46832
910.19094
968.42698
1117.14960
1199.50588

! parapyth17.scl
!
A triple Fokker block of the 2.3.7.11.13 temperament called "parapyth" (TOP tuning)
17
!
58.23604
124.60238
206.95866
265.19471
347.55099
413.91733
472.15337
554.50965
620.87599
703.23227
761.46832
843.82460
910.19094
968.42698
1050.78326
1117.14960
1199.50588

These don't actually contain any of the intervals that would approximate 5-odd-limit ratios in the "pele" temperament that Mike mentioned, so I won't give a version for TOP pele (it would be very similar anyway).

Keenan

--- In tuning@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:
> ! parapyth12.scl
> !
> A triple Fokker block of the 2.3.7.11.13 temperament called "parapyth" (TOP tuning)
> 12
> !
> 58.23604
> 206.95866
> 265.19471
> 413.91733
> 472.15337
> 554.50965
> 703.23227
> 761.46832
> 910.19094
> 968.42698
> 1117.14960
> 1199.50588

Just noting that it would be really sweet to tune a piano (harpsichord? organ?) to this because it has 12 roughly equally spaced notes, and 10 out of 12 fifths are really good, but it also has TWO really good 4:6:7:9:11 chords (on the root and the fifth of the mode shown).

The only way to get more than 10 really good fifths (11 instead of 10) would be to use a 12-note MOS of a temperament whose generator is a really good fifth, meaning Pythagorean[12] (or leapday[12] or whatever you want to call it). Compared to this parapyth-12 scale, that idea is pretty boring.

Keenan

> Using the wonders of mathematics, we can show that this is
> just the same as tempering out 352/351 and 364/363 in
> 2.3.7.11.13, which is rank 3. Of course, nothing stops you
> from, for example, adding 325/324 to the mix and now you have
> 5s.

Thank you for validating that MET-24 is a rank-3 temperament. I
was pretty confident of this, but am glad to get confirmation
from someone who knows the fine points of this theory.

>> This last one is equal to 896/861 plus 352/351, and I propose
>> that it be named the Secorian comma (or superkleisma?),

> I've added it to the Xenwiki comma list.

It may be fun to get George's take on this one; but if there's no
earlier name or precedent, this seems fitting. It's fun to look
back to the time when he got the generating "secor" of the
Miracle Tuning at around 116.7 cents named after him, with the
explanation (SECond minOR).

Best,

Margo

>> In order to give you some sporting chance to read and absorb
>> what I've posted (and the earlier rank-3 article might help),
>> why don't I leave for now with the lattice for MET-24, which may
>> give more clues on the commas tempered out or observed; and
>> another diagram showing some of the ones which are tempered out
>> as one looks at some of the ratios in the rank-3 system:

> So I've finally caught up on everything, and one thing I notice
> right away is that your interval C-G#* is 890 cents, only about 5.4
> cents sharp of a 5/3. If we want to call it 5/3, then we've now
> expanded to a full 13-limit rank-3 temperament, which also tempers
> out 196/195.

Dear Mike,

First, thanks for catching up with all that I've thrown at you in the
last couple of days, and for writing very generous and thoughtful
responses despite your other activities and commitments, including the
moderation of this group.

What I'd like to emphasize is that our dialogue may serve in part, as
Paul Erlich might call it, "A Gentle Introduction to Different
Temperament Theories and Tastes." You are a generous and creative
musician, and I hope that in this dialogue I may follow your example.

The kind of differences which may arise in "mapping" the world of
temperaments -- pun intended -- may be like the difference in
linguistics and anthropology between (phon)etic and (phon)emic
perspectives: what a linguist or anthropologist picks up in a language
or culture, and how a native speaker or informant perceives the
categories (e.g. sounds in a language, and distinctions that make a
difference in meaning). When artistic tastes get into the mix, things
can get marvellously complex, and I hope will stay remarkably civil.

And what I'll propose below is a Cooperative Mapping Project (CMP)
which would look at the pathways and landmarks of temperaments, both
in terms of primes or harmonics and in terms of melodic step sizes and
ratios, with people coming from various approaches noting the schemas,
pathways, and techniques that we find most important.

I'll also try to stumble toward a few possibly helpful concepts as to
tempering schema, effects or pathways, and landmarks.

* * *

Having said that, for MET-24 (2/1, 703.711, 57.422), the augmented
fifth or small neutral sixth plus spacing (C-G#*) is actually 887.109
cents, or 2.751 cents from 5/3, and I have no hesitation in calling it
a representation of 5/3, or a diminished fourth less spacing, C#*-F, a
representation of 6/5. In fact, I was delighted to find a nice version
of Ptolemy's Equable Diatonic, which I'll give in real-world 1024-EDO
with alternating linear generators of 703.125 and 704.297 cents:

J -0.637 -2.751 -2.342 -1.170 -0.533 -1.581 J
0 150.000 312.891 495.703 703.125 853.124 1016.016 1200
B* C# Eb E* F#* G# Bb B*
1/1 12/11 6/5 4/3 3/2 18/11 9/5 2/1
12:11 11:10 10:9 9:8 12:11 11:10 10:9
150.000 162.891 182.812 207.422 150.000 162.891 183.984
-0.637 -2.114 -0.409 +3.512 -0.637 -2.114 +1.581

However, this is the exception that proves the rule of 2.3.7.11.13.
Since I was optimizing for 11:10 as a large neutral second step going
back in this role to al-Farabi (870-950), a side-effect is some ratios
of 5. And in a way it's fitting: since the 14:13:12 and 13:12:11
divisions are at the heart of this tuning, why not include 12:11:10
for the sake of completeness, and as an honor to Ptolemy?

But a few remote instances do not a "5-limit tuning" make; and, given
human nature, I suspect that lots of people might give this priority
over all the other features. It may happen anyway, and of course
anyone is free to expand this so as to get more of these intervals.
You can also get a nice Bohlen-Pierce 3:5:7:9 in four locations, and
3:5:7:9:11 in two. There isn't a full 4:5:6:7:9:11:13, however, one
mark of a compete 13-limit system. Let's say that I'd rather have this
looked on as an "extra bonus."

In a MET-34 with two 17-MOS chains, "full 13-limit" would be more
practically accurate, and Turkish musicians, for example, would have a
perhaps reasonable number of 5/4 thirds plus all the neutral shadings.
So your observation is very astute!

But it's the accuracy of the superparticular neutral steps steps all
consistently within three cents of just (14:13, 13:12, 12:11, 11:10)
that is the point of the design, with an Equable Diatonic as a bonus.
And a tuning with the same variety of neutral steps, but which
happened to place them in between the superparticular ratios, might be
just as nice for maqam music, although I love honoring and following
the traditions of al-Farabi and Ibn Sina: the larger theme is a varied
palette of melodic colors. I'm not saying that all charming neutral
seconds are superparticular, but that these near-just sizes are
charming.

> This temperament apparently already has a name and it's called
> "Pele" temperament:

How about: "From an RMP perspective, your MET-24 is one example of a
temperament pathway called the `Pele' pathway." Then we could do
mutually edifying things like my learning the definition of the
pathway and what that 196/195 is all about, and both of us comparing
notes on your POTE criteria and my criteria for designing MET-24 as I
did, etc. [I'm now confident it's 49/39 vs. 5/4, but it's curious
I actually gave a 39:42:49:52 analysis below of a Hijaz tetrachord without
even thinking of what the comma might be; and looking up 196/195
in Scala didn't tell me much. I discuss this a bit more below -- and
I'm still eager to learn more about the 196/195!]

A CMP might be a kind of multidimensional map with harmonic and
melodic aspects -- like good counterpoint! -- here's the pathway to a
ratio of 5 or 7 or whatever at this point or region, and here's where
your 14/13 or 13/12 might come from if you're into that, etc.

So from one viewpoint, MET-24 might be within the Pele schema, or
maybe better yet, exhibit the "Pele effect" or "Pele pathway to 5."
Saying "MET-24 is an instance of Pele" might raise the issue of who is
defining the salient meaning of the region. But, "Hey, guess what,
MET-24 exhibits the Pele pathway" sounds at worst like an interesting
facet of the tuning, and at best like a high compliment.

> [59]http://x31eq.com/cgi-bin/rt.cgi?ets=58_41_87&limit=13. The
> POTE optimal tuning for this has the fifths at 703.414 cents and
> spacing at 57.468 cents, which leads to 14/11 set to 413.657 cents,
> 13/11 set to 289.756 cents, 5/4 set to 388.196, and 6/5 set to
> 315.218 cents.

Well, the fifth is almost identical to 29-EDO, and this is a perfect
spot for 22:26:33 (let's not have 14/11 rule everything!). Some
Persian musicians use a really small neutral second around 124 cents;
but MET-24 strives to keep 14/13 and 13/7 within three cents of just,
which means a bit more compromise of 3/2, 9/8, and 11/10.

I must admit the equal temperament notation looks complicated to me;
why not define the ways we get different intervals and commas tempered
out? In 2001 or thereabouts, I used two chains of 29-EDO at 59.97
cents spacing as one early instance of emulating HTT; it's a fine
spot, although I've tended to focus a bit north rather than south of
HTT.

The POTE instantiation of the Pele pathway, however, is _not_ an
optimization of 13/11 and 14/11 (in contrast to 33/26), since 13/11 is
still substantially wide of just, so a bit more temperament will bring
them both closer to pure. Beyond 703.597 cents, 13/11 becomes impure
in the narrow direction as 14/11 becomes less narrow, so we're in the
region of compromise as to these two. And Secor's HTT at 703.579 cents
is for all practical purposes identical with that landmark where our
strategy changes. These landmarks might involve an MOJ or "Moment of
Justness" (here 13/11), or maybe also an MOE or "Moment of Equality"
(e.g. 22/21 just, 3/2 and 14/11 equally impure).

Of course, the POTE is going for 5/4 and 6/5, and that makes the
strategy different, too. In MET-24, they're an unintended although not
untoward consequence, and play no role in the optimization. But that
makes the 196/195 relationship you mentioned no less fascinating in
either, and I'd love to have you explain it to me! Or maybe I just
found out in Scala, it's the difference between 49/39 and 5/4!

> As for how you get to these new 5-limit intervals, let's say that C^
> is one of your "spacing" commas sharp of C, and then that Cv is a
> spacing comma flat of it. So 5/3 would be C-G#^, 6/5 would be C-Fbv,
> 5/4 would be C-D#^, and 8/5 would be C-Bbbv.

Yes, in that sense the spacing comma, i.e. limma less spacing,
represents both 81/80 (or maybe 66/65 if the regular major third is
33/26, actually a tad closer than 14/11) and the 64/63 (or also 78/77,
e.g. 13/11 and 7/6). Note that in the Pepper Noble Fifth tuning, if a
chain is carried to a 29-MOS, say, the natural diesis is able to act
like the spacing here, and even more accurately: our near-5/4 is equal
to 21 fifths up, or thrice the apotome at 128.669 cents -- 386.007
cents! I guess this is Graham's diaschismic mapping for 5: a regular
major third minus the 17-note comma.

So MET-24 exhibits the spacing comma or Pele pathway, while rank-2
Pepper approximates 5 through the natural 17-comma.

My JI analysis of Bb-B-C#-Eb or 0-126-391-497 cents in 1024-EDO
(126-265-105 cents) would be 39:42:49:52 or 14:13-7:6-52:49
(128-267-103 cents). So, in a sense, it might be like 2-3-7-13
emulating 5! But it's hard not to think of Ptolemy's Equable Diatonic
as realized above as anything but a remote patch of 5-limit.
[P.S.: I've known for some months of the 49/39 approximation of 5/4,
but hadn't focused on the comma. But if you had said "196/195, for
example 49/39 vs. 5/4," I would have caught on immediately.]

For me, you see, the original motivation to look beyond a 12-MOS, and
specifically to 24, was above all 7 -- this was in 2000, before I had
become involved in maqam music. In the 12-MOS of this region, not only
the 14/11 and 13/11 but the submajor/supraminor thirds around 17/14
and 21/17 or whatever fascinated me for neomedieval European music:
sort of Pythagorean in the reverse direction, and likewise a bit
complex and unstable! But at that time, I wasn't yet aware of the
delights of four different neutral sizes; two were a delight in
themselves, but did give me an expectation that they would generally
be unequal to some degree.

The rank-2 path to 7 came first, and I discovered it quite by accident
in the e-based tuning I had invented at 704.607 cents, playfully
setting the ratio of whole tone and limma to Euler's e. That was in
June, but in October I got a big surprise.

Seeing what would happen in a 24-note chain of e-based, I suddenly
noted that 15 fifths up was a virtually just 7/4! And 2.65 cents was
far less temperament than in what is now called superpyth, or in
meantone, the two well-known routes to 7 other than Pythagorean or
Graham's minutely larger 135-EDO. Search for "e-based" on this list,
and you'll get posts from that era.

I did lots of rank-3 or its equivalent in Pythagorean JI, for example
(2/1, 3/2, 64/63) and delightfully (2/1, 3/2, 7/6), but it was George
and his 29-HTT in late 2001 that showed me the rank-3 path to 7 with a
generator of 704 cents or a bit less tempering.

So for 7, in a sense, my e-based path (15 fifths up, with the natural
diesis of 55.28 cents as the drastically narrowed 28/27) was like
Graham's diaschismic for 5; but the Secor HTT rank-3 pathway to 7,
with the spacing acting as the 28/27, is analogous to the Pele pathway
for getting 5.

Please forgive the history and details, but we may have found an
interesting analogy, and one possibly of parallel evolution from a
rank-2 path to a rank-3 path which spares us some of the tempering
necessary to get a natural 12-note diesis large enough (or 17-note
diesis small enough) to achieve the desired prime.

Maybe we are dealing with a tuning ecology filled with different phyla
of musicians and theorists where tools and concepts adapt for maximum
"fitness" in a given artistic niche or style. And Secor's
cross-pollination is a powerful evolutionary force.

But, as I see it, a pathway is something about which people can
objectively agree: the diaschismatic pathway or the Pele pathway to 5;
the e-based pathway or HTT pathway to 7, etc. Anything reasonably
using the meantone pathway is some kind of meantone family schema. But
schemas can be subjective: some pathways may matter more to some
people than others, but we might be able to sort out some common
schemas or combinations of pathways in a given region, some involving
harmonic and some melodic goals.

> If you don't want to deal with the 5-limit, then parapyth as you
> outlined it has a POTE tuning with fifths at 703.857 cents and
> spacing at 58.338 cents, so all of this is very similar.

That's the same as O3, with a just 22/21 as the defining feature. And I'd add that the POTE is just one possible optimization: musicians
are artistic people, and we can have attachments to different
intervals that can inspire a slight tweak. For example, MET-24 was
designed to make 11/10 a bit more accurate than in O3 or this
synonymous POTE, but 14/13 a bit more accurate than in HTT. It's sort
of a minimax solution.

A POTE can be of interest as another landmark, sort of like Farhat's
suggested tuning for Persian music arrived at by averaging out some
frettings on tars and setars. We read some other sources, by Persian
musicians or otherwise, including objective measurements, and learn
that "optimizations" vary. Still, Farhat's is tasteful and easy to
understand.

A POTE for meantone I might put in the same category as Woodhouse's
7/26-comma that Paul Erlich would often discuss: it's one possible
taste.

And your POTE for meantone is, as I guessed, so close to Kornerup's
Golden Meantone that Scala considered them equal -- 696.214 cents
(Kornerup) and 696.238 cents (POTE). I can see why this might be
considered optimal primewise, and admire some of the details like 13/8
in a 19-note version (not a ratio I usually look for in a meantone,
but delightful if there -- the neutral or "proximate minor" thirds of
1/4-comma are familiar to me).

If someone wants a pure 5/4, however, then 1/4-comma would be best for
as a starting point; the math can predict intervals and cents, but not
tastes.

What these share is the meantone pathway to 5, but not necessarily the
pathway in Kornerup/POTE to 13/8, which might be an essential part of
someone's schema, just as near-equal Vicentino fifthtone steps might
be an essential melodic pathway for another.

If you're tuning 12 notes, then the shading with pure 5/4 thirds
rather than the Vicentino pathway to enharmonic melodic steps is
likely to be part of your schema. But maybe you're thinking of
expanding to 24 or a full circle of 31, and love the enharmonic genus
in Vicentino's sense, so you might want your pure 5/4 thirds now and
maybe enharmonic steps later.

And a schema can also include circulation, as with Salinas, who
evidently agreed with Zarlino that 1/3-comma is not precisely optimal
in euphony -- but wow, it circulates in only 19 notes! That pathway to
circulation was evidently part of his schema.

The CMP seeks to take all of these into account in a mapping of the
continuum. It tries not to prioritize, but to reveal the pathways and
landmarks and let people decide which are most important in a given
region or subregion.

[On project to compose equivalent of
Gesualdo's _Tristis est Anima Mea_]

> There's also 6:9:11 too, which might be a nice counterpart to 4:6:7.
> Maqamic and porcupine have the two both mapping to a type of
> root-fifth-seventh, so that might be a place to start. There's also
> mavila, for those who don't mind really flat fifths.

Well, here's a piece in Peppermint where I used 7:9:11.

<http://www.bestII.com/~mschulter/O_Europae.mid>

[Here you raised the possibility of an analogue to Palestrina
style using 4:6:7 and 14:21:27 in place of 4:5:6 and 10:12:15.
I suggested the latter might be rather tense for this analogy.]

> Yes, that's probably right; and 14:21:27 is even stronger than
> 8:12:15 because the outer dyad starts to (to my ears) get into
> pseudo-octave territory. But I think it would be a nice effect to
> somehow make that chord sound "stable", despite its very harsh
> intonation; I think that if you could pull that off, then you could
> execute various maneuvers which sound like they're going to resolve
> to 4:6:7 and then sneak that chord in instead, which would probably
> be a very strong emotional effect.

Of course, I usually tend to regard 4:6:7 as rather like itself in a
style called "Californian discant" or _sesquisexta_ (literally, "and
again a sixth," i.e. the 7/6 ratio), where the voices move in parallel
4:6:7 or 12:14:18:21 chords, and eventually cadence to a fifth. I
should record this; but of course, it would just be one resource, sort
of like sweet moments of fauxbourdon in Wert or whoever. Kraig Grady
specifically favors parallel 4:6:7 chords, so that is "Gradian
discant," one subclass of the more general sesquisexta.

>> Getting a sense of classical balance like Palestrina's with these
>> materials is a challenging -- and therefore great -- idea!

> I guess I really just wish I had some sort of inkling about how to
> generalize the usual voice-leading rules and rules of tonality to
> other tunings like this (especially things like porcupine). Do you
> (or does anyone) have any idea at all how to start? Or is it
> basically up to us to tinker around for a bit until we piece
> together what the different resources are?

One point is that something like Peppermint or MET-24 or whatever,
from a European point of view, is like two manuals each in a
not-too-far variation on Pythagorean tuning, plus all those neat
septimal and neutral intervals. The voice-leading rules of the
13th-14th centuries more get extended than radically altered: a
neutral third, like a major or minor one, often contracts to a unison
or expands to a fifth. A neutral seventh often contracts to a fifth,
while a neutral sixth expands to an octave. Likewise with the
septimals.

But in something like Porcupine, I'd guess, you _are_ dealing with a
different world of basic voice-leading. Probably, as I've learned,
tinkering is the best way to learn! And there are new things that
emerge even in a tuning system like Secor's 17-WT or parapyth with a
traditional diatonic structure (emphasized by my two-manual
arrangement where anyone accustomed to a wolf at G#-Eb will have few
untoward surprises, at least if they like the tuning).

For example, when George Secor and I were exploring his 17-WT, we both
found that Renaissance or later suspensions took on a new color in
this temperament with its very small semitones often at 64 cents, very
close to 28/27; and I found myself mixing Renaissance-style
suspensions (e.g. 7-6) with some very rich septimal harmony like
12:14:18:21 as the resolution before a cadence to 3:2 or 2:3:4.

>> This is an opportunity for very productive cooperation between
>> those of us who have been exploring these tunings over the last
>> decade from a non-RMP perspective, and RMP people like yourself
>> who would like to delve into them and maybe bring new insights.

> I was talking about this in freenode #xenharmonic last night
> too. So I guess my first question is, what are the things you're
> naming, specifically? When we name temperaments, each individual
> thing that we call a "temperament" which we assign a name to is a
> different way to "map JI" onto some set of generators - or,
> equivalently, to map those generators back to JI.

To me a temperament is a tuning where either just ratios are
deliberately compromised, or a period is divided into an arbitrary
number of parts. Thus 1/4-comma meantone and 31-EDO are two distinct
but very closely neighboring temperaments of the first kind, as are
Secor's 29-HTT and my variations such as MET-24. Something like 13-EDO
is of the second type, where in a sense "existence precedes essence":
the fun is to figure out what ratios or other patterns the arbitrary
division might evoke or even newly create for us.

Here I think temperament "pathway" (an objective way to get some
prime or vertically or melodically relevant ratio or property like
circulation, etc) and temperament "schema" (the relevant pathways or
parameters sought be a given person or style at a given region of the
spectrum) are good choices.

The world, including me, generally still thinks of a "temperament" as
a specific tempered tuning, which may use the same pathways as others
without losing its individuality. And changing this may be like King
Canute commanding the tide to recede (he was precisely demonstrating,
of course, that it wouldn't)!

There might indeed be an "abstract temperament" of which a designer
conceives, that may favor some pathways, discount others, and be
crafted without knowledge of still others that the region offers.
Remember, pathways are objective, but schemas involve a sense of
artistic priorities, which definitely vary.

If I announce a temperament in the year 2000, and someone informs me
that it is actually the "Avant-Garde temperament" discovered in 2011
(whose cited example is discernibly distinct on the continuum -- or
not!), I might get a lot of practice in self-restraint, counting to
ten, and remembering "Newton and Leibniz!"

But if they say, "Hey, back in the year 2000, didn't you describe a
temperament somewhere around 704.6 cents. Well, that shows what we
call the `Avant-Garde pathway.'" Then we have a collegial discussion
about whether this is a new concept found in 2011, something I newly
described back in the year 2000 or so, or something that was well
known because someone had documented it earlier. And we arrive at a
mutually comfortable solution. Like George Secor with the Miracle
Tuning, I might find "Avant-Garde" great. And your telling me an
attribute my tuning has, not offering to name it for me when it
already has a name (although that pathway might not), promotes both
curiosity and cooperation.

Going against established usage can be a needlessly uphill battle,
like my attempt around 1983 to argue that 13th-century cadential
progressions should be regarded as a different form of "functional
harmony." Nice try, but "functional" means "tonal" to most people, not
"tonal or analogous thereto." But "directed verticality" succeeds.

Temperament design for me is something often rather intuitive, rather
like speaking a mothertongue, at least if it's a familiar region. For
example, with MET-24 in 1024-EDO:

(1) Find the solution with maximum evenness where both
14:13 and 11:10 are within 3 cents of just, which
is 600-601-600-601..... (703.125-704.297-703.125...),
indeed the most maximally even choice in the whole
region north of 703.125 and south of 704.297!

(2) Set the spacing so that 7/4 is always within a tuning
unit of just, if possible -- and it is -- 49 units,
or 57.422 cents (yielding either 967.969 or 969.141).

From lots of experience, I know that ratios like 12/11, 13/12, 13/8,
etc., will also do well -- confirmed when I see that 11:12:13 has
all ratios within a cent of just. So it's a done deal, and no other
solution would fulfill condition (1).

Of course I know that I'm tempering out the familiar 896:891, 352:351,
364:363, 10648:10647 -- and let's not forget 28672:28431 or Secor's
comma! Now HTT is precisely and by definition 1/9-Secorian, while here
the 26/21 submajor or large neutral third, _delicious_ for a high
Rast, is off by 0.5657 cents, as close as we come in 1024-EDO, so
let's call this a conceptual 1/8-Secorian comma temperament. And in
1/7-Secorian, we get a just 14/13 with 14/11 and 13/11 equally impure,
more or less synonymous with Peppermint.

Some pathways are mainly harmonic (3/2 about as good or a bit better
than 12-EDO; 13/8, 7/6, 7/4, and interestingly 21/16 near-just); some
are melodic (superparticular neutral steps, with the pathway to 11/10,
where applicable, setting a ceiling on the amount of temperament); and
some both (e.g. melodic 14/13 step and harmonic 13/7, e.g. in 7:11:13
or 7:9:11:13).

That's my schema or "abstract temperament," but your cardinal pathways
and personal schema might be a bit different, maybe with a Pele
pathway in the lower range of this region or a diaschismic pathway
around 704 cents as a main priority. My "just 14/13" may be your
"near-just diaschisma thirds." We can agree on the nature of the
pathways, with or without giving them familiar names, but differ on
the nature of our "abstract temperaments" or schema. This avoids lots
of territorial instincts which, coupled with different artistic and
theoretical tastes and the human tendency to argue over priority
(again, Newton and Leibniz), might not optimally serve our community.

> To be more specific, these are what we call "abstract temperaments,"
> which we usually just call temperaments.

How about temperament schemas, combining a number of pathways --
schemas limited by the available pathways, from which a given musician
may choose a relevant set to optimize and build a given schema. And in
a certain region or subregion, some schemas from an RMP perspective
might be proper or improper, so to speak, like the various properties
(or their absence) we get from SHOW DATA in Scala.

Identifying the pathways and their possible constellations or schemas
is a service to everyone, and everyone can be a part: "This is the
spot where 12:13:14 is just with this fifth and spacing!"

And some people may adopt schemas they are happy with which may not
neatly fit RMP methods and criteria, just as Scala often tells me
"Scale is not proper." The RMP approach as one option, but the
pathways as resources for everyone, and maybe a survey of alternative
schema for a given region or subregion, would be great! And RMP would
be part of the picture.

> This, mathematically, is the same as tempering out a set of commas,
> so you might also say that we name temperaments after the commas
> that they temper out. Of course, just saying what commas are being
> tempered out isn't all that helpful, which is why the first thing
> that we do once we have the set of vanishing commas is to
> mathematically "unpack" that by relating it back to some mapping
> from JI to a set of generator chains.

This sounds great if you're looking for new and unsuspected patterns,
rather like flying around with the best GPS and remote sensing
equipment, but maybe not the strategy a typical musician might use in
a relatively familiar territory, especially someone who "lives there."
I'm fascinated by commatic relationships I discover, or others
discover, in my tunings, or others I use, but tend to reason in terms
of intervals and cents, as simplistic as that may sometimes be.
Maybe it's because I've internalized lots of pathways; and, of course,
part of CMP would be an opportunity for "fluid speakers" in different
regions of the continuum to share their beloved pathways and schemas
or "abstract temperaments."

In such an "inhabited region," maybe it's like Star Trek and the
civilizations they encounter, who usually already have their own sense
of local geography and borders. They would likely be flattered if you
share new information on the astrophysical classification of their
sun, and would be delighted if you took information about local
placenames and landmarks and the histories behind them and made that
part of the Galactic Atlas.

But sometimes different schema names for different folks or purposes
can clarify the diverse uses of a single spot like 1/4-undecimal, to
me "a pure 14/11 eventone," and to others "Leapyear." The latter
schema name implies a 5-limit perspective, since it refers to the
29-MOS as the smallest set to support 5-limit. If we have "Leapyear"
for 29, why not "St. Patrick's" for 17 (St. Paddy's Day, March 17) and
"Lincoln" for 12 (Lincoln's Birthday, February 12)? Everyone can be
happy, and learn about the different schemas (schemata?) and
perspectives.

For example:

"Back in 2000, I tuned 1/4-undecimal and liked it; George Secor, of
course, as I didn't yet know, had similarly used it in 1978 for the
more remote part of his 17-WT, specifically with a pure 14/11 in mind.
But in the older literature, everyone discussing the almost identical
46-EDO seems to be after the 5/4 and 6/5 approximations: they'd be
happiest with the Leapyear schema."

It would also be good to define what pathways people are using, some
unsuspected. For example, around MET-24 or Peppermint, someone who
tunes an MOS of 17, or a set of 24 or 29 in rank-2 could point out
that _interseptimal_ ratios are wonderful for gamelan, one of the
failings of my 24-note schemas ("Where's the 15/13 or 22/19?"), but
not of rank-2! And that broad interseptimal pathway runs from around
29-EDO or a pure 13/11 all the way up to a pure 14/11, where fourth
less natural diesis is still more interseptimal (e.g. 22/17) than
really near 9/7.

Someone playing gamelan with a 704-cent generator should be able to
follow their own schema; and that interseptimal pathway should become
part of the CMP (e.g. markers for a just 15/13 or 26/15, not because
JI is part of traditional gamelan, but because there general sizes are
highly gamelan-compatible and signal landmarks along the continuum).

> There are efficient mathematical ways to do this, but the average
> person on here might not know what those are - so they just see us
> talking about vanishing commas and wonder why anyone would care so
> much about comma pumps. But the goal is to assign every interval in
> the temperament some set of JI intervals that temper down to it.

For your purposes, where you often play traditional major-minor tonal
progressions -- and this goes for 16th-century 5-limit progressions
also -- those pumps are a big concern! I wonder if and when I may have
hit one, at least of an unpleasant nature, in Peppermint or MET-24 or
the like with two, three, or four voices in some neomedieval style. So
far, it hasn't seems a problem. Possibly on these priorities, where
one stands depends on where one sits.

What you describe is like propriety or Myhill's property: people may
take it, leave it, or find it relevant here but not there. The CMP
should show the different options or criteria that people might or
might not adopt.

Suppose, for example, that we're in Lebanon, and have an instrument
with two 12-EDO keyboards at 50 cents apart (24-EDO), which someone
thought was what Arab musicians prefer in reality, as opposed to use
sometimes as a model for counting steps and categorizing intervals in
a generic way.

These keyboards are tunable, but people are so accustomed to 12-EDO
that changing the tuning within either 12-MOS would be a bad idea.

But we're lucky enough to have an option that keeps the relative
intervals on each keyboard, but changes the spacing. So we raise the
upper one by 5 cents -- with neutral seconds at 155-145, neutral
thirds at 355-345, etc. The audience picks up on the subtly unequal
neutral seconds, and is delighted. (We've read Amine Beyhom, not much
disposed to get involved with ratios at least in a modern context, but
very interested in subtly unequal steps as measured in cents.)

Do we need to know exactly ratio 355 cents or 345 cents was
representing (possibly 27/22 and 11/9, or maybe simply 355-ish and
345-ish shadings), or the exact commas tempered out, or whether this
is a true 24-note periodicity block?

I'd say not, just as we don't need to count the exact number of
molecules that made contact between the baseball and bat in order to
know that this was a home run.

And personally, if it's a choice between confirming that a temperament
I design intuitively fits someone's concept of a proper Fokker
periodicity block, or exploring some fine point of intonation in Maqam
Bayyati or Erv Wilson's Marwa Permutations that might give me a great
new idea, I'll take the latter. Indeed, such exploration may give me
new ideas for optimization which an abstract mathematical model might
not.

But to learn how a temperament I love either conforms to or departs
from a periodicity block would be fascinating. The CMP should include
a look at designs which fit this model and others outside it. In other
words, some people's schema are based on this comcept, and others
focus on other things -- but possibly conform anyway, or not. Comparing notes would be fun!

Also, how does one's instrument or setup -- e.g. 24 notes per octave
as a preference resulting from technical considerations -- affect
one's schemas, on the one hand favoring the extension of a 17-NOS by
seven extra notes providing extra comma steps; and on the other hand,
making circulation requiring a larger number of notes irrelevant, and
making intervals "remote" that would be abundant in 29 or 41 or 46 or
whatever. Likewise, a "wonderful system" with abundant neutral and
septimal (or interseptimal) intervals in 24 might be considerably less
opulent in 12 or even 17.

This last just really hit me! Either superpyth or meantone can provide
some septimal intervals in a 12-MOS (lots in superpyth!); but a rank-2
or rank-3 path in parapyth really calls for 24 or more, since 7/4 is
fifteenth generators (with or without help from rank-3 spacing!).

So focusing on things that happen within the first 12 notes might help
as a kind of guide to defining the most characteristic intervals and
pathways in a given region of subregion which will be parts of lots of
schemas. "Meantone" succeeds without controversy not only because it
is traditional, but because it sticks to the obvious and agreed:
approximating 5/4 and 6/5. People have diverse schema in different
parts of the region, but the pathway to 5 is a universally recognized
feature.

> But the key thing is, we classify all tunings of each temperament
> as an instance of that temperament - so quarter-comma meantone,
> third-comma meantone, POTE meantone, etc are all still part of
> meantone temperament.

With meantone, the boundaries of the genre seem fairly clear, at least
in an historical European context: say from 1/3-comma to 12-EDO. It's
the range where we can say that 5/4 and 6/5 are being optimized. But
there are lots of schemas: I'm not aware of anyone in the 16th century
who tuned a bit less than 2/7-comma for 13/8, although it's fun to
imagine Vicentino doing so -- but 45-28 cents might not have been an
ideal enharmonic division for him, while 41-35 at 1/4-comma is
perfect. And for someone with 12 notes per octave, all this is
academic, at least in their immediate situation, but their schema
might include the special charm of 2/7-comma, or what Mark Lindley
calls the "sprightly" melodic character of 1/5-comma.

So Kornerup's Golden Meantone (aka your POTE) is a sweet spot for 13;
but that's absent in 1/4-comma. Both are good for 7, but 1/5 comma or
so is great for 14/11 as an alternative third (here the diminished
fourth). A schema might be something including a path to 13 in
Kornerup; or 2/7-comma as sort of a limit for something suggesting 7/4
rather than clearly interseptimal ratio like 26/15. And 1/3-comma is
interseptimal par excellence.

Thus meantone might be the superschema, but we're going to want
subdivisions, based on traditional criteria (landmarks where just
5-limit intervals appear) and newer ones (various need higher primes).
Pathways come and go as we move through the region. If the idea is to
temper out commas so as to get this or that ratio (e.g. 13/8, 7/4,
15/13), then the meantone pathway describes a region of family of
schemas, but _not_ a single "abstract temperament" or "mapping" in any
practical sense.

And don't forget melody! A Vicentino-style archicembalo needs to be
very close to 1/4-comma or 31-EDO to get those enharmonic fifthtone
steps about equal (and both large enough). Scheidt in 1511 preferred
to have all his major thirds a bit wide, and Kornerup or a POTE likely
wouldn't change his mind.

> So when you name things like "peppermint" and "O3" and so on, are
> you naming temperaments in this sense - e.g. mappings of JI
> intervals onto generator chains which admit a range of valid
> tunings? Or are you naming specific tunings of these temperaments?
> Or both?

Always a specific tuning: Peppermint (2/1, 704.096, 58.060); O3 (2/1, 703.893, 57.148); MET-24 (2/1, 703.711, 57.422), etc.
Of course, there can be device-specific versions for 1024-EDO, say;
and O3 and MET-24 were designed in 1024-EDO, with some kind of more
even or "canonical" version with a single fifth size a later
refinement.

There is a common approach or technique in these tunings, but I wouldn't
try to give it a name that should apply to the region or other people's
systems, which might have quite different goals (like 5-limit around
703.7 cents with a large enough tuning size).

Basically: choose some interval you want to optimize, or maybe two you
want to optimize equally, in a 12-note chain in the region somewhere
between a pure 13/11 and a pure 14/11, since these are two we're
optimizing. And have 14/13 at least 125.3 cents (three cents low), and
somewhat short of the region where it becomes more of a 13/12 -- when
we reach the border, the game changes a bit. These rules aren't
inviolable, and they aren't meant as governing principles, only a
discretionary framework for one style or substyle of art.

Now we set our 12-note chain, and decide on some interval or intervals
as a gauge to set the spacing, which will represent 28/27 (narrow!),
33/32 (a bit wide), and 91/88 (usually within a cent or two of just).
Favorites are 7/6 (Peppermint) or 7/4 (O3 and MET-24 -- at in 29-HTT).

The only EDO that enters this process is the device resolution, for me
1024-EDO, indeed my favorite, (not to put down 49152-EDO either).

And then I'll run it through Scala and find out more precisely about
temperings for different intervals, how close I am to superparticular
neutral second divisions (although I was likely treating that as part
of my plan -- with 14:13, 13:12, 12:11, and often 11:10 as part of the
optimization).

And if it looks good, I'll actually set it on synthesizer.

But if someone comes along with a temperament at 703.65 cents which
they're doing in two 29-MOS sets optimized for 6/5, I wouldn't tell
them that they were using the "Turquoise temperament," or even the
"Turquoise schema," because I have a MET-24 subset called Turquoise
exemplifying some relevant (for me) commas and ratios. Same region, but
different agendas. But to me Turquoise is just a tuning subset I use,
not a placename for everyone who visits or inhabits the region to
adopt as a standard.

Of course, seeing how different people draw up schemas for a given
spot of region can be intriguing. But to paraphrase Admiral Grace
Hopper, the nice thing about schemas is that there are so many of them
from which to choose..

So I'd see RMP as one way of looking at things, and fascinating to
compare with other approaches. And the series of EDO's suggest a
series of mediants or the like (e.g. 104 + 17 = 121, essentially
Pepper's Noble Fifth). I notice that 104 comes up a lot, and in fact
it was a mention of 104 here that led me to design O3 (pure 22/21).
That gave me the idea of the region -- but from there, it was ratios
and cents, and figuring out a scheme to take 8/1024-octave, the best
approximation of 896/891, from each five-generator chain for the best
approximation of the limma.

And I should add that for lots of people, comma pumps are something
very relevant, so it's commendable that you've been focusing on
this. It may be a way both to spare people a lot of frustration, and
to invent wonderful new musics, as some of your group's publications
have made admirably clear.

All this should be part of the CMP, with the RMP as one important
part.

> Thanks,
> Mike

With many thanks,

Margo

--- In tuning@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:

> Here are some triple Fokker blocks you might be interested in.

I'm busily adding to the Xenwiki, but wonder if you could give transversals for these.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, "Keenan Pepper" <keenanpepper@> wrote:
>
> > Here are some triple Fokker blocks you might be interested in.
>
> I'm busily adding to the Xenwiki, but wonder if you could give transversals for these.

! parapyth12trans.scl
!
A JI transversal of parapyth17.scl for use in calculations. If you temper out 352/351 and 364/363 it becomes parapyth17.
12
!
28/27
9/8
7/6
14/11
21/16
11/8
3/2
14/9
22/13
7/4
21/11
2/1

! parapyth17trans.scl
!
A JI transversal of parapyth17.scl for use in calculations. If you temper out 352/351 and 364/363 it becomes parapyth17.
17
!
28/27
14/13
9/8
7/6
11/9
14/11
21/16
11/8
56/39
3/2
14/9
13/8
22/13
7/4
11/6
21/11
2/1

Hello, Keenan and all.

You've asked for some tempered versions of your Fokker block for
parapyth17, and this request provides an occasion both to show
some of the diverse shadings available over the years, and to
honor some colleagues who have shared with me in this
exploration, not the least of them being yourself as the
contributor of a generator which among the systems here surveyed
was until recently the one that best approximates 14/13: the
Pepper Noble Fifth, of course, at 704.096 cents!

</tuning/topicId_12592.html#12592>

I say "until recently," because Jake Freivald has since developed
a "Cantonpenta" tuning where 14/13 may be intended to be just.
Since a just 14/13 is an important landmark, I've decided,
prompted by your system, Jake, to post an optimization based on a
14_13-mint tuning which you are welcome to claim as a 24-note
form of Cantonpenta if this fits your intentions as the designer.

Please let me request that this entire post be included for any
Xenwiki page on parapyth, because it's important that people
should know some of the history. I believe that John Chalmers
suggested the term "para-Pythagorean" to me around early 1998, in
reference to tunings such as 29-EDO or 46-EDO.

For a definition of para-Pythagorean as a region, I would suggest
that it begins around 29-EDO, where augmented and diminished
intervals from 6-10 generators no longer represent ratios of 5
(the schismatic or schismic pathway around Pythagorean), but now
represent small and large neutral or supraminor and submajor
intervals (e.g. 14/13, 63/52, 26/21, 56/39, 21/13, 13/7); and it
ends at 17-EDO, beyond which the 64:63 comma is tempered out is
we move into the domain of the septimal or superpyth region where
regular major and minor thirds represent 9/7 and 7/6.

As a temperament plan or schema, parapyth uses three generators
to approximate ratios of 2.3.7.11.13. These generators are the
period, in these examples the 2/1 octave; the linear generator,
typically a fifth between around 703.5 and 704.1 cents; and the
spacing between two linear chains, often at around 57-59 cents.
The spacing interval acts as a 28/27 (e.g. 9/8-7/6); a 33/32
(e.g. 4/3-11/8); and a 91/88 (e.g. 22/13-7/4).

Commas tempered out include 896/891 (81/64 vs. 14/11, 33/32
vs. 28/27); 352/351 (13/11 vs. 32/27; 39/32 vs. 11/9); 364/363
(33/28 vs. 13/11); and 10648/10647, the tiny difference between
352/351 and 364/363 (273/242 vs. 44/39). Additionally the
28672/28431, equal to 896/891 plus 352/351, is also tempered out
(19683/16384 vs. 63/52).

All of the following are tempered versions of Keenan Pepper's
triple Fokker block; see his tuning list (#105142, 10/24/2012)
</tuning/topicId_105023.html#105142>.

! parapyth17trans.scl
!
A JI transversal of parapyth17.scl for use in calculations.
! If you temper out 352/351 and 364/363 it becomes parapyth17.
17
!
28/27
14/13
9/8
7/6
11/9
14/11
21/16
11/8
56/39
3/2
14/9
13/8
22/13
7/4
11/6
21/11
2/1

Interestingly, this just tuning is similar in its rotation on
Scala step 3 (9/8) to Manuel Op de Coul's pipedum_17c.scl in the
Scala scale archive, which also may be reasonably emulated in the
tunings that follow.

Here we proceed in the direction of increasing temperament of the
fifth, thus appropriately beginning with an explorer of this
region of the spectrum first in precedence, first in versatility
through the years and decades, and first in the esteem of his
many colleagues: George Secor. In 1978, his 29-note High
Tolerance Temperament (29-HTT, secor29htt.scl) included as one of
its subsets a parapyth realization of primes 2.3.7.11.13 -- part
of a much larger scheme supporting primes 2-3-5-7-11-13 and odd
factors of 2-3-5-7-9-11-13-15, providing complete ogdads of
near-just quality at 4:5:6:7:9:11:13:15.

This example of parapyth17 comes from another member of Secor's
HTT family, 41-HTT (secor41htt.scl). In the HTT family, the
period is the 2/1 and the fifth is at 703.579 cents or
(504/13)^(1/9), or a tempering by (28672/28431)^(1/9), or 1.642
cents, so as to obtain a characteristic just ratio of 63/52
(332.208 cents) for the small neutral third. The 13/11 minor
third (289.210 cents) is also virtually just at 289.264 cents.
The generator for a just 13/11 would be 703.597 cents, or
(352/351)^(1/3), to which Secor's fifth is almost identical.

In this 2.3.7.11.13 subset of Secor's HTT, the spacing of the two
chains is at 58.090 cents, almost identical to 91/88 (58.036
cents), to obtain pure 7/4 minor sevenths.

<http://www.bestII.com/~mschulter/secor41htt-parapyth17.scl>

! secor41htt-parapyth17.scl
!
George Secor's 41-HTT, parapyth17 for Fokker block, key 14 like pipedum17_c.scl
17
!
58.08980
125.05086
207.15739
265.24719
347.35372
414.31478
472.40458
554.51111
621.47217
703.57870
761.66850
843.77503
910.73608
968.82588
1050.93241
1117.89347
2/1

We next move to two almost identical tunings created by different
methods, but agreeing on an optimization that preserves many of
the charms of Secor's mild temperament while slightly improving
the most frequent forms of 13/8, 11/6, and 13/7. These
temperaments at around 703.7 cents were independently developed
by Graham Breed and myself. Here I can tell the story of MET-24,
the "Milder Extended Temperament" (by comparison to earlier
shadings I had used such as O3 and Peppermint below, as opposed
to Secor's HTT). With some inspiration from Jacques Dudon and his
JI ideal, I designed this temperament in July, 2011.

My idea was to seek the temperament with maximum evenness in
1024-EDO that would keep both 14/13 and 11/10 within three cents
of just, while approaching HTT's mild degree of tempering for
3/2, 4/3, and 9/8. The perfect and indeed only solution was to
alternate fifths of 703.125 and 704.297 cents (600 and 601 steps
of 1024), rather in the manner of some Neidhardt temperaments,
with an average fifth size of 703.711 cents and a spacing of
57.422 cents (49 steps of 1024) so as to obtain 7/4 minor
sevenths always within a cent of just at 967.969 or 969.141
cents, thus emulating the pure 7/4 of HTT.

I have been delighted with this tuning, and will now post my
MET-24 version in parapyth17 of Keenan Pepper's Fokker block:

<http://www.bestII.com/~mschulter/met24-parapyth17-fokker_g.scl>

! met24-parapyth17-fokker_g.scl
!
MET-24 parapyth for Keenan Pepper's JI block; key 14 like pipedum_17c.scl
17
!
57.42187
125.39062
207.42187
264.84375
345.70312
414.84375
472.26562
553.12500
622.26562
703.12499
760.54687
842.57812
910.54687
967.96875
1050.00000
1117.96875
2/1

Last weekend, I set out to develop a "canonical" version, which I
actually posted to the list. However, as I will explain shortly,
I got a hint yesterday that MET-24 is meant to stay a very subtly
irregular system. Why don't I explain first how I arrived at my
canonical version but had mixed feelings about the result; and
then, thanks to a link supplied by our moderator, Mike Battaglia,
learned of something quite amazing.

For my "canonical" version, I first calculated in Scala the
precise linear generator to make 14/13 and 11/10 equally impure:
703.723 cents (as compared with the average of MET-24's two
alternating generators, 703.711 cents). Then, having noted that
in the 1024-EDO version 7/6 and 13/8 are almost equally impure
(2.027 cents narrow and 2.050 cents wide respectively), I set the
"canonical" spacing so as to make these two precisely equal in
their difference from just. As it happened, the required spacing
was almost identical to that of MET-24 in 1024-EDO: 57.4225 cents
in the canonical version, as compared to 57.421875 cents.

I had mixed feelings about this, because I noted that the 7/4 was
now impure at 968.592 cents, or 0.234 cents narrow. Totally
trivial, and yet somehow not in keeping with my concept as
realized in 1024-EDO that 7/4 should be "as pure as possible"!
For the sake of theoretical completeness, I did post the file
here (which I have since changed on my site to a 2048-EDO version
for those who would like a single size of fifth at 1201 steps of
that temperament or 703.711 cents).

</tuning/topicId_105023.html#105106>

However, there is a wonderful connection of which I was unaware
until yesterday, and learned of only when Mike Battaglia pointed
me to the website of Graham Breed:

</tuning/topicId_105023.html#105109>

One of the files I found at Graham's site and downloaded was for
a temperament in various sizes -- I picked the 17-note version --
described as based on "46 & 17 & 41" and involving three
generators. Since I am accustomed in the regular diatonic
portion of the spectrum (7-EDO to 5-EDO) to think of the fifth as
the generator, and the Scala file on the site had the near-32/21
fifth placed above the 1/1, I wasn't sure of the temperament,
although I realize in retrospect that I might have checked the
fourth also!

Having downloaded it, however, I loaded it in Scala, did a
rotation -- and found something _identical_ to my attempt at a
"canonical" version of MET-24! I mean the two Scala files were
literally identical. Whatever methods Graham had used within the
Regular Mapping Paradigm, he and I had reached precisely the same
result. And that rendered my "canonical version" of MET-24 quite
superfluous: Graham Breed, using the special techniques of the
Regular Mapping Paradigm (evidently involving 46-EDO, 17-EDO,
and 41-EDO), had already arrived at that tuning by another route!

And as I found today, a rotation of Graham's "46 & 17 & 41" set
beautifully realizes Keenan Pepper's Fokker block. While our two
solutions posted here are _almost identical_, his uses a single
size of fifth while mine above in 1024-EDO keeps its subtly
irregular scheme:

<http://www.bestII.com/~mschulter/breed-46-17-41-parapyth17.scl>

! breed-46-17-41-parapyth17.scl
!
Graham Breed 17-note parapyth (key 14 like pipedum_17c.scl)
17
!
57.42250
126.06252
207.44643
264.86893
346.25284
414.89287
472.31537
553.69928
622.33931
703.72322
761.14571
842.52963
911.16965
968.59215
1049.97606
1118.61609
2/1

Next we come to parapyth in the O3 temperament which I designed
in 2010, the name standing for "Ozan, Ottoman, Optimize" and
referring to the Turkish musician and scholar Ozan Yarman. My
goal was to get 22/21 just, or consistently at its best
approximation in 1024-EDO of 69 steps (80.859 cents; just size
80.537 cents), and to obtain a near-just 26/21, a large neutral
or submajor third in historical Ottoman music. The generators, as
in MET-24, are the 2/1 octave; fifths at 703.125 and 704.297
cents; and a spacing between chains of 57.422 cents to set all
7/4 minor sevenths within a cent of just. Here is parapyth17 in
this shading:

<http://www.bestII.com/~mschulter/o3-parapyth17_g.scl>

! o3-parapyth17_g.scl
!
O3 parapyth17 for Fokker block, key 14 like pipedum_17c.scl
17
!
57.42188
127.73437
208.59375
266.01563
346.87501
416.01563
473.43751
554.29688
623.43750
704.29688
761.71876
842.57813
911.71875
969.14063
1050.00000
1119.14063
2/1

Before moving to Peppermint, a system now a decade old, I should
honor Jake Freivald as the designer of Cantonpenta, which appears
to be another independent discovery of this region and
effectively takes the Phi-based definition of Pepper's Noble
Fifth tuning and applies minutely less temperament to make the
14/13 small neutral step just. Whlle I'm not sure if he intends
this ratio to be precisely just, I'll propose a version of our
parapyth17 in this temperament here, and invite Jake to decide if
this should be considered a form of his Cantonpenta. If so, Jake,
it's yours! As in Peppermint, 7/6 is set just. This means
generators of (2/1, 704.043, 58.786).

Note that here the linear generator is (224/13)^(1/7), with each
fifth wide by (28672/28431)^1/7, about 2.088 cents, to make the
apotome or chromatic semitone a just 14/13 -- and likewise the
diminished octave or large neutral seventh a just 13/7, making
possible, for example, some near-just 7:11:13 sonorities. As
George Secor might delight to point out, we also get two
near-just 7:9:11:13 sonorities on the 475-cent and 971-cent steps
(our approximate 21/16 and 7/4 steps):

<http://www.bestII.com/~mschulter/14_13-parapyth17.scl>

! 14_13-parapyth17.scl
!
Interpretation of Jake Freivald's Cantonpenta with just 14/13 ! key 14 like pipedum_17c.scl
17
!
58.78569
128.29824
208.08521
266.87091
346.65787
416.17042
474.95612
554.74309
624.25563
704.04260
762.82830
842.61527
912.12782
970.91351
1050.70048
1120.21303
2/1

Last but not least, we have perhaps one of the first parapyth
temperaments to be designed intentionally to support primes
2.3.7.11.13, in contrast to Secor's 29-HTT of 1978 and the
family of HTT tunings into which he has since expanded this
germinal landmark of our art. The catalyst and indeed in a real
sense the co-author of our final tuning system is Keenan Pepper,
who in September of 2000 proposed his Noble Fifth tuning with a
linear generator of 704.096 cents, thus producing a ratio of Phi
between the logarithmic ratios of the major second and the
chromatic semitone (at 208.191 and 128.669 cents). His purpose
was to find the counterpart in this part of the spectrum to
Thorvald Kornerup's famous Golden Meantone, where the major
second and diatonic semitone have logarithmic sizes at this same
ratio of Phi (696.215 cents), kornerup.scl in the Scala archive.

As Kraig Grady observed on the tuning list shortly thereafter,
this was another example of the independent discoveries and
rediscoveries that often occur: Erv Wilson included the Noble
Fifth tuning, with its distinctive ratio of Phi between the sizes
of whole tone and diatonic semitone, in his Scale Tree, and as
one of his Golden Horograms. This Wilson, with his many
contributions to theory, tunings, and instrument design, also has
a part in the story of Pepper's Noble Fifth tuning.

Waxing very enthusiastic about this shade of temperament on the
list, I decided in 2002 -- having the previous year learned from
George Secor himself about 29-HTT -- to design a 24-note tuning
with two 12-note chains of Pepper's tuning at a spacing of 58.680
cents for just 7/6 minor thirds (2/1, 704.096, 58.680). I loved
it, and appropriately named it Peppermint.

Just as medieval and Renaissance European music often feature a
technique of "successive composition" where one person writes a
piece, and then later either that musician or another adds more
voices, so your tuning, Keenan, was my cantus firmus or theme, to
which I simply added another chain, rather like taking a
Gregorian chant and adding a second voice in parallel fourths or
fifths.

Here is Keenan Pepper's Fokker block in Peppermint's version of
parapyth17:

<http://www.bestII.com/~mschulter/peppermint-parapyth17_g.scl>

! peppermint-parapyth17_g.scl
!
Peppermint parapyth17 from Fokker block; key 14 like pipedum_17c.scl
17
!
58.67969
128.66924
208.19121
266.87091
346.39287
416.38243
475.06212
554.58409
624.57364
704.09561
762.77530
842.29727
912.28682
970.96651
1050.48848
1120.47803
2/1

These are just some of the possible optimizations possible, and
we will likely soon see others. But the saga of parapyth,
involving as it does such artists and scholars as George Secor,
Erv Wilson, Graham Breed, Jake Freivald, and Keenan Pepper, is a
pleasure to participate in as someone who for the past 12 years
has had a special passion for this intriguing portion the
intonational continuum.

My our explorations, and fruitful cooperation, continue.

With much appreciation,

Margo Schulter
24 October 2012

Margo,

I've been underwater, so to speak, for the past week or so and have just started trying to read and absorb the posts here regarding temperaments that are centered on 3/2, 14/11, and 13/11. I'll probably have a lot of questions eventually -- I'm relatively new to microtonality, having only started looking into it about two years ago. That's why my original "Three related temperaments" post covered such familiar territory: It was a newbie explaining obvious things to other newbies. :)

I should note that I didn't name the pentacircle comma, the just Canton scale, or the pentacircle-tempered Cantonpenta scale. I believe all three are due to Gene Smith. I did explore them a bit, though, which led me to understand the value of 14/11 as a major third, even though it's even sharper than the Pythagorean major third, and the 13/11 minor third, which I really like.

Gene derived Canton as a just scale and then tempered it, I believe -- I don't think he iterated a generator, as you've been doing here and as I did in my original post. But maybe there's a subtlety there that would tie the two together, which eludes me.

Anyway, I've found the discussion very interesting, especially since you appear to have a different approach and different goals from many of the others on the list, but since I'm still very busy, I may yet be a few days before I can ask my questions.

Thanks for your generosity with your time.

Regards,
Jake

Sent from my Verizon Wireless BlackBerry

--- In tuning@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:

> Here are some triple Fokker blocks you might be interested in. (A triple Fokker block is a special kind of Fokker block that's a Fokker block in three different senses, which implies that every generic interval comes in no more than three specific sizes. They are the closest things we have to moment of symmetry scales in rank 3.)

It would be nice to have precise definitions of "3-distributionally even" and "triple Fokker black".

--- In tuning@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:

> > I'm busily adding to the Xenwiki, but wonder if you could give transversals for these.
>
> ! parapyth12trans.scl

I converted these to 2.3.7 transversals, and verified their status as Fokker blocks. Scala says the 12-note scale is "3-distributional even", and I don't know what that means, but apparently it's stronger than the trivalence property which is three kinds of interval in each interval class. Keenan gave some conditions on what the step-size structure must look like for a triple Fokker block, and I'm wondering if that's the start of the algorithm for constructing them. What is the algorithm?

! parapyth12-7.scl
!
2.3.7 transversal of parapyth12
! 3-distributional even SLSLSMLSLSLM
! <<1 -2 -6||: -6 to 5; <<1 10 13||: 0 to 11; <<2 8 7||: 0 to 10 even
12
!
28/27
9/8
7/6
81/64
21/16
112/81
3/2
14/9
27/16
7/4
243/128
2/1
! parapyth17-7.scl
!
2.3.7 transversal of parapyth17
! Scale has trivalence property MSLMLSMLSLMLSMLSL
! <<1 -2 -6||: -9 to 7; <<2 13 15||: -1 to 15; <<1 15 21||: 0 to 16
17
!
28/27
2187/2048
9/8
7/6
896/729
81/64
21/16
112/81
729/512
3/2
14/9
3584/2187
27/16
7/41
448/243
243/128
2/1

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
> I converted these to 2.3.7 transversals, and verified their status as Fokker blocks. Scala says the 12-note scale is "3-distributional even", and I don't know what that means, but apparently it's stronger than the trivalence property which is three kinds of interval in each interval class.

No, it's no stronger than that trivalence property. Actually, it's slightly weaker, because "3-distributional even" means *no more than* three kinds of interval in each interval class (so scales with fractional-octave periods are allowed). That's all it means.

> Keenan gave some conditions on what the step-size structure must look like for a triple Fokker block, and I'm wondering if that's the start of the algorithm for constructing them. What is the algorithm?

I haven't been gung-ho enough to work out an actual algorithm yet. I know some properties they must have, which narrows it down, but getting these scales still involved some fiddling around and trial and error. The most important property is that (with one sporadic exception) they must consist of only two chains of some interval; in other words there is some lattice direction such that if you project along that direction the image of the scale is only two points.

Margo, this really ties in to your article about rank-3 scales, because although in general a rank-3 scale can be an arbitrarily large 2D patch of lattice, these triple Fokker blocks (almost) always consist of two chains of a generator separated by a spacing.

(The only kind of exception, which is a triple Fokker block but not two chains of generators, is the 7-note scale with step pattern "aabacab", or the 14-note scale with step pattern "aabacabaabacab", and so on. You need three chains of generators to construct this, not two.)

Keenan

> ! parapyth12.scl
> !
> A triple Fokker block of the 2.3.7.11.13 temperament called "parapyth" (TOP tuning)
> 12
> !

That's a really cool scale. It has the variety that so often seems
only to be in larger scales, but is very compact (I haven't figured
out microtones in Lilypond, so I'm limited to 12 tones); it has plenty
of fourths and fifths that I can use as a crutch if I need them;
although it's a little bit improper, it's not so much as, say,
Wuerschmidt[10]; and it has a lot of great sounds in it. Thanks for
posting it, Keenan.

Regards,
Jake

Thanks to Dan from sonic couture I have a kontakt script that can do the 12 one easily on their replica harpsichords. I'm excited to try this a little later today after seeing Jake's exclamation in his message. Jake it sounds like you think this is pretty special.

Chris
*

-----Original Message-----
From: "Keenan Pepper" <keenanpepper@gmail.com>
Sender: tuning@yahoogroups.com
Date: Wed, 24 Oct 2012 01:58:14
To: <tuning@yahoogroups.com>
Reply-To: tuning@yahoogroups.com
Subject: [tuning] Re: Three similar temperaments

--- In tuning@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:
> ! parapyth12.scl
> !
> A triple Fokker block of the 2.3.7.11.13 temperament called "parapyth" (TOP tuning)
> 12
> !
> 58.23604
> 206.95866
> 265.19471
> 413.91733
> 472.15337
> 554.50965
> 703.23227
> 761.46832
> 910.19094
> 968.42698
> 1117.14960
> 1199.50588

Just noting that it would be really sweet to tune a piano (harpsichord? organ?) to this because it has 12 roughly equally spaced notes, and 10 out of 12 fifths are really good, but it also has TWO really good 4:6:7:9:11 chords (on the root and the fifth of the mode shown).

The only way to get more than 10 really good fifths (11 instead of 10) would be to use a 12-note MOS of a temperament whose generator is a really good fifth, meaning Pythagorean[12] (or leapday[12] or whatever you want to call it). Compared to this parapyth-12 scale, that idea is pretty boring.

Keenan

Hi Margo, this reply I'm writing here is going to be too long for any
human being to ever read, but heck with it...

On Wed, Oct 24, 2012 at 7:40 AM, Margo Schulter <mschulter@...> wrote:
>
> What I'd like to emphasize is that our dialogue may serve in part, as
> Paul Erlich might call it, "A Gentle Introduction to Different
> Temperament Theories and Tastes." You are a generous and creative
> musician, and I hope that in this dialogue I may follow your example.
//snip
> When artistic tastes get into the mix, things
> can get marvellously complex, and I hope will stay remarkably civil.

You're too kind, and I see no reason for anything to get uncivil :)

> However, this is the exception that proves the rule of 2.3.7.11.13.
> Since I was optimizing for 11:10 as a large neutral second step going
> back in this role to al-Farabi (870-950), a side-effect is some ratios
> of 5. And in a way it's fitting: since the 14:13:12 and 13:12:11
> divisions are at the heart of this tuning, why not include 12:11:10
> for the sake of completeness, and as an honor to Ptolemy?

What are you mapping 11/10 to, specifically? If it works out to be the
same as mapping 5/3 to C-G#*, then that means that you've mapped 5 in
a regular way, and I'm curious if it all still matches the Pele
mapping in the end.

> But a few remote instances do not a "5-limit tuning" make; and, given
> human nature, I suspect that lots of people might give this priority
> over all the other features. It may happen anyway, and of course
> anyone is free to expand this so as to get more of these intervals.
> You can also get a nice Bohlen-Pierce 3:5:7:9 in four locations, and
> 3:5:7:9:11 in two. There isn't a full 4:5:6:7:9:11:13, however, one
> mark of a compete 13-limit system. Let's say that I'd rather have this
> looked on as an "extra bonus."

Well, not in the 24-note scale you presented, but there's no reason
why you couldn't theoretically continue the pattern of generators
forever, right? For instance, you can create a hypothetically infinite
(octave equivalent) lattice in which one axis is the slightly sharp
fifth and the other axis is the ~50 cent "spacing" interval, and then
your scale traces out a 2x12 rectangle on that lattice. But I see you
go into more detail about this below, so I'll respond more there. For
now, suffice to say that I find it useful to separate scale generation
from the temperament itself, so that I'd look at your MET-24 as a I've
been thinking of your scale as a generated subset of the underlying
temperament, the mathematical structure of which is an infinite
lattice. From this viewpoint, Keenan's "triple Fokker block" parapyth
scale that he just posted would draw from the same underlying parapyth
temperament, but would utilize a different scale within it (this time,
rather than tracing out a rectangle, it traces out a parallelogram).
It's literally the same thing as how meantone has a 5-note MOS, a
7-note MOS, a 12-note MOS, a 19-note MOS, etc - none of these are
meantone temperament in its entirety, but rather different scales
which can be generated from it. Likewise, parapyth doesn't have MOS's
in the usual sense, but has a whole host of higher-dimensional
generalizations to it. One can cover the lattice by a regular tiling
of hexagons, parallelograms, rectangles, all of which produce scales
that are candidates for higher-dimensional MOS (you can think of a MOS
as a one-dimensional rectangle/parallelogram/hexagon/all-in-one tiling
a one-dimensional lattice).

So indeed, as you point out, 34 notes gives you more 5-limit
intervals, which I view as simply an implication of the underlying
structure of the temperament mapping.

> But it's the accuracy of the superparticular neutral steps steps all
> consistently within three cents of just (14:13, 13:12, 12:11, 11:10)
> that is the point of the design, with an Equable Diatonic as a bonus.
> And a tuning with the same variety of neutral steps, but which
> happened to place them in between the superparticular ratios, might be
> just as nice for maqam music, although I love honoring and following
> the traditions of al-Farabi and Ibn Sina: the larger theme is a varied
> palette of melodic colors. I'm not saying that all charming neutral
> seconds are superparticular, but that these near-just sizes are
> charming.

Sure, nothing wrong with those sorts of tuning optimizations; I'd be
interested in figuring out if you can outline some general rules for
the sorts of things you like when you want to optimize tunings, so
that we can come up with algorithmic methods for computing
"Margo-optimal" tunings for any arbitrary temperament or scale.

> > This temperament apparently already has a name and it's called
> > "Pele" temperament:
>
> How about: "From an RMP perspective, your MET-24 is one example of a
> temperament pathway called the `Pele' pathway." Then we could do
> mutually edifying things like my learning the definition of the
> pathway and what that 196/195 is all about, and both of us comparing
> notes on your POTE criteria and my criteria for designing MET-24 as I
> did, etc. [I'm now confident it's 49/39 vs. 5/4, but it's curious
> I actually gave a 39:42:49:52 analysis below of a Hijaz tetrachord without
> even thinking of what the comma might be; and looking up 196/195
> in Scala didn't tell me much. I discuss this a bit more below -- and
> I'm still eager to learn more about the 196/195!]

We can call it a "temperament pathway" if you'd like. The argument
over the definition of "temperament" has apparently been raging for
years now, predating my joining this list, and people sometimes
disagree on whether 1/4-comma meantone and 1/3-comma meantone are two
different temperaments, or "instances of the same temperament," or
"different tunings of the same temperament family" or whatever. I'm of
the opinion that disagreements over naming can sometimes cause more
problems than they should, and that at this point it's most useful to
simply note what different people call things so we can all
communicate.

So let's, for now, say that don't you want to call that 890 cent
interval 5/3, making this a 2.3.7.11.13 subgroup temperament that
you've named "Parapyth temperament." So then, specifically, the
relationship between your MET-24 scale and parapyth temperament is
exactly the same as the relationship between the quarter-comma
meantone chromatic scale and the overarching meantone temperament. The
quarter-comma meantone scale is a specific scale formed from a
specific tuning of the abstract meantone temperament, and MET-24 is a
specific scale formed from a specific tuning of the abstract parapyth
temperament.

There a few different ways to state this relationship:

- MET-24 is a 24-note scale in a specific tuning of "Parapyth
temperament" (general common usage)
- MET-24 is a 24-note scale in a specific tuning of "the Parapyth
abstract regular temperament" (using Gene's abstract regular
temperament terminology)
- MET-24 is a 24-note scale in a specific temperament within "the
Parapyth temperament family" (Paul's usage)
- MET-24 is a 24-note scale in a specific temperament that belongs to
"the Parapyth temperament class" (Graham's usage)
- MET-24 is a 24-note scale in a specific temperament exhibiting to
"the Parapyth temperament pathway" (what you just proposed)

People have often argued over which of these is "right"; at this point
I'll just call it whatever anyone wants! (If you want to incorporate a
mapping for ratios of 5, then you can replace "Parapyth" with Pele
above.)

> http://x31eq.com/cgi-bin/rt.cgi?ets=58_41_87&limit=13
>
> I must admit the equal temperament notation looks complicated to me;
> why not define the ways we get different intervals and commas tempered
> out?

The "Equal Temperament Mappings" thing is complicated and not very
useful; it's the "Reduced Mapping" that's useful. Each row represents
a different generator, and each column of the matrix represents a
different prime, and this tells you how many of each generator you
need to add to represent a certain prime. But if you have a more
clever way in mind of specifying the temperament, how would you do it?
What's a good systematic way to display, for some arbitrary
temperament, where the relevant tempered commas "turn up" that's
easily readable by a human being?

> The POTE instantiation of the Pele pathway, however, is _not_ an
> optimization of 13/11 and 14/11 (in contrast to 33/26), since 13/11 is
> still substantially wide of just, so a bit more temperament will bring
> them both closer to pure.

Right, this optimizes each interval in order of its complexity
(specified by n*d for some ratio n/d). You can think of this as
optimizing the average tuning of -all- intervals over the entire JI
lattice, with the most weight is placed on really simple intervals
like 3/2, 5/4, and so on. 14/11 and 13/11 have medium complexity and
as such get medium weight, whereas something like 32805/32768 gets
barely no weight assigned to it at all, because it's so complex that
it gets minimal attention. That's one way to think of it, anyway.

If you'd prefer to weight intervals differently, which ones would you
say are most important? Is there some mathematical pattern inherent in
the intervals you'd like to optimize, so we can apply this method of
optimization to all temperaments?

> Of course, the POTE is going for 5/4 and 6/5, and that makes the
> strategy different, too. In MET-24, they're an unintended although not
> untoward consequence, and play no role in the optimization. But that
> makes the 196/195 relationship you mentioned no less fascinating in
> either, and I'd love to have you explain it to me! Or maybe I just
> found out in Scala, it's the difference between 49/39 and 5/4!

I'm not sure how 196/195 turns up, specifically; there's actually an
infinite series of commas tempered out, and 196/195 is apparently one
of them. I was just mentioning the simplest interval that you'd have
to add to the set of vanishing commas which would transform Parapyth
into Pele. You could add 441/440 instead and get the same thing, or
847/845. It was just a technical note.

If we want to not include 5/4 and 6/5 in the optimization, we can just
go back to looking at Parapyth and skip this noise about Pele. So POTE
Parapyth has the fifth at 703.857 cents and the spacing interval at
58.338 cents. This tunes 14/11 to 415.425 cents and 13/11 to 288.431
cents. The general "weighted average of all intervals" optimization
mentioned before still applies, except now we're excluding ratios
where 5 is involved.

> > As for how you get to these new 5-limit intervals, let's say that C^
> > is one of your "spacing" commas sharp of C, and then that Cv is a
> > spacing comma flat of it. So 5/3 would be C-G#^, 6/5 would be C-Fbv,
> > 5/4 would be C-D#^, and 8/5 would be C-Bbbv.
>
> Yes, in that sense the spacing comma, i.e. limma less spacing,
> represents both 81/80 (or maybe 66/65 if the regular major third is
> 33/26, actually a tad closer than 14/11) and the 64/63 (or also 78/77,
> e.g. 13/11 and 7/6).

Looks right to me; limma minus spacing represents 81/80, 66/65, 64/63,
and 78/77, by my calculations.

> Note that in the Pepper Noble Fifth tuning, if a
> chain is carried to a 29-MOS, say, the natural diesis is able to act
> like the spacing here, and even more accurately: our near-5/4 is equal
> to 21 fifths up, or thrice the apotome at 128.669 cents -- 386.007
> cents! I guess this is Graham's diaschismic mapping for 5: a regular
> major third minus the 17-note comma.

Hmm, if C-C### is mapped to 5/4, that corresponds to a 5-limit
temperament tempering out 10737418240/10460353203. See here:
http://x31eq.com/cgi-bin/rt.cgi?ets=46_29&limit=5

Looks like the 13-limit version of this is called "Leapday"
temperament: http://x31eq.com/cgi-bin/rt.cgi?ets=46_29&limit=13

What limit is the Pepper Noble Fifth tuning in?

> But, as I see it, a pathway is something about which people can
> objectively agree: the diaschismatic pathway or the Pele pathway to 5;
> the e-based pathway or HTT pathway to 7, etc. Anything reasonably
> using the meantone pathway is some kind of meantone family schema. But
> schemas can be subjective: some pathways may matter more to some
> people than others, but we might be able to sort out some common
> schemas or combinations of pathways in a given region, some involving
> harmonic and some melodic goals.

Sorry, I'm confused now - what's the end specification of the e-based
pathway, and the HTT pathway? Which limits are these temperaments in,
and how are the various primes mapped? If you posted something back in
2000 or so that we've assigned a different name to, then you get
precedence, but I'm still not sure which of HTT, e-based, MET, O3,
parapyth, etc you consider to be generalized mappings, or very
specific tunings, or specific tunings of specific scales, etc. It
seems like some of these are different names for different tunings of
the same underlying temperament or temperament "pathway", whereas
others are names for different "pathways" themselves.

> And your POTE for meantone is, as I guessed, so close to Kornerup's
> Golden Meantone that Scala considered them equal -- 696.214 cents
> (Kornerup) and 696.238 cents (POTE).

The fact that this is the case is, I think, one of the great
coincidences that makes meantone as powerful as it is. Roughly
speaking, golden tunings tend to be optimal for a certain type of
melodic optimization, one in which all the intervals appearing in the
MOS series are as far apart from one another as possible, and for
meantone this happens to coincide with the tuning where the intervals
are, on average, as close to JI as possible.

For instance, consider an extreme tuning of the diatonic scale where
the half steps are only 10 cents, and the whole steps are 236 cents.
It's obviously going to be difficult to tell, in the context of a
rapid-fire melodic line, the difference between when someone's played
a major third and a perfect fourth, or a major second and a minor
third, and so on - the difference between these intervals is only 10
cents! Every pair of intervals differing by a limma will become
"ambiguous" in this way as the size of the limma shrinks to nothing.

Now consider, instead, a tuning where the half steps are 165 cents and
the whole steps are 174 cents. Now we have the opposite situation:
intervals differing by a diatonic semitone are easy to tell apart, but
intervals differing by a chromatic semitone, like the major vs minor
third, or major vs minor second, etc, are going to be very difficult
to tell apart in context, because the chroma is now only 9 cents. Now
it's every pair of intervals differing by a chroma that becomes
ambiguous, and that's a lot of pairs.

(Side note: try doing this with the half steps set to 150 cents and
the whole steps set to 180 cents, and marvel in the fact that these
150 cent half steps are still somehow "half steps" and not "neutral
seconds". That blows my mind.)

So I think we can both agree that these two situation create a certain
type of aural confusion that isn't present in any of the more familiar
tunings of the scale, such as 12-EDO, or third/fourth/fifth/sixth
comma meantone, and so on. These are extreme cases, and as the tuning
tends more towards these cases, this behavior becomes more prevalent.
What we really want is a tuning where pairs of intervals are as far
apart from one another as humanly possible, where intervals are
weighted by how often they appear in the scale.

Keenan Pepper and I are working on something to measure this property
of scales called "categorical entropy," so named because it
(hopefully) measures how difficult it is for a listener to come up
with a categorical perception for that scale. And one nice rule of
thumb for finding scales that are low in categorical entropy is to
tune it to a nearby "golden" tuning, for precisely the same reason
that your work with Dave Keenan on metastable intervals tends to zero
in on intervals that are local maxima of harmonic entropy. The results
disagree with categorical entropy a bit, for instance, which places
the melodically optimal tuning of the diatonic scale closer to
1/6-comma or 12-EDO (with some models having a second "optimal" region
around 17-EDO), but it's still a good rule of thumb. We want the most
"jagged" scales possible, because that's a good thing when it comes to
differentiating between intervals, so whereas metastable intervals can
lead to very discordant harmony, using metastable generators is
precisely what the doctor ordered.

So long story short, this means that the meantone tuning which is
melodically and categorically optimal is very close to the meantone
tuning which is harmonically optimal. There are a few other
temperaments which exhibit this effect, like machine temperament and
its "Schulter hexatonic scale" in 11-EDO, but some of the best tunings
DON'T have this effect. For instance, mohajira doesn't, and it has a
sort of nice cloudy effect that stands in opposition to the crystal
clarity of the Schulter hexatonic, diatonic, etc scales (to my ears
maqam music exploits this cloudiness very well). Porcupine's another
one which is more hazy and "pops out" less.

> The CMP seeks to take all of these into account in a mapping of the
> continuum. It tries not to prioritize, but to reveal the pathways and
> landmarks and let people decide which are most important in a given
> region or subregion.

Wait, what are you calling a pathway here? I thought a temperament
pathway was an abstract temperament, but I guess not... Does pathway
just mean a mapping?

> > There's also 6:9:11 too, which might be a nice counterpart to 4:6:7.
> > Maqamic and porcupine have the two both mapping to a type of
> > root-fifth-seventh, so that might be a place to start. There's also
> > mavila, for those who don't mind really flat fifths.
>
> Well, here's a piece in Peppermint where I used 7:9:11.
>
> <http://www.bestII.com/~mschulter/O_Europae.mid>

OK, is it the A-C#-F that's the 7:9:11, about 3/5 of the way into the
piece? That was interesting when I first heard it; it was so
acoustically strong that it destroyed my context of the tonality and
sounded just like a raw sonic object (like a car horn or something),
and I had to go back and listen over it again to be able to "frame"
the scale in the proper context. I think if I were an expert
Peppermint listener this wouldn't have happened though; I'd know how
to frame it correctly the first time around. That's a really
interesting effect; I'd like to explore more music which manages to
"break through" that interpretational layer occasionally.

> > Yes, that's probably right; and 14:21:27 is even stronger than
> > 8:12:15 because the outer dyad starts to (to my ears) get into
> > pseudo-octave territory. But I think it would be a nice effect to
> > somehow make that chord sound "stable", despite its very harsh
> > intonation; I think that if you could pull that off, then you could
> > execute various maneuvers which sound like they're going to resolve
> > to 4:6:7 and then sneak that chord in instead, which would probably
> > be a very strong emotional effect.
>
> Of course, I usually tend to regard 4:6:7 as rather like itself in a
> style called "Californian discant" or _sesquisexta_ (literally, "and
> again a sixth," i.e. the 7/6 ratio), where the voices move in parallel
> 4:6:7 or 12:14:18:21 chords, and eventually cadence to a fifth. I
> should record this; but of course, it would just be one resource, sort
> of like sweet moments of fauxbourdon in Wert or whoever. Kraig Grady
> specifically favors parallel 4:6:7 chords, so that is "Gradian
> discant," one subclass of the more general sesquisexta.

This is exactly what I've been imagining! This is like fauxbourdon but
with 7/4, you mean? That sounds like a good place to start. Do you
have a recording of this?

> > I guess I really just wish I had some sort of inkling about how to
> > generalize the usual voice-leading rules and rules of tonality to
> > other tunings like this (especially things like porcupine). Do you
> > (or does anyone) have any idea at all how to start? Or is it
> > basically up to us to tinker around for a bit until we piece
> > together what the different resources are?
>
> One point is that something like Peppermint or MET-24 or whatever,
> from a European point of view, is like two manuals each in a
> not-too-far variation on Pythagorean tuning, plus all those neat
> septimal and neutral intervals. The voice-leading rules of the
> 13th-14th centuries more get extended than radically altered: a
> neutral third, like a major or minor one, often contracts to a unison
> or expands to a fifth. A neutral seventh often contracts to a fifth,
> while a neutral sixth expands to an octave. Likewise with the
> septimals.

When you say the neutral third expands to a fifth, do you mean like
D-F^ -> C-G, where ^ is a half-sharp? And would a neutral seventh
contracting to a fifth be like Bv,-A -> C-G?

> But in something like Porcupine, I'd guess, you _are_ dealing with a
> different world of basic voice-leading. Probably, as I've learned,
> tinkering is the best way to learn! And there are new things that
> emerge even in a tuning system like Secor's 17-WT or parapyth with a
> traditional diatonic structure (emphasized by my two-manual
> arrangement where anyone accustomed to a wolf at G#-Eb will have few
> untoward surprises, at least if they like the tuning).

I guess I just don't understand, for instance, why certain rules are
what they are. Why can't I have parallel fifths? Is it because the
fifth is the generator, or because it's 3/2, or both? Is the correct
generalization "you can't have parallel generators" or "you can't have
parallel low-complexity intervals" or...? I still seem to think in
12-mode when I write stuff in porcupine; it's a challenge to snap my
brain out of that sometimes...

> To me a temperament is a tuning where either just ratios are
> deliberately compromised, or a period is divided into an arbitrary
> number of parts. Thus 1/4-comma meantone and 31-EDO are two distinct
> but very closely neighboring temperaments of the first kind, as are
> Secor's 29-HTT and my variations such as MET-24. Something like 13-EDO
> is of the second type, where in a sense "existence precedes essence":
> the fun is to figure out what ratios or other patterns the arbitrary
> division might evoke or even newly create for us.

I note that 13-EDO makes a lot of sense as a 2.7/6.11.13 temperament,
with perhaps 9 and 5 thrown in there if you don't mind a bit more
error. There's been a lot of useful work done in making harmonic use
of tunings like 11-EDO and 13-EDO. 11-EDO is a real gem; it has
4:7:9:11 chords which are about as accurate as 12 has 4:5:6, and you
can use this to write quite "tonal" sounding music. I don't think
there's an EDO that exists that one couldn't make legit harmonic use
of at this point - whatever you can think of, there's definitely
something which works.

> Here I think temperament "pathway" (an objective way to get some
> prime or vertically or melodically relevant ratio or property like
> circulation, etc) and temperament "schema" (the relevant pathways or
> parameters sought be a given person or style at a given region of the
> spectrum) are good choices.

Is a temperament pathway your name for what I called an "abstract
temperament?" What would be an example of a temperament schema? Is
porcupine a temperament schema or a pathway?

> The world, including me, generally still thinks of a "temperament" as
> a specific tempered tuning, which may use the same pathways as others
> without losing its individuality. And changing this may be like King
> Canute commanding the tide to recede (he was precisely demonstrating,
> of course, that it wouldn't)!

If "abstract temperament" is a bit too sterile, there's always Paul's
"temperament family" to make things clearer.

> But if they say, "Hey, back in the year 2000, didn't you describe a
> temperament somewhere around 704.6 cents. Well, that shows what we
> call the `Avant-Garde pathway.'" Then we have a collegial discussion
> about whether this is a new concept found in 2011, something I newly
> described back in the year 2000 or so, or something that was well
> known because someone had documented it earlier. And we arrive at a
> mutually comfortable solution. Like George Secor with the Miracle
> Tuning, I might find "Avant-Garde" great. And your telling me an
> attribute my tuning has, not offering to name it for me when it
> already has a name (although that pathway might not), promotes both
> curiosity and cooperation.

I didn't realize I was overwriting your established name for an older
temperament. But my understanding is that we aren't naming the same
sorts of objects; the things we're trying to assign names to are these
birds-eye view "temperament families" or "abstract temperaments" or
"temperament classes" or whatever the name du jour is, which group
together an entire family of scales and things that you just call
"temperaments" and so on. It seems like the things you're naming are
specific tunings of those families, but not temperament families
themselves.

The way we've been operating is, if someone discovered it first, they
get precedence in terms of naming. So if you think that Pele should
actually be named MET and want to claim precedence that's fine, but I
thought you wanted the abstract temperament underlying MET to be
called "parapyth" and said it was a 2.3.7.11.13 temperament? I'm
confused where the conflict is.

Which specific temperament families are you saying that you have had
existing names for that we've just recently rediscovered and renamed?
I don't think anyone would mind updating a name in an instance where
there's historical precedence, but my understanding was that it seems
like we're just naming different sorts of things - there are some
things you've named which I thought are names for specific useful
tunings of the superpyth family, I thought (like Peppermint and so
on). So if a temperament of yours implies an entirely new temperament
family (as was the case with parapyth) then it's easy to name the
family after your temperament, but if you have two names for two
temperaments that fit into the same family (e.g. same mapping, commas,
etc, but a different tuning), which gets the family name in that case?

> How about temperament schemas, combining a number of pathways --
> schemas limited by the available pathways, from which a given musician
> may choose a relevant set to optimize and build a given schema. And in
> a certain region or subregion, some schemas from an RMP perspective
> might be proper or improper, so to speak, like the various properties
> (or their absence) we get from SHOW DATA in Scala.

Is a schema just an identifying feature of a temperament? So we have a
pathway, a schema, and a mapping as three characteristics of a
temperament (or is it temperament family?)?

> This sounds great if you're looking for new and unsuspected patterns,
> rather like flying around with the best GPS and remote sensing
> equipment, but maybe not the strategy a typical musician might use in
> a relatively familiar territory, especially someone who "lives there."
> I'm fascinated by commatic relationships I discover, or others
> discover, in my tunings, or others I use, but tend to reason in terms
> of intervals and cents, as simplistic as that may sometimes be.
> Maybe it's because I've internalized lots of pathways; and, of course,
> part of CMP would be an opportunity for "fluid speakers" in different
> regions of the continuum to share their beloved pathways and schemas
> or "abstract temperaments."

I think it would be useful if we had some sort of algorithmic way to
tell a musician what the "meaning" of these commas being tempered out
is. The mapping matrix is a useful way of sort-of-kind-of-ehh-maybe
doing this, but a systematic way to point out "hey, these are the
intervals getting equated" would be useful. We just haven't figured
out how to do it yet, short of taking the time to play and analyze
every single temperament and working out some basic insights of its
structure. For instance, I know lots of "practical" information about
porcupine temperament, and what it "means" for 250/243 to be tempered
out (it means three minor thirds = two fourths, among other things),
but I've never played Pele before so I didn't have much insight to
share on it.

This is, I guess, the curse of having a mathematical framework that's
able to yield information about an infinite number of temperaments,
but not an infinite amount of time to unpack all of that information.
What's really needed is some systematic way of yielding useful
information about what a temperament "means" to a musician, but I'm
not sure how to best do that. If you have any insights in this regard,
maybe we can generalize them to a systematic approach for "unpacking"
any temperament.

> But sometimes different schema names for different folks or purposes
> can clarify the diverse uses of a single spot like 1/4-undecimal, to
> me "a pure 14/11 eventone," and to others "Leapyear." The latter
> schema name implies a 5-limit perspective, since it refers to the
> 29-MOS as the smallest set to support 5-limit. If we have "Leapyear"
> for 29, why not "St. Patrick's" for 17 (St. Paddy's Day, March 17) and
> "Lincoln" for 12 (Lincoln's Birthday, February 12)? Everyone can be
> happy, and learn about the different schemas (schemata?) and
> perspectives.

I have no problem with other ways to classify musical tuning systems.
RMP has just proven to be the most conceptually simple for me, but at
this point, actually, I'd really like to see some others - especially
ones that deal with melody as the basis for music with harmony as a
secondary consideration, which is the opposite of how RMP does things.
But I think leapday isn't just 29-EDO itself, it's the name of a
temperament family supported by 29-EDO and 46-EDO, much like meantone
is the name of a temperament family supported by 19-EDO and 31-EDO,
etc...

> What you describe is like propriety or Myhill's property: people may
> take it, leave it, or find it relevant here but not there. The CMP
> should show the different options or criteria that people might or
> might not adopt.

I agree. I actually think that in general, to make an even more
extreme case, ratios themselves are things that people sometimes don't
find relevant, as Igs was just mentioning. It would be nice to come up
with some sort of framework that left them out, perhaps, and came up
with a framework for scales and interval categorization instead,
regardless of which ratio we're assuming the intervals represent.

> Do we need to know exactly ratio 355 cents or 345 cents was
> representing (possibly 27/22 and 11/9, or maybe simply 355-ish and
> 345-ish shadings), or the exact commas tempered out, or whether this
> is a true 24-note periodicity block?
>
> I'd say not, just as we don't need to count the exact number of
> molecules that made contact between the baseball and bat in order to
> know that this was a home run.

I agree!

> And personally, if it's a choice between confirming that a temperament
> I design intuitively fits someone's concept of a proper Fokker
> periodicity block, or exploring some fine point of intonation in Maqam
> Bayyati or Erv Wilson's Marwa Permutations that might give me a great
> new idea, I'll take the latter. Indeed, such exploration may give me
> new ideas for optimization which an abstract mathematical model might
> not.

Fair enough, but to spin it the other way, I gain more information
from learning how these regular temperaments interrelate than anything
else at this point. Maybe that's just me.

> But to learn how a temperament I love either conforms to or departs
> from a periodicity block would be fascinating. The CMP should include
> a look at designs which fit this model and others outside it. In other
> words, some people's schema are based on this comcept, and others
> focus on other things -- but possibly conform anyway, or not.
> Comparing notes would be fun!

Well, this brings up the point from before - the things we're calling
temperaments aren't individual scales, so periodicity blocks can exist
-within- them, but they're not blocks themselves... So there's three
things here

1) A specific scale with a specific tuning and mapping
2) An infinite lattice generated by a few generators with a specific
tuning and mapping
3) An infinite lattice generated by a few generators -without- a
specific tuning, but still with a specific mapping

I think some of the communication breakdown might be that you use the
word "temperament" to refer to #1 and #2, whereas we toss it around
for #3 (when maybe we should say "temperament family" instead, which
is probably nicer than "abstract temperament" at this point).

> So focusing on things that happen within the first 12 notes might help
> as a kind of guide to defining the most characteristic intervals and
> pathways in a given region of subregion which will be parts of lots of
> schemas. "Meantone" succeeds without controversy not only because it
> is traditional, but because it sticks to the obvious and agreed:
> approximating 5/4 and 6/5. People have diverse schema in different
> parts of the region, but the pathway to 5 is a universally recognized
> feature.

Well, sometimes 12-EDO listeners seem to think that the 5/4 of 31-EDO
is a bit on the flat side...

> With meantone, the boundaries of the genre seem fairly clear, at least
> in an historical European context: say from 1/3-comma to 12-EDO.

There's 26-EDO too, which I have to say I find a bit charming, though
definitely not optimal, though it has roughly the same amount of error
as 12-EDO.

> It's the range where we can say that 5/4 and 6/5 are being optimized. But
> there are lots of schemas: I'm not aware of anyone in the 16th century
> who tuned a bit less than 2/7-comma for 13/8, although it's fun to
> imagine Vicentino doing so -- but 45-28 cents might not have been an
> ideal enharmonic division for him, while 41-35 at 1/4-comma is
> perfect. And for someone with 12 notes per octave, all this is
> academic, at least in their immediate situation, but their schema
> might include the special charm of 2/7-comma, or what Mark Lindley
> calls the "sprightly" melodic character of 1/5-comma.

I'm sorry, I'm very lost on this schema thing... I'll probably have to
go through this again...

> > So when you name things like "peppermint" and "O3" and so on, are
> > you naming temperaments in this sense - e.g. mappings of JI
> > intervals onto generator chains which admit a range of valid
> > tunings? Or are you naming specific tunings of these temperaments?
> > Or both?
>
> Always a specific tuning: Peppermint (2/1, 704.096, 58.060);
> O3 (2/1, 703.893, 57.148); MET-24 (2/1, 703.711, 57.422), etc.
> Of course, there can be device-specific versions for 1024-EDO, say;
> and O3 and MET-24 were designed in 1024-EDO, with some kind of more
> even or "canonical" version with a single fifth size a later
> refinement.

OK, so the idea is that Peppermint, O3, MET, etc all fit into the
parapyth temperament family, right?

> But if someone comes along with a temperament at 703.65 cents which
> they're doing in two 29-MOS sets optimized for 6/5, I wouldn't tell
> them that they were using the "Turquoise temperament," or even the
> "Turquoise schema," because I have a MET-24 subset called Turquoise
> exemplifying some relevant (for me) commas and ratios. Same region, but
> different agendas. But to me Turquoise is just a tuning subset I use,
> not a placename for everyone who visits or inhabits the region to
> adopt as a standard.

That's what I've been trying to say - we're not naming what you'd call
specific temperaments, but rather overarching temperament families. So
if someone wants to play 1/3-comma meantone, we wouldn't say "oh,
that's 1/4-comma meantone" just because it's similar. But we might say
"this is part of the meantone family." Of course, anyone is able to
reject that classification if they don't like it, but I find it
useful.

I think that sums up my response, but in short - RMP isn't the
end-all-be-all of music. RMP is a tool that gives us an immense amount
of mathematical information about which temperament families are
useful. That information can be unpacked and decoded and made musical
sense of by a musician, and that's what I think is the fun part -
playing around in porcupine and mavila and figuring out "what it all
means." And there's so many temperament families that I'm almost never
let down, and we've still barely figured out rank-3 tuning systems
yet. But the result is, we have far more mathematical information than
we've managed to unpack so far. Some day I hope that we have tomes
written about lots of these temperament families and what it "means"
that comma xyz vanishes - but in the meantime, we just have some
pointers that have been divined out of nowhere by mathematics telling
us where to look, and it's up to us to do the looking and make sense
of it.

Part of the making sense of it is figuring out which tunings of the
various temperament families you might like. For instance, RMP says
nothing about whether one listener is going to prefer 1/4-comma
superpyth or 1/3-comma superpyth, other than to say which one has more
overall average weighted error. But what if you don't mind having more
error on the 3/2 because you really want a pure 9/7? Then that's for
each musician to figure out for his/her own personal enrichment, using
the theory to find out which temperament families might be musically
useful and delving into them. For instance, I've found that my
favorite porcupine tuning isn't exactly the harmonically optimal one,
but one which is a bit close to what I'd consider melodically optimal,
with that 7L1s MOS having a slightly bigger "s" step than optimal
would suggest.

I wasn't around when these names were given, but it seems like at
times you're frustrated at having found some great temperament, only
for us to have come up with a different name for it 10 years later and
not attribute it to you, and then tell you that your temperament is
"just" our renamed version of the same thing minus a specific tuning.
I could see how that would be frustrating. I do think it would be good
to go back and rename things if you've named some useful temperaments
and we've run over them with our own names because we didn't know you
had them. And if you come up with a good temperament and we can
extrapolate from that back to an entire abstract temperament family,
there's no reason to not name the family after your specific
temperament (which is what just happened with parapyth).

But I think the things you name aren't the same as the things we name;
from a pure RMP perspective many of your temperaments are simply
different tunings of superpyth[17] or leapday[17] something like that.
That obviously loses a lot of the information that you put into making
these scales, such as which intervals are optimized, so you may find
it a bit of a coarse approach that loses something that you were
after. But I don't think there's any real conflict between these two
approaches; as you continue to name specific useful tunings, there's
no harm in saying that one of these tunings happens to also be a
specific tuning of a more generic temperament family, which also
admits these other scales, and other tunings as well, etc. It's simply
a way of trying to place your work along a spectrum of mathematically
similar works and connect it to other ideas, so we can continue to
build the big picture of this infinite-dimensional musical space we're
trying to get our heads wrapped around. Nobody's forced to accept this
version of "the big picture" if they don't want to, but it's at least
useful to note such connections when they occur.

What I think is a good goal for would be to take some of your ideas,
find out what the underlying musical motivation is, and automate them,
so we can apply them in general to any temperament family we want. For
instance, your work places a heavy emphasis on specific tuning, and
not just the mapping. If I had a better idea of the sorts of tuning
optimizations you like, then I could perhaps come up with a
"Schulter-optimal tuning" which we could apply to arbitrary
temperaments such as porcupine, machine, etc, instead of always using
TOP or POTE or so on. I think that would be a good accomplishment.

Thanks, and if anyone reads this whole thing it'll be a miracle,
Mike

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:
> [Gene Ward Smith (#105118) wrote:]
> > Using the wonders of mathematics, we can show that this is
> > just the same as tempering out 352/351 and 364/363 in
> > 2.3.7.11.13, which is rank 3. Of course, nothing stops you
> > from, for example, adding 325/324 to the mix and now you have
> > 5s.

I devised my high-tolerance temperament (HTT) in the spring of 1975. (This was around the same time as my article in Xenharmonikon 3 appeared, describing the temperament that would later be named "Miracle".) My original objective was to map a set of tones in 13-limit otonal JI to a 29- or 31-tone octave as one of the permanently programmed (i.e., hard-wired) tunings in my generalized-keyboard Scalatron. I found that a 29 mapping offered more possibilities, since tones differing by a 5-comma (80:81) can be mapped to separate keys, but with JI I could map only three 15-limit ogdoads, totaling 18 tones. I found that if I tempered out 351:352, I could have 11's and 13's in three more keys with very little error, and likewise the 7's in each of those keys, by tempering out 363:364 with <3.2 cents error for all 15-limit consonances. Thus, HTT is a mapping of a mapping: 15-limit JI mapped to a 4-D temperament, which is in turn mapped to a 29-tone octave.

I did try tempering out 324:325 to include the 5's in the 7-11-13 chain of regular fifths, but decided against it because it more than doubled the maximum error (from ~1.6c to ~3.6c) at the 7-limit (where beating is most noticeable) and didn't give me much of anything in return. (I just now roughed this out and arrived at a 3.6442 cents 13-limit max error with two chains of fifths 1.5670 cents wide exactly 4:7 apart with untempered octaves; 5:13 is exact, and 15-limit max error is 5.2112 cents, although this is probably not optimal.) Of course, this will be of interest if your're seeking a 3-D temperament with an unspecified number of tones; note that, unlike HTT, both otonal & utonal 13-limit harmony will be available.

As for documentation, the 29-HTT tuning appeared in Xenharmonikon 4 (fall of 1975) in an article describing the specifications of the generalized-keyboard Scalatron. The 17-HTT set (15-limit otonal ogdoads in 3 keys) was devised in the winter of 1978 and appeared in Jonathan Glasier's publication Interval in that same year. I didn't come up with the 41-HTT mapping (15-limit otonal ogdoads in 11 keys) until 2001 and included it in this message that gives Scala listings for all of the HTT sets:
/tuning-math/message/7574

When I wrote my paper about the Miracle temperament and decimal keyboard,
http://www.anaphoria.com/SecorMiracle.pdf
I neglected to mention that the 41-HTT superset is an excellent way to get 13-limit harmony on the decimal keyboard, inasmuch as any tuning that can be mapped to a 41-tone octave can readily be played on that keyboard. This is to correct the mistaken impression that the decimal keyboard, which is constructed using the Miracle generator (a/k/a the "secor"), is not good for much of anything besides Miracle.

> ...
> >> This last one is equal to 896/861 plus 352/351, and I propose
> >> that it be named the Secorian comma (or superkleisma?),
>
> > I've added it to the Xenwiki comma list.
>
> It may be fun to get George's take on this one; but if there's no
> earlier name or precedent, this seems fitting. It's fun to look
> back to the time when he got the generating "secor" of the
> Miracle Tuning at around 116.7 cents named after him, with the
> explanation (SECond minOR).

Do whatever you like. As for whoever else may have precedence for a temperament with a wide-fifths generator that tempers out both 363:364 and 351:352, you'll have to look prior to 1975. As for who gets to name it (if no one else has done so), then be my guest. After all, I didn't get to name Miracle!

--George

--- In tuning@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:
> I haven't been gung-ho enough to work out an actual algorithm
> yet. I know some properties they must have, which narrows it
> down, but getting these scales still involved some fiddling
> around and trial and error. The most important property is
> that (with one sporadic exception) they must consist of only
> two chains of some interval; in other words there is some
> lattice direction such that if you project along that direction
> the image of the scale is only two points.

How far apart can the points be?

> (The only kind of exception, which is a triple Fokker block but
> not two chains of generators, is the 7-note scale with step
> pattern "aabacab", or the 14-note scale with step pattern
> "aabacabaabacab", and so on. You need three chains of generators
> to construct this, not two.)

Interesting exception...

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "Keenan Pepper" <keenanpepper@> wrote:
> > I haven't been gung-ho enough to work out an actual algorithm
> > yet. I know some properties they must have, which narrows it
> > down, but getting these scales still involved some fiddling
> > around and trial and error. The most important property is
> > that (with one sporadic exception) they must consist of only
> > two chains of some interval; in other words there is some
> > lattice direction such that if you project along that direction
> > the image of the scale is only two points.
>
> How far apart can the points be?

Technically they can be arbitrarily far apart, but if there are any lattice points in between the two lines, that means every note of the scale is a note of the same non-trivial sublattice. IOW the intervals of the scale don't saturate the lattice; they're all restricted to some subgroup. So in that case why not call the sublattice the whole lattice, making the distance between the lines as small as possible in this new lattice?

> > (The only kind of exception, which is a triple Fokker block but
> > not two chains of generators, is the 7-note scale with step
> > pattern "aabacab", or the 14-note scale with step pattern
> > "aabacabaabacab", and so on. You need three chains of generators
> > to construct this, not two.)
>
> Interesting exception...

Indeed. This pattern is also an exceptional Disjoint Covering System of Rational Beatty Sequences, if you want a phrase to search for. Great examples of this pattern are http://xenharmonic.wikispaces.com/dimeanporc and http://xenharmonic.wikispaces.com/dimavenipu (or its marvel tempering).

Keenan

--- In tuning@yahoogroups.com, "gdsecor" <gdsecor@...> wrote:

> I did try tempering out 324:325 to include the 5's in the 7-11-13 chain of regular fifths, but decided against it because it more than doubled the maximum error (from ~1.6c to ~3.6c) at the 7-limit (where beating is most noticeable) and didn't give me much of anything in return.

Did you consider tempering put 540/539 instead? But I don't see much error increase from 325/324 by L2 measure.

> Margo,

> I've been underwater, so to speak, for the past week or so and have
> just started trying to read and absorb the posts here regarding
> temperaments that are centered on 3/2, 14/11, and 13/11. I'll
> probably have a lot of questions eventually -- I'm relatively new
> to microtonality, having only started looking into it about two
> years ago. That's why my original "Three related temperaments" post
> covered such familiar territory: It was a newbie explaining obvious
> things to other newbies. :)

Dear Jake,

Thank you for your kind note, and I totally understand about your
being "underwater" -- in fact, that might be a good hint to me
that there are other things to consider at least now and then :)
But I was immensely impressed with your readable style and your
interest in this style of temperament. Please take your time.

> I should note that I didn't name the pentacircle comma, the
> just Canton scale, or the pentacircle-tempered Cantonpenta
> scale. I believe all three are due to Gene Smith.

Thank you for this explanation: I noticed you had lots of posts
and music in Canton or Cantonpenta, and assumed these might be
your tunings. Thank you for clarifying this, and my apologies to
Gene for getting it wrong!

As to what you are terming the "pentacircle comma," or in other
words the comma used in these tunings, I'd say that there are
actually four usual ones, and that a group of us discussed some
possible names in 2002, when George Secor and Dave Keenan were
collaborating with others on their Sagittal Notation project.
Since I use 14/11 and 13/11 as my everyday major minor thirds,
and was doing it back in 2002, obviously I was very interested in
this process, which played out in good part on tuning-math.

[However, after writing this and then actually checking out
Cantonpenta.scl, I'd guess it might involve some additional
commmas, since it's not regular! That would likely be something
very new! This should be fun to sort out, and I need to check out
more posts that might have addressed this. What follows only
addresses the usual commas we'd get if we had a 56/39 tritone and
a 21/11 major seventh. I missed this point until I tried Scala.]

Here's a post from October 8, 2002, that discusses the four
likeliest commas you have in mind: the 896:891, 352:351, 364:363,
and 10648:10647. The last and smallest one I named the harmonisma
in honor of Kathleen Schlesinger and her _harmoniai_ or
subharmonic scales which often included 14/11 and 13/11; that
became recognized in Scala. For the others, I favor the naming
conventions of George Secor, more or less, as explained in the
first part of the article, with some diagrams in the second:

</tuning/topicId_39367.html#39367>
</tuning/topicId_39371.html#39371>

You'll note that writing a decade apart, we both make the same
kinds of comparisons between what you term "Three Similar
Temperaments," each going about it in our own way, but following
the same logic. It's sort of like when two civilizations from
different stellar systems meet, and find that their mathematics
agree! And we're all newbies at this on some level, with lots
more to learn!

> I did explore them a bit, though, which led me to understand
> the value of 14/11 as a major third, even though it's even
> sharper than the Pythagorean major third, and the 13/11 minor
> third, which I really like. Gene derived Canton as a just
> scale and then tempered it, I believe -- I don't think he
> iterated a generator, as you've been doing here and as I did in
> my original post. But maybe there's a subtlety there that would
> tie the two together, which eludes me.

Of course, there are at least two perspectives on these
tunings. For someone doing tonal harmony, classical or rock or
whatever, a major third at 14/11 or even Pythagorean is
stretching it. For someone like me doing a neomedieval style,
it's the norm: somewhere between Pythagorean and 9/7 or so!
This has been true for me since the year 2000. It's a different
universe. Here's a great piece in a related temperament,
Peppermint, recorded by Aaron Johnson:

</tuning/topicId_67062.html#67062>

Another clue to this: you may remember some posts about a "flat
major third" at 369 cents or so -- it's natural people oriented
to 5/4 would consider it "flat," but in my curious universe, it's
called a near-just 26/21 or maybe 21/17 submajor or large neutral
third. (The 442:441 is tempered out.) We can talk about this
more: I don't want to throw too much at you at once, especially
given my posts already.

But as to Gene's tempered Cantonpenta, what I did, to borrow an
old phrase, was to run it up the flagpole and see if Scala would
salute it. Surprise! It uses a regular temperament with a fifth
of 704.059 cents, which would not be considered part of the
parapyth family of temperaments -- but isn't a regular chromatic
scale, which would have 11 regular fifths! The very small minor
sixth at 752.768 cents -- a great feature! -- is something else.
So this is a "one of a kind," just or tempered!

> Anyway, I've found the discussion very interesting, especially
> since you appear to have a different approach and different
> goals from many of the others on the list, but since I'm still
> very busy, I may yet be a few days before I can ask my
> questions.

True: different goals, different approaches, and mutual aid and
cooperation! And I'm looking forward to those questions, when
it's right for you. In the meantime, please enjoy!

> Thanks for your generosity with your time.
> Regards,
> Jake

And thank you likewise, with best wishes,

Margo

> Hi Margo, this reply I'm writing here is going to be too long for any
> human being to ever read, but heck with it...

Dear Mike,

In fact I did read it, and I'd call it both heroic and heroically
successful! Bravo! It was no longer than necessary, and given the
length of lots of my posts, I should be last person to complain,
especially in view of the fact that the length of my post sort of
set the framework!

And your trimming was impeccable! I'll try to follow your
example, and especially the commandment: "Thou shalt trim thine
own previous comments, to put thy neighbor's new ones in better
view."

Since I'm a bit tired, I may only begin tonight, and address more
of your _wonderful_ post tomorrow. Before getting into specifics,
here are a few ideas in line with your absolutely marvelous
example of fairness, empathy with my concerns, and eagerness for
openminded dialogue and cultural exchange!

First, "temperament family" is exactly right! It fits!

Secondly, to keep this simple, I do feel that back in 2000-2002 I
was not merely developing individual temperaments, but also
prototypes for families: one might say that I was indeed in a
family way :)

And the civilized solution is to keep all the present families
involving intentional use or optimization of prime 5 in place,
e.g. Pele and Leapday, and additionally recognize three families
involving 2.3.7.11.13 that coexist in the same region, with the
new Parapyth being one of them!

Repeat: current families in the region of 703.447-705.0 cents
stay as they are (Pele, Leapday, and any others), but are
understood to imply a desire or intent for 5; while Parapyth
implies a desire for 2.3.7.11.13, with 5, if present, as a side
effect rather than an intended part of the design. The 11:10
neutral second presents a special case, and may be considered
consistent with either Pele or with Parapyth (e.g. MET-24, O3).

This should be as graceful and gracious and painless a transition
as possible for everyone involved.

While I'll get some rest and then get to actually replying point
by point to your letter -- it deserves my best effort and more --
what I might do in the meantime is offer some links to posts from
2000-2002 both to document the three families, and to let you
read my perspectives back then on commas and optimization, etc.
Some of the terminology is obviously dated, etc., but my purpose
is to communicate some of the ethos and worldview behind these
families and tunings. Retrospectively, maybe it's a kind of FAQ
that needs, of course, to be updated, with your questions and
ideas an invaluable resource for this!

Please don't be bothered by my custom back then of sometimes
speaking of things like "11-flavor" meaning (ratios of 11 and 13),
maybe I was concerned that "11-limit" or "13-limit" might imply
the presence of 5.

Quickly, and this can be fleshed out at leisure, the region for
these three families runs from 29-EDO (703.447 cents) to an even
upper limit of 705.0 cents. We can think of all three as
belonging to the "undecimal-tridecimal-Zalzalian" -- or more
compactly, "neomedieval" superfamily. They share two themes in
common, both revealed in a simple 12-MOS chain.

The first is some reasonable representation of intervals in the
JI divisions of the fifth 22:26:33 (near the bottom of the
superfamily range) or 22:28:33 (near the top), with 14/11 and
13/11 both being near-just in the middle of the region. Note that
33/26 is close to just around 29-EDO or Secor's HTT; while 33/28
is close to just at 705 cents (also 80-EDO). Thus the first part
of the superfamily name, undecimal-tredecimal.

The second is the presence of Zalzalian, i.e. submajor/supraminor
or central neutral intervals, from a chain of 6-11 generators,
the equivalent of the schismatic 5-limit approximations at or
near Pythagorean. In the lower part of the superfamily range, we
get things like 63/52 and 26/21, or 17/14 and 21/17 (14:17:21
being the lowest-odd-factor division of the fifth into two
neutral thirds, as George Secor once pointed out to me).

In the upper part of the range, these Zalzalian intervals (named
after the great `oudist Mansur Zalzal, who lived in Baghdad in
the 8th century) subtly shift from submajor/subminor to the
central neutral range, e.g. 360 and 345 cents, or 16/13 and 11/9,
at 705.0 cents, the upper limit of the superfamily region.

Note that Zalzalian is distinct from terms such as Maqamic, which
can remain as they are or evolve as people using them wish. Specifically, Zalzalian implies a _medieval_ Near Eastern
outlook, which ties in with the medieval European aspects of the
region when viewed through a 2.3.7.11.13 lens.

Quickly, within this region, I propose three families, one of
which we've already agreed upon and adopted. That family, Parapyth,
runs from 29-EDO (703.447 cents) up to around the regular tuning
where 8 generators up provide a pathway to a precise Phi at
833.090 cents, that is around 704.136 cents. Indeed, Parapyth's
pathway of 8 generators up is targeted not at 13/8, but complex
supraminor or small neutral thirds such as 21/13 (also involving
prime 13) or Phi itself! With rank-3, we use a pathway of minor
sixth plus spacing to get 13/8, and so have our intriguing 21/13
or Phi while also enjoying our delicious harmonic neutral sixth.

Here Peppermint is a prototype of the Parapyth family, with O3,
MET-24, and Graham Breed's 46-17-41 tuning almost identical to
MET-24 as other rank-3 members. There's also an extended rank-2
form aimed at another class of complex and beautiful intervals
(guess which -- I'll reveal all in my next post), but rank-3 is
the prototype and the classic 2.3.7.11.13 form. Some
documentation on commas and optimization a la 2002, for starters:

<http://tech.groups.yahoo.com/groups/tuning/message/38721>
<http://tech.groups.yahoo.com/groups/tuning/message/41244>
<http://tech.groups.yahoo.com/groups/tuning/message/39367>
<http://tech.groups.yahoo.com/groups/tuning/message/39371>
<http://tech.groups.yahoo.com/groups/tuning/message/38689>

A distinguishing mark is the near-just realization in this
classic form of 33:36:39:42:44 (12:11-13:12-14:13-22:21) or
1/1-12/11-13/11-14/11-4/3, as noted in my first announcement of
this new tuning -- and also family we now call Parapyth:

<http://tech.groups.yahoo.com/groups/tuning/message/38440>

The second family, centered on a just 14/11 tuning (704.377
cents) is called, as gesture of friendship and neighborliness to
Herman Miller and Leapday (hope I got the authorship right this
time, Gene and Jake!), ChristmasEve or 12/24, whose full form has
a single 24-note rank-2 chain seen as two 12-MOS sets with a
"spacing" of the natural 12-diesis! Note that with a tuning size
of up to 17, Leapday and ChristmasEve look identical: sort of
like light as a wave or particle in physics. Here's an article on
the 12-MOS in its ideal form at 704.377 cents, optionally used
with five additional notes for an irregular circulating system I
now call Accession-17, in honor of the Accession Day of Queen
Elizabeth I (November 17, 1558) -- although this would have
sounded a few centuries out of date in one direction or the
other! Note that while the second post has an interval chart
referring to all of Accession-17, lots of the intervals are also
found in the basic 12-MOS at 704.377 cents, and also my
correction of a goof about ratios that Paul Erlich caught:

<http://tech.groups.yahoo.com/groups/tuning/message/12284>
<http://tech.groups.yahoo.com/groups/tuning/message/12378>
<http://tech.groups.yahoo.com/groups/tuning/message/12432>
<http://tech.groups.yahoo.com/groups/tuning/message/12612>

This was in September 2000 -- and only next summer, while reading
an old Tuning List digest from the days before I got involved did
I learn that George Secor had designed his 17-tone
well-temperament or 17-WT in 1978, so Accession-17 wasn't such a
new genre! But the simple 12-MOS within it is the epitome of
ChristmasEve: use of 1/4-undecimal kleisma, the pure 14/11
tuning, for 2.3.7.11.13 rather than the prime 5 of Leapday.

The third and final family is Biapotomic-7/6, where, unlike
rank-3 Parapyth, the 169:168 is tempered out, and the apotome is
at or close to (7/6)^1/2. Moving from ChristmasEve, we enter this
zone where I would say that 13-14-15 generators are clearly and
accurately pointed at 7 somewhere around 704.5 cents. In the
lower portion, 7/4 is at or near pure (15 fifths up); at the
center, around 704.776 cents, 7/6 is just (14 generators up), and
the apotome precisely equal to its geometric mean. At the upper
limit, 705.0 cents, 9/7 (13 generators up) is virtually just.
For people who like EDO landmarks, the just 7/4 tuning is close
to 109-EDO; the 7/6 tuning virtually identical to 63-EDO; and the
9/7 tuning to 80-EDO.

The prototype for Biapotomic-7/6 (sounds almost like Potomac!) is
the e-based temperament, which I enthusiastically explored as a
"Neo-Gothic" tuning (later "neomedieval" became standard, to
include the Near Eastern or Zalzalian elements) in a 12-MOS
during the summer and 2000. Then, suddenly, in October 2000, I
realized that 15 fifths up would form a virtually just 7/4 minor
seventh! Here's the initial excited post about the 24-note rank-2
form, and a series of articles exploring the pathways of 13-14-15
generators to the family of 2-3-7-9 intervals (9/7, 7/6, 7/4):

<http://tech.groups.yahoo.com/groups/tuning/message/14361>
<http://tech.groups.yahoo.com/groups/tuning/message/20573>
<http://tech.groups.yahoo.com/groups/tuning/message/23881>
<http://tech.groups.yahoo.com/groups/tuning/message/24448>

Using the region of 704.5-705.0 cents specifically for
2.3.7.11.13, the Biapotomic ethos, was the topic of a discussion
I had with Graham Breed in January 2000 comparing different
families and the pathways to 7 they offer:

<http://tech.groups.yahoo.com/groups/tuning/message/18098>

Really, Parapyth is most characteristic, but the other forms are
also part of the neomedieval worldview. I would guess that people
approaching this part of the spectrum from a 2.3.7.11.13
perspective as our everyday reality (Parapyth especially as our
_participatio communis_ or "common temperament") find it hard to
understand what it is like to take prime 5 for granted, and vice
versa. But our dialogue and cultural exchange will help!

With warmest wishes,

Margo

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:

> Here's a post from October 8, 2002, that discusses the four
> likeliest commas you have in mind: the 896:891, 352:351, 364:363,
> and 10648:10647. The last and smallest one I named the harmonisma
> in honor of Kathleen Schlesinger and her _harmoniai_ or
> subharmonic scales which often included 14/11 and 13/11; that
> became recognized in Scala.

The Xenwiki comma page has it iisted as "chalmersia", since Chalmers seems to have been the first to discuss it. Should that be changed? I'm soliciting opinions here from anyone who cares to give one.

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:
>
> First, "temperament family" is exactly right! It fits!

I find it extremely confusing, and often am left unsure as to what people mean. But I'm probably unusual in that regard.

On Fri, Oct 26, 2012 at 8:06 AM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:
> >
> > First, "temperament family" is exactly right! It fits!
>
> I find it extremely confusing, and often am left unsure as to what people mean. But I'm probably unusual in that regard.

We can head back to "temperament class" if you like.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Fri, Oct 26, 2012 at 8:06 AM, genewardsmith
> <genewardsmith@...> wrote:
> >
> > --- In tuning@yahoogroups.com, Margo Schulter <mschulter@> wrote:
> > >
> > > First, "temperament family" is exactly right! It fits!
> >
> > I find it extremely confusing, and often am left unsure as to what people mean. But I'm probably unusual in that regard.
>
> We can head back to "temperament class" if you like.

Good idea. You can start by translating this:

"Here Peppermint is a prototype of the Parapyth family, with O3,
MET-24, and Graham Breed's 46-17-41 tuning almost identical to
MET-24 as other rank-3 members. There's also an extended rank-2
form aimed at another class of complex and beautiful intervals
(guess which -- I'll reveal all in my next post), but rank-3 is
the prototype and the classic 2.3.7.11.13 form."

into terms you know I'll understand.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, "gdsecor" <gdsecor@> wrote:
>
> > I did try tempering out 324:325 to include the 5's in the 7-11-13 chain of regular fifths, but decided against it because it more than doubled the maximum error (from ~1.6c to ~3.6c) at the 7-limit (where beating is most noticeable) and didn't give me much of anything in return.
>
> Did you consider tempering put 540/539 instead? But I don't see much error increase from 325/324 by L2 measure.

I need to correct my statement, because I was making the assumption that ratio you suggested would have led to the most "obvious" way to include 5. Now that I've taken the time to make a calculation, I see that it was 5103:5120 that I tried (and rejected).

--George

--- In tuning@yahoogroups.com, "gdsecor" <gdsecor@...> wrote:

> I need to correct my statement, because I was making the assumption that ratio you suggested would have led to the most "obvious" way to include 5. Now that I've taken the time to make a calculation, I see that it was 5103:5120 that I tried (and rejected).

Do either you r Margo have a name for the rank 3 temperament tempering out 325/324, 352/351 and 364/363?

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "gdsecor" <gdsecor@> wrote:
>
> > I need to correct my statement, because I was making the assumption that ratio you suggested would have led to the most "obvious" way to include 5. Now that I've taken the time to make a calculation, I see that it was 5103:5120 that I tried (and rejected).
>
> Do either you r Margo have a name for the rank 3 temperament tempering out 325/324, 352/351 and 364/363?
>

While we are at it, what about the comma 179200/177147, which otherwise I'll go and call "bunrod" (bunya-rodan) or something like that?

>> When artistic tastes get into the mix, things can get
>> marvellously complex, and I hope will stay remarkably civil.

> You're too kind, and I see no reason for anything to get
> uncivil :)

Nor I! In fact, I think we've found the perfect solution, which
will take shape in a fruitful way so that the new neomedieval
families get added without any current ones getting displaced.

And I'm having lots of fun learning some of the terminology here,
which will also help communication. For example, two of my rules
of neomedieval optimization in a nutshell:

(1) When in doubt, go for the Eigenmonzo/minimax solution!

(2) Optimize the Swiss tetrad (12:14:18:21), which is in no
way "anomalous" when 2.3.7.9.11.13 is seen as a usual
full set. This is a favorite and very sweet as well as
rich cadential sonority resolving by stepwise contrary
motion in all voices to the fifth. Some articles from
way back when on "Neo-Gothic" and "sesquisexta" discuss
this.

One very important thing: I discovered that there's already a
name for the overall family or class: Gentle, along with a Gentle
comma at 364:363. So Parapyth, ChristmasEve, and Bi-apotomic-7/6
are all subclasses or subfamilies of Gentle.

My warmest thanks to Gene or whoever did the Xenwiki pages on
Gentle and its comma of 364:363! I'm delighted by these names,
and find it simplest and best to regard the three families I
originally proposed as in fact subfamilies of Gentle.

My apologies for the length of what follows, with much of your
letter left to reply to, including the question of 196/195, which
I think I may understand, at least in the context of MET-24.

Often it seems that I may be saying too much, and threatening to
turn a dialogue into a monologue; yet saying too little might not
communicate the clearest answer. There's a real dilemma here, and
I'm resisting what might be the wise inclination to spend an hour
or two trimming this. Please feel free to tell me that my
inclination would have pointed me to the right choice :)

> What are you mapping 11/10 to, specifically? If it works out
> to be the same as mapping 5/3 to C-G#*, then that means that
> you've mapped 5 in a regular way, and I'm curious if it all
> still matches the Pele mapping in the end.

Good question. The 11/10 is the rank-2 diminished third or
"dilimma," e.g. C#-Eb, essentially equal to twice the limma at
around 22/21. In MET-24, the usual 1024-EDO version, however, we
do a bit better than 441/440 narrow, because the slightly and
unavoidably irregular tempering of the fifths results in two
limmas at 80.859 and 82.031 cents, and thus 11/10 at 162.891
cents.

There's another comma tempered out: 441/440!

Note that the purpose is to get 11/10 neutral steps, and
associated intervals such as al-Farabi's tetrachord, which we
would now call a bright Rast, at 9/8-99/80-4/3 (0-204-369-498
cents) or 9:8:11:10-320:297 (204-165-129 cents). The 5-limit, in
the sense of 5-odd, isn't the focus, but rather large and small
neutral intervals.

And the 11/10 occurs in the basic 12-MOS from 10 generators down,
a size where there are no 5-limit thirds -- and likewise in a
17-MOS.

I should add that a convincing 11/10 is possible in Parapyth from
29-EDO up to O3 or 1/5-kleisma (i.e. 896/891^1/5 with 22/21
just), but by Peppermint it's not really there: we have more of a
23/21 or whatever (159.044 cents). So it's in lower Parapyth that
we have 11:10 as well as 12:11, 13:12, and 14:13 all as goals in
the optimization.

[On the potential for an expanded MET-24 to represent Pele]

> Well, not in the 24-note scale you presented, but there's no reason
> why you couldn't theoretically continue the pattern of generators
> forever, right?

Agreed! And there is a possible "diplomatic solution" here: to
recognize both the original intent and actual capabilities or
potentials of system that's 2.3.7.11.13 in intent but
"Pele-compatible" or "Pele-expandable" in structure.

> For instance, you can create a hypothetically infinite (octave
> equivalent) lattice in which one axis is the slightly sharp
> fifth and the other axis is the ~50 cent "spacing" interval,
> and then your scale traces out a 2x12 rectangle on that
> lattice. But I see you go into more detail about this below,
> so I'll respond more there.

Well put! As you say, we'll pick this up more below, but I just
want to say that so far we're in total agreement.

> For now, suffice to say that I find it useful to separate scale
> generation from the temperament itself, so that I'd look at
> your MET-24 as a I've been thinking of your scale as a
> generated subset of the underlying temperament, the
> mathematical structure of which is an infinite lattice. From
> this viewpoint, Keenan's "triple Fokker block" parapyth scale
> that he just posted would draw from the same underlying
> parapyth temperament, but would utilize a different scale
> within it (this time, rather than tracing out a rectangle, it
> traces out a parallelogram).

So that's what PIPEDUM means! In Scala, it's called a
parallelepiped, which I seem to recall from looking into 3-D
graphics means a 3-D extension of a parallelogram.

But we're totally agreed: it could be 2x17, or 2x29, etc.

> It's literally the same thing as how meantone has a 5-note MOS,
> a 7-note MOS, a 12-note MOS, a 19-note MOS, etc - none of these
> are meantone temperament in its entirety, but rather different
> scales which can be generated from it. Likewise, parapyth
> doesn't have MOS's in the usual sense, but has a whole host of
> higher-dimensional generalizations to it. One can cover the
> lattice by a regular tiling of hexagons, parallelograms,
> rectangles, all of which produce scales that are candidates for
> higher-dimensional MOS (you can think of a MOS as a
> one-dimensional rectangle/parallelogram/hexagon/all-in-one
> tiling a one-dimensional lattice).

This is a beautiful and awesome description, at once
mathematically accurate and poetic! And your remark that parapyth
"doesn't have MOS's in the usual sense" makes me wonder if a
comment I read about Starling might apply here also. It was
something like, "We usually don't give scales, because Starling
is like an open JI lattice."

> So indeed, as you point out, 34 notes gives you more 5-limit
> intervals, which I view as simply an implication of the
> underlying structure of the temperament mapping.

True, of course. We might say that MET-24, and temperaments of
the Parapyth class or family in its lower range especially (here
at 703.711 cents), are "Pele-compatible" or "Pele-expandable."
The 5-limit isn't part of the design: these are some rare and
remote intervals cast at one's feet by the equations, adding a
bit of extra variety to what was already a complete system based
on 2.3.7.11.13. But someone who wants to make it part of their
integral design, using the same "underlying structure," can!

Similarly, MET-24 happens to have two major sixths at 924.609
cents, a virtually just 29/17 (924.622 cents). But I'd be
hesitant to describe the system as prime-29 in its design,
although those sixths are there, and actually more "ordinary"
from a neomedieval perspective than 5/3, although they all are
part of the picture.

> Sure, nothing wrong with those sorts of tuning optimizations; I'd
> be interested in figuring out if you can outline some general rules
> for the sorts of things you like when you want to optimize tunings,
> so that we can come up with algorithmic methods for computing
> "Margo-optimal" tunings for any arbitrary temperament or scale.

For the Gentle family, I'll try for once to be concise while
giving a sense of the different subfamilies. Note that my remarks
are descriptive of how I approach things, not prescriptive:
tastes may well vary, including my own on different occasions.

My apologies for possibly diverting our dialogue into a monologue
of over 100 lines, which I'll thus set off with asterisks so that
people will feel free to jump over it if they'd like to rejoin
our general conversation. But it may be helpful to document a bit
about my sense of the three subfamilies of Gentle. And the
boundaries I propose are open to criticism, feedback,
negotiation, and revision!

* * *

Parapyth in its classic rank-3 form is the most intricate, the
range running from 29-EDO (703.447 cents) to the temperament for
a precise Phi from 8 fifths up (704.136 cents):

(1) We want the effect of 3/2 and 4/3 to be "almost"
Pythagorean, so MET-24 in 1024-EDO with its 6 fifths in
each 12-note chain at 703.125 cents is especially
attractive, and the ones at 704.297 cents aren't that
much more impure than 12n-EDO.

(2) We want to temper enough so that 14/11 and 13/11 are
both within three cents of just, but the former might
still be a bit closer to 33/26, as in MET-24; however,
as in meantone, the nice thing about shadings is that
there are so many from which to choose!

(3) A major reason for tempering is to make the diminished
fourth a large neutral third at around 26/21 or 21/17,
for example; and the augmented fourth a small neutral
third at around 63/52 or 17/14, etc. In other words, we
are deliberately tempering _away_ from the 5-limit!
Both our regular and "alternative" thirds (8-9
generators) are relatively blending but active and
rather complex, like medieval Pythagorean thirds.

(4) Melodically, the whole tone will represent both 9/8 and
44/39, while the apotome will represent 14/13, and the
limma 22/21. We want a better 9/8 than 12n-EDO, and at
the same time an excellent 44/39. We also want, in
MET-24, 14/13 and 11/10 (the double limma or augmented
third, e.g. C#-Eb) about equally impure, a defining
parameter of MET-24.

(5) The spacing should be set for an eigenmonzo of 7/4 in
this range around 703.7-703.9 cents (MET-24, O3), but
for an eigenmonzo of 7/6 around Peppermint (704.096
cents). Right around 704 cents, as in my "Friendly
Introduction" article, we might split the difference
and have both 7/4 and 7/6 about a cent impure.

(6) In MET-24 or O3, we'll have representations of 14:13,
13:12, 12:11, and 11:10 all within three cents or so of
just -- with MET-24 at the point of equal impurity for
14:13 and 11:10, In the Peppermint region, we won't
really have anything close to 11/10 -- but rather a
just or virtually just 12:13:14.

(7) Swiss Tetrads, in Xenwiki parlance, 12:14:21:24, are of
the essence: the 9/7 may be up to 4 cents or a bit more
wide in a 1024-EDO version of MET-24 or O3, with 7/4 as
the eigenmonzo; but Peppermint with a 7/6 eigenmonzo is
the minimax solution. With the first two tunings, we
put on our "352/273 is Beautiful!" t-shirt, and
speculate that Marchettus of Padua may have sung it the
same way.

While we're at it, let's not forget parapyth rank-2 either, whose
special attraction is the wonderful family of interseptimal
intervals, or interseptimals for short, located somewhere between
8/7 and 7/6; 9/7 and 21/16; 32/21 and 14/9; and 12/7 and 7/4.
These are neat for some exuberant neo-14th century European
cadences, gamelan, and much else. With rank-2 Parapyth, 13-14-15
generators are a pathway and passport to interseptimal space.

<http://mschulter@.../~mschulter/IntervalSpectrumRegions.txt>

Here a starting point might be set 8 fifths up at or near Phi
(833.090 cents), with Peppermint (704.096 cents) very close to
the precise Phi tuning of 704.136 cents. In the Phi tuning that
marks the upper border of Parapyth we'll get interseptimals at
446.228 cents and 257.908 cents; and a small minor seventh of
962.044 cents that moves into the septimal region. The 12-tone
diesis at 49.635 cents is great for some exuberant 14th-century
European cadences, while there are also some gamelan potentials
for the division of the tempered 4/3 into 258-238 cents, etc.
Note how 13-14 generators are pointed not at prime 7, but at the
open reaches of interseptimal space.

How about the rank-2 families: ChristmasEve or 12/24, and
Bi-apotomic-7/6?

For ChristmasEve, running from 704.136 cents to 704.5 cents:

(1) Go for the eigenmonzo, 704.377 cents, making 14/11 and
11/7 just.

Note that we're on a cusp between interseptimal and septimal: 13 generators gives us 443.009 cents, the better part of a full
896:891 wide of 9/7, but very close to 128/99 (444.772 cents).
Commawise, the 17-comma is close to 64:63 (27.264 cents), the
difference between the pure 14/11 and 128/99. The 7/6 is still
narrow by over five cents at 261.278 cents, with only 7/4 really
getting close at 965.655 cents (3.171 cents narrow).

That's why, from a neomedieval perspective, the ChristmasEve
region around a pure 14/11 tuning is transitional rather than
clearly interseptimal or septimal, and Bi-apotomic-7/6 is a
separate family. This starts around 704.5 cents or so, and its
trademark is accurate septimal approximations from 13-14-15
generators, with twice the apotome (e.g. C-Cx) representing 7/6.

For Bi-apotomic-7/6, at 704.5-705.0 cents:

(1) Go for your favorite eigenmonzo, or something near
it: a pure 7/4 (704,588 cents) or the nearby e-based
tuning (704.607 cents) where the tone and diatonic
semitone have a logarithmic ratio equal to Euler's e,
2.71827, a kind of honorific pun since Euler
championed 7/4 as a new consonance; or the minimax
tuning with a pure 7/6 (704.776 cents); or the tuning
with a pure 9/7, which we'll minutely overshoot by
rounding to an even 705.0 cents.

Finally, like meantone between 1/4-comma and 2/7-comma, every
shade is worth exploring and there's no "wrong" choice: I'm just
pointing out some favorites of my own as possible starting
points.

* * *

> We can call it a "temperament pathway" if you'd like. The
> argument over the definition of "temperament" has apparently
> been raging for years now, predating my joining this list, and
> people sometimes disagree on whether 1/4-comma meantone and
> 1/3-comma meantone are two different temperaments, or
> "instances of the same temperament," or "different tunings of
> the same temperament family" or whatever. I'm of the opinion
> that disagreements over naming can sometimes cause more
> problems than they should, and that at this point it's most
> useful to simply note what different people call things so we
> can all communicate.

True! Temperament families and subfamilies are very natural for
me, and classes and subclasses pretty much equivalent. I like
families because it reminds me of biological taxonomy, and has a
friendly sound. A "pathway," to me, would mean a strategy for
getting a specific prime or other desired type of interval: "The
pathway to 13/8 in rank-3 Parapyth is a minor sixth (4 fifths
down) plus spacing; while an augmented fifth or 8 fifths up is
the pathway to 21/13 or Phi."

> So let's, for now, say that don't you want to call that 890
> cent interval 5/3, making this a 2.3.7.11.13 subgroup
> temperament that you've named "Parapyth temperament."

A good question that invites me to clarify my view. First, I
would playfully note that the remote sixth we're discussing is actually 887.1 cents: the near-21/13 augmented fifth C-G# at
829.7 cents, plus the spacing of 57.4 cents. That's 2.751 cents
wide of 5/3, or a bit more accurate than Zarlino's 2/7-comma
meantone.

But let's cut to the chase here, from a neomedieval perspective.

In Parapyth, we've slipped the silken bonds of 5, and find
ourselves in a different space where 14/13 and 13/11, and various
kinds of neutral and septimal intervals are the new luminaries.
This is a beautiful experience, but maybe not so easy to explain
to those who haven't yet made the journey.

And, vitally important, 16th-century European modal polyphony
with its 5-limit ideal or major/minor tonality is _not_ the
intended style, but something like 13th-14th-century European
polyphony or Near Eastern styles emphasizing neutral intervals.
Saying "13-limit" creates misleading expectations. In contrast,
saying 2.3.7.11.13 alerts people that this is something else
again, with George Secor's article a classic on some of the new
as well as old sounds possible:

<http://anaphoria.com/Secor17puzzle.pdf>

Suppose someone is looking for a vacation of "surf and sun,"
which may mean basking in Sol's tangible warmth at Malibu in
California.

But Parapyth is more like a journey to Ursa Major, where it
remains possible to admire Sol -- as a charmingly remote star in
some constellation.

Now Sol is Sol, no matter where we view it from; and I'd
certainly consider 887.1 cents to represent 5/3.

But Parapyth, or a journey to Ursa Major, may disappoint someone
who sees mention of "basking in the sun" and expects to look out
the window of our spaceship and see sand, surf, and beach
blankets!

Of course, as you observe, 24-note Parapyth is Pele-compatible,
or more precisely Pele-expandable, since someone oriented to 5
will likely want way more than three 5/4 locations!

> So then, specifically, the relationship between your MET-24
> scale and parapyth temperament is exactly the same as the
> relationship between the quarter-comma meantone chromatic
> scale and the overarching meantone temperament. The
> quarter-comma meantone scale is a specific scale formed from a
> specific tuning of the abstract meantone temperament, and
> MET-24 is a specific scale formed from a specific tuning of
> the abstract parapyth temperament.

Yes, I understand this usage, but suspect that "quarter-comma is
a a specific scale or tuning of the meantone family, and MET-24
of the parapyth subfamily of the gentle family" might be more
intuitive. Either is logically correct, however.

> There a few different ways to state this relationship:

This is a nice summary and set of alternatives to consider!

> - MET-24 is a 24-note scale in a specific tuning of "Parapyth
> temperament" (general common usage)

Yes, just as we speak of "meantone temperament."

> - MET-24 is a 24-note scale in a specific tuning of "the
> Parapyth abstract regular temperament" (using Gene's abstract
> regular temperament terminology)

Gene's concept is very clear, although not necessarily the way I
would speak myself as a liberal arts major. :)

But with meantone, for example, that definitive article "_the_
abstract regular temperament" might raise certain complications.

Specifically, around Kornerup's Golden Meantone or the POTE,
about 696.2 cents, we get 13/8 from the regular major sixth less
the 12-note diesis -- is this 15 fifths up? Yes, Scala confirms
that it is.

But in 1/5-comma or thereabouts at 697.7 cents, we get 13/8 from
minor sixth plus a much smaller 12-note comma, which I guess
would be 16 fourths up. Yes, Scala confirms this also.

They are the same "abstract temperament" for 5 at 4 fifths up,
but evidently different for 13.

If we focus only on 5, they share this common pathway; but the
pathways for 13 are a bit different, so it's not so clear that
they are "the same abstract temperament," although they both
belong to the meantone family (a concept which really developed
before European theorists were much concerned with 13).

> - MET-24 is a 24-note scale in a specific temperament within
> "the Parapyth temperament family" (Paul's usage)

Very natural and intuitive -- my favorite!

> - MET-24 is a 24-note scale in a specific temperament that belongs to
> "the Parapyth temperament class" (Graham's usage)

Quite intuitive also; but somehow family seems a bit more relaxed
and down at home.

> - MET-24 is a 24-note scale in a specific temperament exhibiting to
> "the Parapyth temperament pathway" (what you just proposed)

Here I should clarify that a "pathway" as I would see it is a
strategy for getting one variety of interval: e.g. "10 fourths up
is the pathway to 11/10 in HTT or MET-24 or Graham's 46-17-41
temperament; but by 704 cents or higher, when this pathway yields
an interval or 160 cents or less, we have really moved out of the
11/10 zone."

So Parapyth would be a set of pathways that are part of the
intentional design or schema, plus others that are unintended but
also present in the structure; e.g., in MET-24, 9 fourths up plus
spacing (867.2 + 57.4 cents) yields at two locations 924.609
cents, a virtually just 29/17 (924.622 cents). And we know about
the near-5/3 at 887.1 cents.

> People have often argued over which of these is "right"; at
> this point I'll just call it whatever anyone wants! (If you
> want to incorporate a mapping for ratios of 5, then you can
> replace "Parapyth" with Pele above.)

Yes, and "Pele-compatible" would be my term to convey this -- or,
more specifically, "Pele-expandable" (from 24 to 34 or 58, say).

[To be continued]

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:

> My warmest thanks to Gene or whoever did the Xenwiki pages on
> Gentle and its comma of 364:363! I'm delighted by these names,
> and find it simplest and best to regard the three families I
> originally proposed as in fact subfamilies of Gentle.

Of course, I got the idea from you. :)

> There's another comma tempered out: 441/440!

Which tunings or scales have 441/440 tempered out?

> I should add that a convincing 11/10 is possible in Parapyth from
> 29-EDO up to O3 or 1/5-kleisma (i.e. 896/891^1/5 with 22/21
> just), but by Peppermint it's not really there: we have more of a
> 23/21 or whatever (159.044 cents). So it's in lower Parapyth that
> we have 11:10 as well as 12:11, 13:12, and 14:13 all as goals in
> the optimization

Does O3 assume 352/351 and 364/363 are tempered out, and what else might it assume? In particular, is it a 2.3.7.11.13 system?

> For ChristmasEve, running from 704.136 cents to 704.5 cents:
>
> (1) Go for the eigenmonzo, 704.377 cents, making 14/11 and
> 11/7 just.

Can you say what ChristmasEve tempers out?

> For Bi-apotomic-7/6, at 704.5-705.0 cents:

Same question for Bi-apotomic.

> True! Temperament families and subfamilies are very natural for
> me, and classes and subclasses pretty much equivalent. I like
> families because it reminds me of biological taxonomy, and has a
> friendly sound.

I like things whose precise definition I understand, which is why I like abstract temperaments.

> Yes, I understand this usage, but suspect that "quarter-comma is
> a a specific scale or tuning of the meantone family, and MET-24
> of the parapyth subfamily of the gentle family" might be more
> intuitive. Either is logically correct, however.

But what does it mean? What, precisely, are the definitions involved?

> Specifically, around Kornerup's Golden Meantone or the POTE,
> about 696.2 cents, we get 13/8 from the regular major sixth less
> the 12-note diesis -- is this 15 fifths up? Yes, Scala confirms
> that it is.

If 13 is not a part of the abstract regular temperament, it says exactly nothing about what 13/8 is. It isn't anything, since the definitions are in no way based on actual tuning. The idea here is that "abstract" really does mean abstract. You have notes, but to play the notes you need a tuning. It's like a score--if you have a score for a piece in meantone, you can't actually play it until you decide on a tuning.

Margo Schulter <mschulter@...> wrote:

> Here I should clarify that a "pathway" as I would see it
> is a strategy for getting one variety of interval: e.g.
> "10 fourths up is the pathway to 11/10 in HTT or MET-24
> or Graham's 46-17-41 temperament; but by 704 cents or
> higher, when this pathway yields an interval or 160 cents
> or less, we have really moved out of the 11/10 zone."

I have to disclaim ownership here. It was produced by a
program I wrote but it doesn't belong to me. It's the
outcome of certain mathematical processes that depend on
what the user feeds in. That diminishes the excitement of
finding temperament classes, of course, but makes it easier
to enjoy the excitement of writing in them.

Graham

Dear Graham,

Thank you for sharing with me news that's very exciting, and
maybe merits a joint article or paper somewhere!

>> Here I should clarify that a "pathway" as I would see it is
>> a strategy for getting one variety of interval: e.g. "10
>> fourths up is the pathway to 11/10 in HTT or MET-24 or
>> Graham's 46-17-41 temperament; but by 704 cents or higher,
>> when this pathway yields an interval or 160 cents or less,
>> we have really moved out of the 11/10 zone."

> I have to disclaim ownership here. It was produced by a program
> I wrote but it doesn't belong to me. It's the outcome of
> certain mathematical processes that depend on what the user
> feeds in. That diminishes the excitement of finding temperament
> classes, of course, but makes it easier to enjoy the excitement
> of writing in them.

Here the totally independent and identical results are
intriguing: maybe like a chess match between Susan Pulgar and
Deep Blue (a famous computer program of the mid-1990's) which
ends in a classic draw by perpetual check. What I was going for
was to take my usual 1024-EDO version of MET-24, and define a
"smoothed" version with three generators instead of the four that
we have with the two unequal fifths in 1024-EDO (600 and 601 steps).

Maybe we should jointly write an article or paper on this. One
thing it tells me is that maybe MET-24 is a mathematically
identifiable "sweet spot" in its own right rather than just my
impressionistic tweak of George Secor's HTT generator (703.579
cents).

Anyway, for the MET-24 version that I soon learned, when Mike
pointed me to your site, was identical to your program's earlier
result for 46-17-41, here's the algorithm I followed:

1. Find the linear generator where 14/13 (7 fifths up)
and 11/10 (10 fourths up) are equally impure,
i.e. 703.723 cents.

2. Now find the spacing generator or distance between
two chains of (2/1, 703.723) so that 7/6 and 13/8
are equally impure (57.423 cents).

Query: could our results be mutually validating in some way? Is
there something to be learned from this about optimization
algorithms?

> Graham

With many thanks,

Margo

>> My warmest thanks to Gene or whoever did the Xenwiki pages on
>> Gentle and its comma of 364:363! I'm delighted by these names, >
>> and find it simplest and best to regard the three families I >
>> originally proposed as in fact subfamilies of Gentle.

> Of course, I got the idea from you. :)

Great, and I love that naming of 364:363 as the "gentle comma"!

To provide a framework for answering some of your questions, I should maybe start by defining a few characteristics of the Gentle
family in general that apply to its subfamilies as well.

(1) The tempering of the fifths is "gentle," which means
not more than 3.045 cents at the upper bound for the linear
generator, 705.0 cents, where 9/7 is virtually pure.
The lower limit is 29-EDO or 703.447 cents.

(2) All Gentle subfamilies and their tunings will temper out
896:891, 352:351, 364:363 (the gentle comma), and
10648:10647. The pathway to 14/11 is 4 fifths up, while the
pathway to 13/11 is 3 fourths up.

If I'm correct, any regular tuning that meets (1) should also meet
(2). At the low end of Gentle and the Parapyth subfamily, we are
closest to 22:26:33 in JI; in the upper range of Parapyth near
Peppermint or your Cantonpenta, 14/11 and 13/11 are about equally
impure; in the ChristmasEve family (which I'm leaning to revise to
704.2-704.5 cents), 14/11 is at or near just and 3 generators is about
midway between 13/11 and 33/28; in Bi-apotomic (704.5-705.0 cents), we
are closest to 22:28:33 in JI.

Interestingly, 288-EDO supports Gentle with its available generator of
704.167 cents, or 169 steps. Two chains of these generators with a
spacing of 58.333 cents (14 steps) gives a rank-3 system quite similar
to Peppermint.

So Gentle is basically the undecimal/tredecimal eventone continuum for
14/13 and 13/11, analogous to meantone for 5/4 and 6/5, or superpyth
for 9/7 and 7/6.

By eventone I mean a generalized term for meantones (5/4 and 6/5) and
other regular tunings with an analogous structure but different ratios
sought with 3 and 4 generators: thus an Archyan eventone seeks 9/7 and
7/6, while an undecimal/tredecimal eventone seeks 14/11 and 13/11.

>> There's another comma tempered out: 441/440!

> Which tunings or scales have 441/440 tempered out?

First, I should admit with due humility that while I take an interest
in commas, I'm hardly a tuning-math expert, and welcome the chance to
get an analysis of the commas in these systems from people who really
know what they are doing on the fine points. But some dialogue may
help in answering some of these questions and sharing the information
which may help you and me in seeking the answers to others.

What I had in mind was the region around Peppermint, where a single
tempered interval of 367.235 cents, for example, represents both 26/21
and 21/17, a difference of 441/440.

While both these ratios will be represented by a single tempered
interval more generally through all of Parapyth, if I'm right, the
441/440 unison vector might be especially relevant where 8 fifths up
is somewhere between 26/21 and 21/17.

And likewise, in MET-24, for example (703.711 cents), in its smoothed
2048-EDO version (generator 1201/2048 octave), 9 fifths up or 333.398
cents represents both 63/52 and 17/14.

Also, from another perspective, in MET-24 we have Ibn Sina's comma of
2080/2079 tempered out, here the difference between 63/52 and 40/33,
both again represented by 333.390 cents. In Ibn Sina (980-1037), the
very small difference is noted between 14/13 and 320/297, although
without giving the actual ratio for this comma.

>> I should add that a convincing 11/10 is possible in Parapyth from
>> 29-EDO up to O3 or 1/5-kleisma (i.e. 896/891^1/5 with 22/21 just),
>> but by Peppermint it's not really there: we have more of a 23/21 or
>> whatever (159.044 cents). So it's in lower Parapyth that we have
>> 11:10 as well as 12:11, 13:12, and 14:13 all as goals in the
>> optimization

>> Does O3 assume 352/351 and 364/363 are tempered out, and what else might it
>> assume? In particular, is it a 2.3.7.11.13 system?

Absolutely yes on both! All Gentle family tunings temper our 352/351
and 364/363, by definition; and O3 is a member of the Parapyth
subfamily of Gentle, defined in its "smoothed" version as the rank-3
Parapyth tuning (2/1, 703.893, 57.148), i.e. a just 22/21 and 7/4.

Any rank-3 Parapyth system is 2.3.7.11.13 by definition, with a linear
generator range for which I'm now inclined to set the upper bound
slightly higher, say 703.447-704.2 cents. That includes the 288-EDO
version of (2/1, 704.167, 58.333).

In these tunings, 12:13:14 and 11:12:13 are generally quite accurate,
with 169/168 and 144/143 definitely observed.

Let me list a few Gentle temperaments, to which your Cantonpenta could
be added as a kind of subset of a 34-note or 17x2 Parapyth with
generators in 271-EDO of 271, 159, and 13 steps. If this is correct,
the rank-3 version would be (2/1, 704.059, 57.565 cents).

---------------------------------------------------------------
Gentle temperament family tunings (703.447 cents - 705.0 cents)
----------------------------------------------------------------
Subfamily 1: Parapyth, usually rank-3 (703.447-~704.200 cents)
................................................................

George Secor's 29-HTT, secor29htt.scl (2/1, 703.579, 58.090)
pure 63/52 and 7/4
<http://www.bestII.com/~mschulter/secor29htt.scl>

MET-24, my Milder Extended Temperament (2/1, 703.771, 57.422)
<http://www.bestII.com/~mschulter/met24-canonical.scl>
14/11 and 11/10 about equally impure, 7/4 near-just (2048-EDO)

Graham Breed's computer 46-17-41 (2/1, 703.723, 57.423)
<http://x31eq.com/cgi-bin/rt.cgi?ets=46-17-41&limit=2_3_7_11_13>

O3, my just 22/21 tuning with pure 7/4 (2/1, 703.893, 57.148)
<http://www.bestII.com/~mschulter/O3-reg-24.scl>

Cantonpenta(?) rank-3 (2x17, 271-EDO) (2/1, 704.059, 57.565)
Superset of Gene Ward Smith Cantonpenta
<http://www.bestII.com/~mschulter/cantonpentalike34.scl>

Peppermint, Kennan Pepper's Noble Fifth (2/1, 704.096, 58.680)
in rank-3 (2x12) with just 7/6 schulter_pepr.scl
<http://www.bestII.com/~mschulter/schulter_pepr.scl>

288-EDO rank-3 tuning (2/1, 704.167, 58.333)
(virtually just 7/6, 266.667 cents)
<http://www.bestII.com/~mschulter/parapyth_288-edo.scl>

----------------------------------------------------------------
Subfamily 2: ChristmasEve (704.2-704.5 cents), mostly rank-2
................................................................

Pure 14/11 tuning (2/1, 704.377 cents)
Just above 46-EDO
<http://www.bestII.com/~mschulter/christmas_eve24.scl>

----------------------------------------------------------------
Subfamily 3: Bi-apotomic, usually rank-2 (704.5 - 705.0 cents)
................................................................

e-based tuning, Blackwood's R = e (2/1, 704.607)
15 fifths up is near-pure 7/4
<http://www.bestII.com/~mschulter/neogeb24.scl>

minimax 2-3-7-9 tuning with pure 7/6 (2/1, 704.776)
Minutely above 63-EDO
<http://www.bestII.com/~mschulter/biapotomic_septimal24.scl>

tuning for virtually pure 9/7 (2/1, 705.000)
synonymous with 80-EDO
<http://www.bestII.com/~mschulter/reg705_24.scl>

===============================================================

> For ChristmasEve, running from 704.136 cents to 704.5 cents:
>
> (1) Go for the eigenmonzo, 704.377 cents, making 14/11 and
> 11/7 just.

> Can you say what ChristmasEve tempers out?

First, I'm inclined, as just mentioned, to make that 704.2-704.5
cents. Exactly what it tempers out is an interesting question --
beyond, of course, what all of Gentle tempers out, namely the 896/891,
352/351, 364/363, and 10648/10647.

This is a rank-2 system, whereas Parapyth is classically rank-3
although a rank-2 variation is possible, where the mapping beyond a
12-MOS is different. I often tend to think of cents for a given
shading, or nearby ratios, rather than commas -- but maybe we can pin
down some commas together.

The easy answer is that, if we consider ChristmasEve to be septimal
(as it is for 14 or especially 15 generators), then what gets tempered
out is the 169/168, observed in rank-3 Parapyth (14/13 vs. 13/12).

With ChristmasEve, I'd say that 13 generators down map to a tempered
128/99, while 14 are in the suburbs of 7/6, and 15 give a much more
accurate 7/4. Note that 128/99 is equal to 9/7 plus 896/891.

Now 128/99 is equal to 4/3 less 33/32. And if this were Pythagoren,
then 13 fifths down or fourths up would give 4/3 less a Pythagorean
comma, of course. So this ChristmasEve comma at around a pure 14/11
tuning (704.377 cents) or the almost identical 46-EDO may be defined
as the 180224/177147, which I note has the same denominator as the
famous Pythagorean wolf fifth 262144/177147.

For 14 or 15 generators at a pure 14/11 tuning, where the pathway is
to 7, our comma is equal to the difference between 28/27 and the
Pythagorean comma, 531441/524288, or 14680064/14348907 (39.501
cents).

ChristmasEve is the 2.3.7.11.13 version of lower Leapday, more or less
-- because from a neomedieval perspective, 443 cents is better viewed
as a good 128/99 than a quite inaccurate 9/7; while the 7/6 at this
point, around 704.377 cents, is still well over 5 cents narrow. So
it's a mixture of interseptimal and septimal, in contrast to the
optimized septimal of Bi-apotomic-7/6. On interseptimal intervals, see

<http://www.bestII.com/~mschulter/IntervalSpectrumRegions.txt>

>> For Bi-apotomic-7/6, at 704.5-705.0 cents:

> Same question for Bi-apotomic.

The four basic commas of Gentle (896/891, 352/351, 364/363,
10648/10647) plus, above all, the 14680064/14348907!

This region of 704.5-705.0 cents has 13-14-15 generators as accurate
pathways to 9/7, 7/6, and 7/4. In Pythagorean, these would be,
respectively, 21/16, 7/4, and 7/6 impure in each case by a 3-7 schisma
(64/63 less 531441/524288) at around 3.804 cents. So the comma is
equal to 49/48 plus the 3-7 schisma, or 14680064/14348907.

Another very important comma tempered out, as with ChristmasEve, is
the 169/168, maybe best illustrated by the minimax tuning at 704.776
cents, where 7/6 is just, formed from two apotomes at (7/6)^(1/2).

>> True! Temperament families and subfamilies are very natural for
>> me, and classes and subclasses pretty much equivalent. I like
>> families because it reminds me of biological taxonomy, and has a
>> friendly sound.

> I like things whose precise definition I understand, which is why I
> like abstract temperaments.

This is an area where you are much more acquainted with these commas,
let alone Fokker blocks, that I am. So deciding which commas are
tempered out might be a dialogue, where I can explain about the
tunings but you might identify new commas of which I wasn't aware.

>> Yes, I understand this usage, but suspect that "quarter-comma is a
>> a specific scale or tuning of the meantone family, and MET-24 of
>> the parapyth subfamily of the gentle family" might be more
>> intuitive. Either is logically correct, however.

> But what does it mean? What, precisely, are the definitions
> involved?

Actually it's a bit impressionistic. But I'd say that the Gentle
family tempers enough to get intervals of 7-11 generators to their
values in 29-EDO, fairly near 26/21 and 63/52 and at least 13-14 cents
from 5/4 and 6/5, for example, with 13/11 close to just; and no more
than required for 13 generators to produce a just 9/7 (allowing a tiny
overshoot to draw the boundary neatly at 705.0 cents).

So, to me, it's more like defining an historical era or period, for
example in music history, than an exact mathematical science. But it
is, of course, subject to mathematical analysis.

It may be analogous to a partnership between the composer who writes
by intuition in good part, and the analyst who looks for patterns.

>> Specifically, around Kornerup's Golden Meantone or the POTE,
>> about 696.2 cents, we get 13/8 from the regular major sixth less
>> the 12-note diesis -- is this 15 fifths up? Yes, Scala confirms
>> that it is.

> If 13 is not a part of the abstract regular temperament, it says
> exactly nothing about what 13/8 is. It isn't anything, since the
> definitions are in no way based on actual tuning. The idea here is
> that "abstract" really does mean abstract. You have notes, but to
> play the notes you need a tuning. It's like a score--if you have a
> score for a piece in meantone, you can't actually play it until you
> decide on a tuning.

I get the idea that the "abstract temperament" may of course choose to
map some primes and not others, even if they accidentally arise, like
a near-13/8 in 1/5-comma.

And that's analogous, then, to MET-24, where the abstract temperament
is 2.3.7.11.13, although it happens that 5 does come up at a few
locations, and 29/17 also, as Mike and I have been discussing.

Best,

Margo

Hi, Mike,

Here I'm picking up the dialogue from my last post, with the small
revision that I might draw the line between Parapyth and ChristmasEve
at around 704.2 cents. Among other things, this permits the 288-EDO
realization of rank-3 Parapyth at (2/1, 704.167, 58.333 cents).

Please forgive me if I'm being a bit repetitive on the Gentle family
(as aptly proposed by Gene!) and its subfamilies, but your question
about what's a name for a tuning or a family suggested that it might
not hurt to go over this. Maybe a bit of dialogue may help clarify or
possibly revise certain boundaries.

> The "Equal Temperament Mappings" thing is complicated and not very
> useful; it's the "Reduced Mapping" that's useful. Each row
> represents a different generator, and each column of the matrix
> represents a different prime, and this tells you how many of each
> generator you need to add to represent a certain prime. But if you
> have a more clever way in mind of specifying the temperament, how
> would you do it? What's a good systematic way to display, for some
> arbitrary temperament, where the relevant tempered commas "turn up"
> that's easily readable by a human being?

Certainly mapping primes is one useful appraach. The complications
come, for example, when we're aiming for something other than a prime,
like "an interseptimal major third around 128/99" or "a Phi-sixth."
For optimization, it would be nice to recognize these as goals in
themselves, rather than inaccurate forms of 7 or 13.

In Parapyth, of course, we are aiming mostly at primes, and quite
accurately at that! But things like the Phi-sixth or a bright neutral
or submajor third around 26/21 or 21/17 might perhaps be served by
some notation which could be used together with that of primes.

>> The POTE instantiation of the Pele pathway, however, is _not_ an
>> optimization of 13/11 and 14/11 (in contrast to 33/26), since 13/11
>> is still substantially wide of just, so a bit more temperament will
>> bring them both closer to pure.

> Right, this optimizes each interval in order of its complexity
> (specified by n*d for some ratio n/d). You can think of this as
> optimizing the average tuning of -all- intervals over the entire JI
> lattice, with the most weight is placed on really simple intervals
> like 3/2, 5/4, and so on. 14/11 and 13/11 have medium complexity and
> as such get medium weight, whereas something like 32805/32768 gets
> barely no weight assigned to it at all, because it's so complex that
> it gets minimal attention. That's one way to think of it, anyway.
> If you'd prefer to weight intervals differently, which ones would
> you say are most important? Is there some mathematical pattern
> inherent in the intervals you'd like to optimize, so we can apply
> this method of optimization to all temperaments?

Well, POTE (or TOP) seems one way to seek out a likely solution which
can serve both as a good tuning and a point of departure for others.
Another approach might be to "go for the eigenmonzo," or for some
point where two intervals of interest are equally impure, or a for a
minimax which could also be an eigenmonzo point, like a just 7/6
making 9/7 and 7/4 as impure as the fifth.

An interesting point is the 13-limit Leapday family divides into two
families for 2.3.7.11.13: ChristmasEve (704.2-704.5 cents), where 13
generators up is at least as close to 128/99 or the like as to 9/7;
and Bi-apotomic (704.5-705.0 cents) where 9/7, 7/6, and 7/4 are all
more accurate from 13-14-15 generators. The ideal Bi-apotomic tuning
is 704.776 cents, where an apotome is equal to (7/6)^(1/2).

Now when I first read of Leapday, I wondered how a pure 14/11 major
third (a great eigenmonzo in itself, of course) or 46-EDO could be an
optimization for 7 when tempering a bit more would improve 9/7, 7/6,
and 7/4 as well. Then I realized: these people are also weighing 5,
and maybe more than either 14/11 or 9/7 or even 7/6! That quickly
explained a lot!

To put the matter concisely, for Parapyth, ChristmasEve, or
Bi-apotomic, it's a matter of indifference whether 21 fifths up gives
a third of 392 cents (as around a pure 14/11, 704.377 cents), or 400
cents (as around the pure 7/6 tuning). But in Leapday, which is
definitely going for 5, that's a counterbalance to going much beyond
46-EDO.

> I'm not sure how 196/195 turns up, specifically; there's actually
> an infinite series of commas tempered out, and 196/195 is
> apparently one of them. I was just mentioning the simplest interval
> that you'd have to add to the set of vanishing commas which would
> transform Parapyth into Pele. You could add 441/440 instead and get
> the same thing, or 847/845. It was just a technical note.

The 196/195 is something I think I understand as a fortuitous aspect
of MET-24 -- the tempering out of this comma, that is.

Note that 14/13 (128.298 cents) is always within three cents of just
in the 1024-EDO version (125.4 or 126.6 cents), Likewise 7/6 is
consistently just over two cents narrow (264.844 cents). So 14/13 and
7/6 are each a little bit narrow.

Now there's a form of the Hijaz tetrachord -- here 126.6-264.8-105.5
cents or Bb-B-C#*-Eb -- that's rather close a just 14/13-7/6-52/49 or
128.3-266.9-103.9 cents, or 1/1-14/13-49/39-4/3 (0-128.3-395.2-498.0
cents). This is the series 39:42:49:52.

126.6 264.8 105.5
MET-24 0 126.6 391.4 496.9
Bb B C#* Eb
39 42 49 52
1/1 14/13 49/39 4/3
JI: 0 128.3 395.2 498.0
128.3 266.9 103.9

Now the 196/195 is the difference between 49/39 and 5/4, about 8.855
cents. Because both 14/13 and 7/6 are narrow, it happens that this
absorbs some of the comma, making 5/4 and especially 6/5 considerably
closer to just. It's really an accident. although someone wanting to
focus on 5 and expand the temperament structure could get Pele from
it.

Now I've found that 352/351 + 441/440 equals 196/195, although I'm not
sure how this would be used in an actual tuning. But I would say that
evidently MET-24 does temper out the 196/195, and is thus a
Pele-compatible or Pele-expandable member of Parapyth.

> If we want to not include 5/4 and 6/5 in the optimization, we can
> just go back to looking at Parapyth and skip this noise about
> Pele. So POTE Parapyth has the fifth at 703.857 cents and the
> spacing interval at 58.338 cents. This tunes 14/11 to 415.425 cents
> and 13/11 to 288.431 cents. The general "weighted average of all
> intervals" optimization mentioned before still applies, except now
> we're excluding ratios where 5 is involved.

This is very close to a just 22/21 or O3, and also seems to be the
region of Parapyth that Gene, I believe, has termed Pentacenter.
I'm not sure if Pentacenter is a specific tuning or more of a
subregion of Parapyth where 22/21 is close to just.

>> Yes, in that sense the spacing comma, i.e. limma less spacing,
>> represents both 81/80 (or maybe 66/65 if the regular major third is
>> 33/26, actually a tad closer than 14/11) and the 64/63 (or also
>> 78/77, e.g. 13/11 and 7/6).

> Looks right to me; limma minus spacing represents 81/80, 66/65,
> 64/63, and 78/77, by my calculations.

Great!

> Hmm, if C-C### is mapped to 5/4, that corresponds to a 5-limit
> temperament tempering out 10737418240/10460353203. See here:
> [48]http://x31eq.com/cgi-bin/rt.cgi?ets=46_29&limit=5 Looks like
> the 13-limit version of this is called "Leapday" temperament:
> [49]http://x31eq.com/cgi-bin/rt.cgi?ets=46_29&limit=13 What limit
> is the Pepper Noble Fifth tuning in?

Well, the basic Pepper Noble Fifth is rank-2 at (2/1, 704.096), so
with an open tuning, the "limit" maybe in the eye or ear of the
beholder, as you've pointed out also. But this is very close an
eigonmonzo of 5/4, with the apotome at (5/4)^(1/3). The magic spot is
704.110177 cents. Of course, as with Leapday, this implies a useful
MOS for 5 of 29 or larger. And in something like Peppermint-58, this
would happen also within each 29-MOS chain.

But it wouldn't in Peppermint-34, unlike MET-34, because here the
best pathway to 5 is within a single chain (21 fifths up), while, as
we've discussed, in MET-34 it would be the spacing comma (Pele's
pathway, if I'm correct).

> Sorry, I'm confused now - what's the end specification of the
> e-based pathway, and the HTT pathway? Which limits are these
> temperaments in, and how are the various primes mapped? If you
> posted something back in 2000 or so that we've assigned a different
> name to, then you get precedence, but I'm still not sure which of
> HTT, e-based, MET, O3, parapyth, etc you consider to be generalized
> mappings, or very specific tunings, or specific tunings of specific
> scales, etc. It seems like some of these are different names for
> different tunings of the same underlying temperament or temperament
> "pathway", whereas others are names for different "pathways"
> themselves.

One important thing is that it's easy to say, "I've designed what
might be a new tuning." It may be a bit harder, however, to conclude,
"Well, this seems to be not only a new tuning, but a new temperament
family as well." HIstorical perspective often helps.

To clarify: George Secor's High Tolerance Temperament family beginning
with 29-HTT (1978) has a 2.3.7.11.13 subset which served as a
prototype for rank-3 Parapyth: Peppermint (2002, based on the Pepper
Noble Fifth linear generator of 704.096 cents), O3 (2010), MET-24
(2011), Graham's computer-created 46-17-41 tuning, etc. These are all
members of the Parapyth subfamily of the larger family Gene has very
felicitously named Gentle (703.447-705.0 cents).

Now what all rank-3 Parapyth tunings share is a pathway to ratios of
2-3-7-9 through 1-2-3 generators plus or minus spacing for 9/7, 7/6,
and 7/4 respectively. That is what I termed the "HTT pathway" to honor
George Secor's germinal example which I learned about from George in
2001, and in 2002 eagerly applied to Peppermint, and since to other
shadings of Parapyth: it's more generally the Parapyth pathway to 7.

In 2000, I announced the e-based tuning, a rank-2 system (2/1, 704.607), and a few months later excitedly discovered that 15 fifths up is a virtually just 7/4. Now I propose that e-based
is one prototype for the Bi-apotomic subfamily of Gentle, where the
apotome or chromatic semitone is very close to (7/6)^(1/2). This runs
from around 704.5 to 705.0 cents, with 13-14-15 generators in a single
chain yielding approximations of 9/7, 7/6, and 7/4.

In 2000, I also announced a 12-MOS with a just 14/11 major third
(704.377 cents), and an optional 17-note circulating version adding
five remote notes to this 12-MOS. Here our interest is in a regular
rank-2 tuning at around 704.2-704.5 cents, with the 14/11 eigenmonzo
as an obvious point of attraction, I now call ChristmasEve, suggesting
a full tuning set of 24 and a 12-MOS, or 12/24 (the date style in
the UK of 24/12 for Christmas Eve is fine also).

Now ChristmasEve is a playful allusion to Leapday -- the difference is
a focus on 2.3.7.11.13 rather than 13-limit. From my perspective, this
ChristmasEve subfamily of Gentle is transitional between Parapyth,
where an independent spacing generator is needed to get near 7; and
Bi-apotomic, where the 12-tone diesis can serve as the natural spacing
and get us there rather nicely, especially from 704.6 cents on up.

The distinctive feature of rank-2 ChristmasEve is that 13 generators
actually yields a large major third at around 443 cents for the pure
14/11 tuning, nicely representing 128/99 or the like rather than 9/7;
but 14 and 15 generators are reasonabl close to 7/6 and 7/4. So it's a
kind of charming compromise between septimal and what I term
interseptimal, with each part of the spectrum having its own value.

<http://www.bestII.com/~mschulter/IntervalSpectrumRegions.txt>

From the informal term "gentle temperaments" that I used in 2002 to
describe systems with fifths tempered around 704 cents, regular thirds
near 14/11 and 13/11, and lots of neutral and septimal (or sometimes
interseptimal) intervals in rank-2 or rank-3 sizes like 17 or 24, Gene
suggested the name Gentle, which I enthusiastically embrace!

A quick description of the Gentle family as a whole would note a range
of about 703.447-705.0 cents; for those oriented to EDO landmarks,
this would be from 29-EDO to 80-EDO. A 2.3.7.11.13 orientation is the
other defining element of this family, with its undecimal-tredecimal
regular intervals (e.g. 14/11, 13/11, 22/21); neutral intervals
(e.g. 14/13, 26/21, 13/12, 13/8); and septimal or interseptimal
intervals (e.g. 9/7, 7/6, 7/4, 21/16; or, in rank-2 Parapyth or
ChristmasEve, shadings around 128/99, 297/256, 15/13, 13/10, etc.).

Gentle is the heartland of what in 2000 I termed "neo-Gothic tunings,"
and after George Secor in 2001-2002 had directed my attention to
medieval and later Near Eastern music, now more broadly term
"neomedieval." The variety of neutral steps, with four sizes in rank-3
Parapyth and two in rank-2 systems, makes it very attractive for Near
Eastern music.

One neomedieval motto: "Neomedieval needs 5 like a fish needs a
bicycle." This isn't to exclude the extra diversity of 5 when it
occurs, only to say that it's incidental, and that Gentle tunings are
not likely to suit 5-limit tonal harmony.

So the ethos of Gentle and its sufamilies is to be gentle in tempering
the fifth and fourth (especially in Parapyth, where the tempering is
about equal to 12n-EDO, or a bit less in HTT and MET-24, for example);
and to favor different types of rather active and complex thirds and
sixths that can resolve to stable intervals such as unisons or fifths
in directed progressions of a kind emblematic of 13th-14th century
European music based on Pythagorean tuning.

Best,

Margo

[to be continued]

On Mon, 29 Oct 2012, Margo Schulter wrote:

> Here the totally independent and identical results are
> intriguing: maybe like a chess match between Susan Pulgar and
> Deep Blue (a famous computer program of the mid-1990's) which
> ends in a classic draw by perpetual check.

That should be Susan Polgar; I don't how I managed to spell
the name of this Grandmaster wrong, but the least I can do
is a quick correction.

Best,

Margo

Hi All

I got to say this didn't work out too well to my ears. YMMV

http://micro.soonlabel.com/various/20121027_organ.mp3

Though the problem is probably at my end...

Chris

On Thu, Oct 25, 2012 at 3:27 PM, <chrisvaisvil@gmail.com> wrote:

> **
> Thanks to Dan from sonic couture I have a kontakt script that can do the
> 12 one easily on their replica harpsichords. I'm excited to try this a
> little later today after seeing Jake's exclamation in his message. Jake it
> sounds like you think this is pretty special.
>
> Chris
> *
> ------------------------------
> *From: * "Keenan Pepper" <keenanpepper@...>
> *Sender: * tuning@yahoogroups.com
> *Date: *Wed, 24 Oct 2012 01:58:14 -0000
> *To: *<tuning@yahoogroups.com>
> *ReplyTo: * tuning@yahoogroups.com
> *Subject: *[tuning] Re: Three similar temperaments
>
>
>
> --- In tuning@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:
> > ! parapyth12.scl
> > !
> > A triple Fokker block of the 2.3.7.11.13 temperament called "parapyth"
> (TOP tuning)
> > 12
> > !
> > 58.23604
> > 206.95866
> > 265.19471
> > 413.91733
> > 472.15337
> > 554.50965
> > 703.23227
> > 761.46832
> > 910.19094
> > 968.42698
> > 1117.14960
> > 1199.50588
>
> Just noting that it would be really sweet to tune a piano (harpsichord?
> organ?) to this because it has 12 roughly equally spaced notes, and 10 out
> of 12 fifths are really good, but it also has TWO really good 4:6:7:9:11
> chords (on the root and the fifth of the mode shown).
>
> The only way to get more than 10 really good fifths (11 instead of 10)
> would be to use a 12-note MOS of a temperament whose generator is a really
> good fifth, meaning Pythagorean[12] (or leapday[12] or whatever you want to
> call it). Compared to this parapyth-12 scale, that idea is pretty boring.
>
> Keenan
>
>
>

--- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@...> wrote:
>
> Hi All
>
> I got to say this didn't work out too well to my ears. YMMV

I liked the harmony, if that's the problem.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "gdsecor" <gdsecor@> wrote:
>
> > I need to correct my statement, because I was making the assumption that ratio you suggested would have led to the most "obvious" way to include 5. Now that I've taken the time to make a calculation, I see that it was 5103:5120 that I tried (and rejected).
>
> Do either you r Margo have a name for the rank 3 temperament tempering out 325/324, 352/351 and 364/363?

I know of no name for it.

--George

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:

> There's another comma tempered out: 441/440!

Together with 352/351 and 364/363, that's pele.

> And the 11/10 occurs in the basic 12-MOS from 10 generators down,
> a size where there are no 5-limit thirds -- and likewise in a
> 17-MOS.

Again, pele.

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:

> 2. Now find the spacing generator or distance between
> two chains of (2/1, 703.723) so that 7/6 and 13/8
> are equally impure (57.423 cents).

I wonder what 13-limit intervals temper to this 57.423 cent interval? The POTE pele tuning of 28/27 is 57.468 cents for what that is worth.

>> There's another comma tempered out: 441/440!

> Together with 352/351 and 364/363, that's pele.

I'd say pele-compatible or pele-expandable, as discussed in some
discussions with Mike; but parapyth is not pele.

The difference, as you neatly explained for meantone, is that the
abstract temperament of pele is 13-limit (with 5), while the
abstract temperament of parapyth is 2.3.7.11.13.

>> And the 11/10 occurs in the basic 12-MOS from 10 generators down,
>> a size where there are no 5-limit thirds -- and likewise in a
>> 17-MOS.

> Again, pele.

Again, parapyth (for something like MET-24 or O3), because the
abstract temperament is 2.3.7.11.13.

The abstract temperament concept which you and Mike have taught
me nicely permits 2.3.7.11.13 parapyth to be recognized as a
distinct entity, even though any set of 24 or larger will have
some kind of pathway (pele or leapday) to 5 that could be quite
deliberately and systematically exploited in a larger tuning
set.

Best,

Margo

> I wonder what 13-limit intervals temper to this 57.423 cent
> interval? The POTE pele tuning of 28/27 is 57.468 cents for what
> that is worth.

Dear Gene,

Please let me address your question about 57.423 cents or a similar
spacing generator in parapyth, after taking note, just to avoid any
possible misunderstandings, that I regard 2.3.7.11.13 abstract
temperament members such as MET-24 and O3 (I can only speak for my own
tunings) as definitely parapyth, rather than 13-limit pele, although
they are indeed pele-compatible or pele-expandable.

Certainly 57.423 cents represents 28/27, for example if we think of
two fifths up as the 9/8, and two fifths up plus spacing as 7/6. So
that's part of the answer.

But remember, parapyth and gentle more generally temper out 896:891,
352:351, 364:363, and 10648:10647. So 57.423 cents (in my first
"canonical" version of MET-24, of which I now suspect I could have
mistakenly had a 17-note subset loaded in Scala when I thought it was
Graham's computed parapyth POTE or the like) represents not only
28/27, but lots else.

For one, 91/88: the difference between 44/39 and 7/6, or 22/13 and
7/4. A beautiful example is Secor's 29-HTT, where 22/13 is within
1/20-cent or so of just, and 7/4 is pure, so the spacing of 58.090
cents is also a near-just 91/88. That's the textbook illustration,
where everything is just about infinitessimally close to JI.
Note that the difference of 28/27 and 91/88 is 352/351, a very
important unison vector for the whole gentle family.

For another, 33/32, very notably, for example, the difference between
4/3 (one fourth up) and 11/8 (one fourth up plus spacing). Now 33/32
is equal to 28/27 less 896/891, another basic unison vector for all of
gentle.

For yet another, 121/117, the difference between 13/11 and 11/9, both
of which these temperaments tend to get quite accurately as three
fourths up, and three fourths up plus spacing.

And the 121/117 gives a great illustration of three of our four basic
commas for the gentle family. It differs from 28/27 by 364/363, the
gentle comma; from 33/32 by 352/351; and from 91/88 by 10648/10647.

So "What does a spacing generator like 57.423 cents in rank-3 parapyth
stand for?" is a great introduction to the four gentle family commas.

(13/11-11/9)
58.198
121:117
|-----------------------|---|-----------------------|
33:32 91:88 28:27
53.273 58.036 62.961
(4/3-11/8) (22/13-7/4) (9/8-7/6)
|------------------------|---|-----------------------|
364:363 10648: 364:363
4.763 10647 4.763
0.163
|----------------------------|
352:351 |---------------------------|
4.925 352:351
4.925
|----------------------------------------------------|
896:891
9.688

Here's one possible JI version of MET-24 to illustrate these intervals
and commas. This is a Constant Structure and JI epimorphic, as it
happens, so I'm going with this one. Note that 10/9 and 5/3 are not
part of the abstract 2.3.7.11.13 tuning structure, but do arise under
it, with the question left open as to whether 49/44 and 147/88 at
441/440 larger (respectively 14/13 and 21/13 plus 91/88), for example,
might better express the precise rational structure. But musically,
a JI version with 10/9 and 5/3 has the advantage of avoiding a fifth
a full 896/891 wide (189/128-49/44); so the motivation for the simpler ratios is to keep all 22 usual fifths no more than 5120/5103 from pure
(a ratio I just noticed, and have seen mentioned on this group, with
the fifth 189/128-10/9 tempered wide by this amount).

<http://www.bestII.com/~mschulter/met24-ji1.scl>

! met24-ji1.scl
!
Possible JI interpretation of MET-24
24
!
91/88
14/13
10/9
9/8
7/6
13/11
11/9
14/11
21/16
4/3
11/8
63/44
189/128
3/2
14/9
21/13
5/3
22/13
7/4
39/22
11/6
21/11
63/32
2/1

And here's the 2048-EDO version of MET-24:

<http://www.bestII.com/~mschulter/met24-canonical.scl>

! met24-canonical.scl
!
Smoothed MET-24 in 2048-EDO, generators (2/1, 703.711c, 57.422c)
24
!
57.42188
125.97656
183.39844
207.42188
264.84375
288.86719
346.28906
414.84375
472.26563
496.28906
553.71094
622.26563
679.68750
703.71094
761.13281
829.68750
887.10938
911.13281
968.55469
992.57812
1050.00000
1118.55469
1175.97656
2/1

Best,

Margo

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:
>
> >> There's another comma tempered out: 441/440!
>
> > Together with 352/351 and 364/363, that's pele.
>
> I'd say pele-compatible or pele-expandable, as discussed in some
> discussions with Mike; but parapyth is not pele.

Parapyth also doesn't temper out 441/440--do that, and it turns into pele.

> The difference, as you neatly explained for meantone, is that the
> abstract temperament of pele is 13-limit (with 5), while the
> abstract temperament of parapyth is 2.3.7.11.13.
>
> >> And the 11/10 occurs in the basic 12-MOS from 10 generators down,
> >> a size where there are no 5-limit thirds -- and likewise in a
> >> 17-MOS.
>
> > Again, pele.
>
> Again, parapyth (for something like MET-24 or O3), because the
> abstract temperament is 2.3.7.11.13.

It can't be, because 11/10 has a 5 in it. (4/3)^10/(16*(11/10)) = 655360/649539, and adding that comma to parapyth adds 5 to the group and converts it to pele.

> The abstract temperament concept which you and Mike have taught
> me nicely permits 2.3.7.11.13 parapyth to be recognized as a
> distinct entity, even though any set of 24 or larger will have
> some kind of pathway (pele or leapday) to 5 that could be quite
> deliberately and systematically exploited in a larger tuning
> set.

Right. But if you exploit it, you own it. I get the feeling we are talking past each other--you are discussing approximations which appear in parapyth, but not with an eye to using them?

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:

> Please let me address your question about 57.423 cents or a similar
> spacing generator in parapyth, after taking note, just to avoid any
> possible misunderstandings, that I regard 2.3.7.11.13 abstract
> temperament members such as MET-24 and O3 (I can only speak for my own
> tunings) as definitely parapyth, rather than 13-limit pele, although
> they are indeed pele-compatible or pele-expandable.

Since your JI version, or "transversal", of MET-24 contains 5-limit intervals it seems to me that "definately" is awfully strong. Your scale has three 5/4 intervals in it, for instance--one pure, one off by 352/351, and one off by 2080/2079. Add that last, and once again we get pele, not parapyth.

> For one, 91/88: the difference between 44/39 and 7/6, or 22/13 and
> 7/4.

The POTE parapyth tuning of 91/88 is 58.338 cents, again for what that's worth, but I see POTE pele was much closer to your value for this interval.

>> Again, parapyth (for something like MET-24 or O3), because the
>> abstract temperament is 2.3.7.11.13.

> It can't be, because 11/10 has a 5 in it. (4/3)^10/(16*(11/10)) =
> 655360/649539, and adding that comma to parapyth adds 5 to the
> group and converts it to pele.

In your worldview, yes; but in mine, your reasonable point about
11/10 gives rise to this: 2.3.11/10.7.11.13. I'd call it a
variant of parapyth, because the object is isn't 5-limit thirds,
but a tetrachord like al-Farabi's 1/1-9/8-99/80-4/3 or
9:8-11:10-320:297.

Note that this tetrachord, or pentachord 9:8:11:10-320:297-9:8 has
the 11/10 superparticular neutral step, but no 5-limit
thirds. Rather, it has the submajor and supraminor thirds that
are a big theme of parapyth. And it's been no secret since 2002
that almost any parapyth rank-3 tuning, if carried to 24 or more
notes, will have some approximations of 5 via either the pele or
leapday route: one can't avoid it.

>> The abstract temperament concept which you and Mike have
>> taught me nicely permits 2.3.7.11.13 parapyth to be recognized
>> as a distinct entity, even though any set of 24 or larger will
>> have some kind of pathway (pele or leapday) to 5 that could be
>> quite deliberately and systematically exploited in a larger
>> tuning set.

> Right. But if you exploit it, you own it. I get the feeling we
> are talking past each other--you are discussing approximations
> which appear in parapyth, but not with an eye to using them?

We probably are talking past each other, but I would say that the
incidental 5-limit approximations that do occur in a parapyth
tuning like MET-24 or O3, etc., should be used with caution and
discretion, as opposed to the 11:10 neutral second which is meant
to be used routinely, for example in a bright Syrian Rast where
the relevant thirds are submajor and supraminor (around 333.4 and
370.3 cents in MET-24 for 2048-EDO, which has the convenience of
letting me quote single values for each interval type).

Before we get to the three 5/4 thirds, we have four other
interval families: regular (e.g. 14/11 and 13/11),
submajor/supraminor (e.g. 26/21 and 63/52), septimal (e.g. 7/6
and 9/7) and central neutral (e.g. 16/13 and 11/9).

All of these other common families are a lot more complex than
5/4, and trying to rely too much on the incidental 5-limit
thirds may make almost everything else seem out of tune --
although using them as remote intervals now and then can add
further to the diversity of the tuning. For example, a
39:42:49:52 Hijaz of the kind I mentioned to Mike is fine as a
special effect; but attempts to use MET-24 for any kind of
5-limit style are apt to run into the rather comical scarcity of
these intervals for that purpose.

While 11:10 neutral thirds were part of the design, 5/4 and 5/3
were incidental effects neither optimized for nor deliberately
avoided. And note that 11/10 (162.891 cents) is considerably
closer to just than 5/4 (390.820 cents), whereas an RMP scheme
going for 5/4 would presumably optimize differently.

And from my point of view, the results here and in O3 are mostly
musically interchangeable. The main difference is that MET-24 is
kinder on the fifths, especially in 1024-EDO where six of 11 in
each 12-MOS are impure by only 1.17 cents.

And since 11/10 is a part of the optimization while 5/4 isn't,
applying a POTE model for purposes of classification could
distract from the musically most relevant analysis. This happens
to theorists and composers of various eras, for example when
modern writers try to read major/minor tonality into 13th-14th
century European music.

The "pele-compatible" or "pele-expandable" label neatly resolves
this and avoids lots of arguments.

Best.

Margo

> Since your JI version, or "transversal", of MET-24 contains
> 5-limit intervals it seems to me that "definately" is awfully
> strong. Your scale has three 5/4 intervals in it, for
> instance--one pure, one off by 352/351, and one off by
> 2080/2079. Add that last, and once again we get pele, not
> parapyth.

Actually, that may be an artefact of starting on C as reckoned on
my two Halberstadt MIDI keyboards, where it's rather natural to
choose 10/9 and 5/3 rather than look for some creative schismas
or whatever <grin>. And, of course, this is ex post facto: when I
designed MET-24, I was looking for the point of maximum evenness
in 1024-EDO (600-601-600-601-600-601-600-601-600-601-600 steps
for the fifths in a 12-MOS).

Let's do our traversal from A instead, and we actually get a
better JI tuning from the viewpoint of each of the 22 usual
fifths being tempered by no more than 352:351.

<http://www.bestII.com/~mschulter/met24-ji3_A.scl>

! met24-ji3_A.scl
!
JI interpretation of MET-24, 1/1 is A or 22/13 of C-C version
24
!
91/88
22/21
13/12
9/8
7/6
13/11
11/9
14/11
21/16
4/3
11/8
88/63
13/9
3/2
273/176
11/7
13/8
22/13
7/4
39/22
11/6
21/11
63/32
2/1

Here's a similar traversal, if that's the right term, that I
posted here in the dialogue that Mike and I are having:

<http://www.bestII.com/~mschulter/met24-lattice1.jpg>

And any which way we slice this, there are going to be some
rather close 5-limit approximations: in order to avoid them, if
that were my intent, I would have had to compromise other goals
of MET-24. That's why it's indeed pele-compatible, although
someone who wants it to be pele-effective would probably expand
the design to 34 or more notes.

> The POTE parapyth tuning of 91/88 is 58.338 cents, again for
> what that's worth, but I see POTE pele was much closer to your
> value for this interval.

Let's analyze what's happening here from a neomedieval
perspective, and I think that we'll find there's no independent
intent to optimize 5-limit thirds -- as well as maybe find some
criteria that may suggest when such intent may be present --
criteria not met here. I'll address first the general tuning
design, then the question of the spacing for 91/88.

With pele, as I understand it, there is in effect a compromise
between two important ratios for the third or spacing generator
of the rank-3 design.

For 7, we want 22/13 (typically close to just in this range for
either parapyth or pele) plus 91/88 to equal 7/4, as it does in
Secor's 29-HTT (703.579 cents). That's true in either parapyth or
pele.

For the 5 of Pele, however -- not 11/10 as the main focus, but
5-limit thirds -- we want 63/52 (just in 29-HTT) plus 65/63 to
equal 5/4.

And between 91/88 and the smaller 65/63 of pele, there's a comma
of 441/440.

Now there's a way to lean toward 5 as we wouldn't do in Parapyth,
and we don't do in MET-24 -- temper the fifth very lightly, so
that 63/52 is actually narrower than just! The reason is that the
63/52 grows (and the spacing for a just 5/4 shrinks) at nine
times the rate of growth for the fifth, while 22/13 grows toward
7/4 at only three times this rate. In my conclusion, I'll get
into this a bit more.

So POTE pele, if I have it right, has a fifth way lower than
MET_24: around 703.4 cents as opposed to 703.7 cents.

In terms of the 28672/28431, which is actually the most relevant
comma for MET-24 as designed (this 441/440 discussion is
fascinating, but very much an afterthought), we have a tempering
of about 1/8 of this Secorian comma -- meaning that 26/21 and
21/13 are very close to just (370.312 and 829.688 cents, with
just values at 369.747 and 830.253 cents).

For POTE pele, we want less than 1/9-Secorian comma. And that
would be contrary to one of the main optimization criteria for
MET-24: both 14/13 and 11/10 should be within 3 cents of just in
the 1024-EDO version. In the 2048-EDO version, 11/10 is impure by
2.114 cents, and 14/13 by 2.322 cents.

That's one good choice for parapyth, but not so good for pele,
because it causes us to overshoot 5/4 by 4.507 cents. Note that
the relevant 2.3.7.11.13 ratios, and also 11/10, generally do
considerably better.

Since the linear generator is simply wrong for POTE pele, why set
91/88 at 57.422 cents rather at 58.594 cents (the next available
size in 1024-EDO), say?

For me, the main reason was to favor 11/8 and 13/8, albeit at the
expense of 9/7 (often more of a 352/273 in the 1024-EDO version,
as reflected by my JI traversal above). One result is that 7/6
and 13/8 are about equally impure -- these sizes, 264.844 cents
and 842.578 cents, are built from an even number of linear
generators plus spacing, and so are constant in 1024-EDO also,
respectively impure by 2.027 cents in the narrow direction and
2.050 cents in the wide direction. So it's a classic balance for
usual 2.3.7.11.13 purposes.

The POTE pele compromises lots of intervals, with 13/7 more than
four cents wide, and 14/13 equally narrow -- not bad, but
different from MET-24 (and a "compromise" only from that point
of view!). And likewise with 14/11, and more subtly
with 13/12 and 12/11.

Please let me emphasize that none of these things are faults,
only signs that we're exploring a slightly different part of the
continuum than MET-24. In fact, the POTE pele has a virtually
just third that I'll mention below.

Also, this may be a rather trivial point, but Mike Battaglia's
parameters for the POTE at (2/1, 703.414, 57.468) result in 7/4
at always a bit more than a cent narrow -- a compromise to
improve 5/4, as the POTE algorith would lead us to expect.

At MET-24 (2/1, 703.711, 57.422), the 22/13 is a bit larger
rather than smaller than pure, so 57.422 (a bit narrow of a just
91/88) is actually close to optimal for 7/4. In 1024-EDO, it is
the only spacing to keep 7/4 always within a cent of just. That,
plus the almost perfect balance between 7/6 and 13/8, are more
than ample justification for its use.

In short, the MET-24 criteria of "tune all four superparticular
neutral seconds within three cents of just in 1024-EDO, and keep
7/4 always within a cent of just" would not be the best criteria
if this were pele.

Another test to confirm this musical reality is to look at
results with SHOW LOCATIONS 4:6:7:9:11:13, where both POTE pele
and MET-24 (1024-EDO or 2048-EDO) do quite well. Then try this:
SHOW LOCATIONS 4:5:6:7:9:11.

In POTE pele, 5/4 will be wide by 1.88 cents -- a tribute to the
optimization for 5.

In MET-24, 5/4 will be wide by 3.921 or 5.093 cents in the
1024-EDO version, and consistently by 4.507 cents in the 2048-EDO
version. This makes sense for an incidental interval which the
equations -- rather fascinating ones -- cast at our feet, but not
for 5/4 in a 5-limit optimization of near-just quality, as MET-24
seeks to maintain for 2.3.11/10.7.11.13.

To sum up, "keep 7/4 always within a cent of just in 1024-EDO,
and 13/8 within 2 cents or so of just" is sufficient to account
for the 57.422-cent spacing.

Now for the last point I promised to cover: the rates of growth
for 22/13 and 63/52. Let's compare POTE pele and MET-24:

5th spacing Maj6 Maj6-7/4 Aug2 Aug2-5/4
POTE Pele 703.414 57.468 910.242 58.584 330.726 55.588
MET-24 703.711 57.422 911.133 57.693 333.398 52.916

Note how in POTE pele, the spacing of 57.468 cents is a good
compromise between the 58.584 cents we'd need for a just 7/4, and
the 55.588 cents we'd need for a just 5/4.

In MET-24, however, an almost identical spacing of 57.422 cents
is almost exactly the 57.693 cents we need for a just 7/4 with
our slightly larger 22/13 -- but way beyond the 52.916 cents we'd
need for a just 5/4, what with the growth of our 63/52 by over
2.5 cents. It fits 2.3.11/10.7.11.13 -- but 13-limit, not so
much.

However, MET-24 is only one possible shading or region to
explore, and the POTE pele tuning has an interval of special
interest: a virtually just supraminor third at 23/19, 330.726
cents as compared with a just 330.761 cents.

Indeed. we could define a near-POTE pele generator of 703.418
cents or (736/19)^(1/9). Here we are tempering out the difference
between the Pythagorean augmented second 19683/16384 and 23/19, a
comma of 376832/373977 (13.166 cents). Thus the tempering is
1.463 cents.

And here we get a superparticular comma or schisma temperaed out
by 29-EDO, for example: the difference between 23/19 and 63/52,
or 1197/1196 (1.447 cents). If this hasn't been named, I might
propose "29-EDO supraminor comma," since the actual supraminor
third or augmented second (9 fifths up) is 331.034 cents, about
0.273 cents wide of 23/19 and 1.174 cents narrow of 63/52
(332.208 cents).

In fact, both POTE pele and MET-24 have good approximations of
63/52, from different directions. Each tempering is beautiful,
and I'm just pointing out that the purposes and designs are a bit
different.

Best,

Margo

Dear Gene,

Here's another querying of the idea that MET-24 "tempers out
the 441/440" in such a way as to make 5/4 significantly less
impure than 441/352 (390.244 cents), which would be one
meaningful criterion for a pele optimization. However, as I
discuss below, the matter of Ptolemy's Equable Diatonic suggests
that MET-24 may have some interesting properties which are yet
distinct from those of pele. Let's focus for the moment on
441/352.

In the 1024-EDO version of MET-24, we find the relevant interval
sizes at 390.234 cents (almost identical to 441/352, 0.010 cents
narrow); and 391.406 cents (1.162 cents wide).

In the 2048-EDO version, we have 390.820 cents, or 0.576 cents
wide.

So, from the standpoint of optimizing prime 5 (i.e. 5/4, which I
understand is what prime 5 especially refers to in POTE), we get
either a virtually just 441/352, or something a tad wider. This
is not an "improvement" of 5/4, quite unlike pele.

Now are there 2.3.7.11.13 explanations -- and keeping 11/10
within three cents of just is not even a necessary explanatory
factor here -- that can account for a 441/352 without any intent
to optimize 5/4 and possibly sacrifice the accuracy of other
intervals in order to do so (as is done in POTE pele with 14/11,
certainly, and more subtly with 7/4)?

As it happens, 441/352 is the difference between 22/21 and 21/16,
for example -- both 2.3.7.11.13 intervals which a neomedieval
outlook and MET-24 in particular do seek to optimize. The 22/21,
occuring in Ptolemy's Intense Chromatic and in various medieval
Near Eastern sources, is a superparticular or epimoric limma much
sought; and the 21/16 is beautiful in 16:21:24:28, for example,
as found in the 1-3-7-9 hexany.

<http://www.bestII.com/~mschulter/met24-lattice1.jpg>

Now with 441/352, and fifths that are wide by 1.756 cents on the
average (or consistently in the 2048-EDO version), we're going to
be very close to 10/9 and 5/3, as appears in that JI transversal
from C. But it carries the possible inference of a just 5/4 which
simply doesn't hold because the fifths are actually tempered!
When the impurity is around 441/440 or greater, this becomes in
my view a nontrivial point in distinguishing parapyth from pele.

Note that the JI sequence 39:42:49:52 has 49/39 (395.169 cents)
reduced to 441/352 -- as we might expect in a system which, by
definition for parapyth and more broadly the gentle family,
tempers out the 352/351. This is different from pele, which seeks
a more accurate 5/4 than 441/352.

MET-24 may be a just about perfect optimization not only for my
intended 1/1-9/8-99/80-4/3 (9/8-11/10-320:297) or whatever, but
also for Ptolemy's Equable Diatonic 12:11:10:9, in a mode of two
disjunct tetrachords 1/1-12/11-6/5-4/3-3/2-18/11-9/5-2/1.

Note that there is no 5/4 interval above the 1/1, while 12/11 and
18/11 are virtually just. Here's a study on some epimores as
represented in MET-24, with a section on the Equable Diatonic:

<http://www.bestII.com/~mschulter/Epimores_temper.txt>

From this standpoint, both the accuracy of the Equable Diatonic
and the relatively lower accuracy of 5/4 suggest that the
notation 2.3.11/10.7.11.13 may best represent the nature of the
MET-24 tuning.

Best,

Margo

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:

> 5th spacing Maj6 Maj6-7/4 Aug2 Aug2-5/4
> POTE Pele 703.414 57.468 910.242 58.584 330.726 55.588
> MET-24 703.711 57.422 911.133 57.693 333.398 52.916

I get this:

POTE Pele: 703.414 57.468
13-limit minimax Pele: 703.522 57.750 eigenmozos: 9/5, 13/9
15-limit minimax Pele: 703.410 57.078 eigenmonzos: 5/3, 13/9

POTE Parapyth: 703.856 58.338
13-limit minimax Parapyth: 703.579 58.090 eigenmonzos: 7, 13/9
15-limit minimax Parapyth: 703.579 58.090 eigenmonzos: 7, 13/9

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:

> > It can't be, because 11/10 has a 5 in it. (4/3)^10/(16*(11/10)) =
> > 655360/649539, and adding that comma to parapyth adds 5 to the
> > group and converts it to pele.
>
> In your worldview, yes; but in mine, your reasonable point about
> 11/10 gives rise to this: 2.3.11/10.7.11.13. I'd call it a
> variant of parapyth, because the object is isn't 5-limit thirds,
> but a tetrachord like al-Farabi's 1/1-9/8-99/80-4/3 or
> 9:8-11:10-320:297.

In my lexicon, something like "2.3.11/10.7.11.13" is on a different level of abstraction than it seems to be in yours, and involves group theory. The result is that 2.3.11/10.7.11.13 and 2.3.5.7.11.13 mean exactly the same thing. In Graham's temperament finder they don't, but someone else will have to unravel the mysteries of what, exactly, it does and why.

> We probably are talking past each other, but I would say that the
> incidental 5-limit approximations that do occur in a parapyth
> tuning like MET-24 or O3, etc., should be used with caution and
> discretion, as opposed to the 11:10 neutral second which is meant
> to be used routinely, for example in a bright Syrian Rast where
> the relevant thirds are submajor and supraminor (around 333.4 and
> 370.3 cents in MET-24 for 2048-EDO, which has the convenience of
> letting me quote single values for each interval type).

You can see how from my point of view this distinction can't really be drawn, since 2.3.11/10.7.11.13 and 2.3.5.7.11.13 are the same. I'm not sure what the point of drawing it is. If your scale has 5/4s or 5/4s in it, why be cautious in their use other than as a matter of style? I can't see how style should figure in these definitions.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, Margo Schulter <mschulter@> wrote:

> In my lexicon, something like "2.3.11/10.7.11.13" is on a different level of abstraction than it seems to be in yours, and involves group theory. The result is that 2.3.11/10.7.11.13 and 2.3.5.7.11.13 mean exactly the same thing.

However, a minimax or least-squares target which consists of the no-5s 13 or 15 limit plus {11/10, 20/11} or {11/10, 13/10, 20/13, 20/11} does *not* mean the same thing as the full 13 or 15 limit. In fact, if you only add 11/10, 13/10 and their inversions, you get exactly the same minimax tuning as parapyth for pele, except that pele has a tuning (2790.298 cents) for 5. So just toss 390.298 cent major thirds in wherever it's convenient, don't change the tuning at all, and call that pele.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:

> POTE Pele: 703.414 57.468
> 13-limit minimax Pele: 703.522 57.750 eigenmozos: 9/5, 13/9
> 15-limit minimax Pele: 703.410 57.078 eigenmonzos: 5/3, 13/9
>
> POTE Parapyth: 703.856 58.338
> 13-limit minimax Parapyth: 703.579 58.090 eigenmonzos: 7, 13/9
> 15-limit minimax Parapyth: 703.579 58.090 eigenmonzos: 7, 13/9
>

Both pele and parapyth have a 13/9 eigenmonzo in the 13 and 15 limits. If we keep that feature descending to rank 2 from pele, we get rodan, mystery, misty and cassandra in the 13-limit, and rodan, misty and cassandra in the 15-limit. Cassandra has a generator of a fifth; so does misty, but it divides the octave into three for a period of 1/3 octave. Cassandra, however, is a little less accurate than the others.

On Fri, Oct 26, 2012 at 9:23 AM, genewardsmith <genewardsmith@...>
wrote:
>
> Good idea. You can start by translating this:
>
> "Here Peppermint is a prototype of the Parapyth family, with O3,
> MET-24, and Graham Breed's 46-17-41 tuning almost identical to
> MET-24 as other rank-3 members. There's also an extended rank-2
> form aimed at another class of complex and beautiful intervals
> (guess which -- I'll reveal all in my next post), but rank-3 is
> the prototype and the classic 2.3.7.11.13 form."
>
> into terms you know I'll understand.

Yeah, so, I may be MIA for a little bit because of this hurricane.
I'll respond to everything once I can figure out what's going on with
my apartment in NYC (I'm in Philly right now).

-Mike

>> 5th spacing Maj6 Maj6-7/4 Aug2 Aug2-5/4
>> POTE Pele 703.414 57.468 910.242 58.584 330.726 55.588
>> MET-24 703.711 57.422 911.133 57.693 333.398 52.916

> I get this:

> POTE Pele: 703.414 57.468
> 13-limit minimax Pele: 703.522 57.750 eigenmozos: 9/5, 13/9
> 15-limit minimax Pele: 703.410 57.078 eigenmonzos: 5/3, 13/9
> POTE Parapyth: 703.856 58.338
> 13-limit minimax Parapyth: 703.579 58.090 eigenmonzos: 7, 13/9
> 15-limit minimax Parapyth: 703.579 58.090 eigenmonzos: 7, 13/9

This looks very reasonable to me, and I'd like to do my part to
de-escalate any tensions that may have arisen in this thread. This
should be a friendly cultural exchange, in which we seek to understand
our different worldviews and approaches to tuning design and
optimization. It should be at least mutually amusing, and perhaps
mutually edifying.

Let's look at your numbers above as one way to understand why I find
it very curious that MET-24 would be described as an instance of pele,
although I would agree that someone looking for a pele pathway will
find it. The simple fact, as I see it, is that MET-24 uses a way
larger linear generator or 703.711 cents, way beyond POTE pele and
somewhat beyond George Secor's HTT at 703.579 cents.

One indication of this is that in 1978, George _didn't_ rely on pele
to get 5/4: instead, he used an ingenious irregular 29-note structure
to make 5/4 just! That to me is the dog that didn't bark, alerting
us that 703.711 cents in MET-24 is simply too high for this purpose
when George, about the most ingenious designer out there, evidently
found 703.579 cents a bit high for a pele path to 5/4.

A simple sense of scale and place will show that MET-24 is much closer
to POTE parapyth than to POTE pele:

POTE Secor's POTE
pele HTT MET-24 parapyth
|------------------------|---------------|--------------|
703.414 703.579 703.711 703.856

Commas or dieses provide another view of this. At the POTE pele tuning
we have a near-eigenmonzo of 23/19 for the augmented second or
supraminor third. Now from 23/19 to 5/4 is 95/92 or 55.552 cents, as
compared to the 91/88 or 58.036 cents -- a comma of 2093/2090, or
2.483 cents. That isn't so hard to finesse so that 5/4 and 7/4 are
both at or very close to just, as happens in POTE pele.

But even at Secor's HTT generator, 703.579 cents, we essentially have
91/88 as the spacing for 7/4, but the considerably smaller 65/63 (from
the eigenmonzo of 63/52 to 5/4) for 5/4. That's a full 441/440 comma.

Now let's quickly recap what happens in MET-24 at 703.711. Here 22/13
is in 2048-EDO a bit larger than just, so the spacing for a pure 7/4
is actually a bit less than 91/88: 57.693 cents. But the ideal spacing
for 5/4 has shrunk to 52.916 cents. Now the 7/4 spacing is a bit
narrow of 91/88, and the 5/4 spacing almost equally narrow of 33/32,
so the comma has effectively grown to about 364/363.

In my view, that's simply not a good optimization strategy for someone
following the pele agenda and seeking 5/4. We seem to agree that pele
optimizes below HTT, and MET-24 deliberately goes a bit above it.

Now, as I see it, it's impossible to optimize for 2.3.7.11.13 in this
general region and resolve to make the fifths a bit closer to pure
than in 12n-ED2, as in MET-24, and not get a pele pathway. It
happens in POTE Parapyth, also: carry this to 34 notes (2x17-MOS), and
we get 12 thirds closer to pure than in 1/6-comma meantone. Maybe
we might seek out fine points of the exact name for the pathway at
different temperings, or the precise commas involved, but in practice
we have quite recognizable approximations of 5/4 all around the parapyth
region (around 703.447-704.2 cents). And likewise for ChristmasEve
(704.2-704.5 cents), known from a 13-limit perspective as leapday.

In fact, POTE Parapyth in 24 notes (393.043 cents), to take the set
analogous to MET-24, isn't that far from a pure 14/11 tuning at
704.377 cents in the size of its closest 5/4 approximation, there
391.917 cents. It's like different shades of meantone. And if we
carry POTE Parapyth to 58 notes (2x29-MOS), we get 380.976 cents
(a bit closer than 1/3-comma meantone or 19-EDO) via the leapday
route (21 fifths up, or thrice the apotome, on a single chain).

In deciding whether something is actually an intentional example of
pele or leapday, for example, or rather a pele-compatible or
leapday-compatible temperament, here are three questions which I find
helpful:

(1) Is the tuning size large enough supply a reasonable number
of 5-limit thirds? This would mean a 29-MOS for leapday
(as that name suggests), or a 34-note set for pele.
If the set is smaller, it's probably trying to do other
things, with a pathway to 5 as a side effect.

(2) Is the linear generator reasonably optimized for the
relevant pathway to 5? If POTE pele is 703.414 cents, and
MET-24 is at 703.711 cents, that may be a hint that pele is
not the point, although pele-compatibility might be a
consequence.

(3) What's the tuning ethos or culture: what styles does it
seek to support (not that anyone is bound to those, just
that they may reflect the designer's intent)? When a tuning
gives high priority to optimizing 14/13 and 13/11 as usual
thirds, the presence of a few remote 5/4 approximations
less precise than a long list of 2.3.7.11.13 ratios which are
intentionally optimized may again suggest the distinction
between the design and the interesting consequences.

Best,

Margo

> Yeah, so, I may be MIA for a little bit because of this hurricane.
> I'll respond to everything once I can figure out what's going on
> with my apartment in NYC (I'm in Philly right now).

Dear Mike,

Thank you for reminding us of the tragedy and devastation which has
taken place -- and also of the wise precautions taken to minimize the
damage and protect people in many places. My prayers and warmest
wishes to you and your family!

Of course, thank you also for the very friendly and educational
dialogue we've been having! May your spirit of friendship and
cooperation, shared by many others, help you in NYC.

In peace and love,

Margo

On Tue, 30 Oct 2012, Margo Schulter wrote:

> (3) What's the tuning ethos or culture: what styles does it
> seek to support (not that anyone is bound to those, just
> that they may reflect the designer's intent)? When a tuning
> gives high priority to optimizing 14/13 and 13/11 as usual
> thirds, the presence of a few remote 5/4 approximations
> less precise than a long list of 2.3.7.11.13 ratios which are
> intentionally optimized may again suggest the distinction
> between the design and the interesting consequences.

Obviously here I meant 14/11 and 13/11 as usual thirds: 14/13 is
absolutely wonderful, but not quite a third <grin>.

Best,

Margo

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:
>
> This looks very reasonable to me, and I'd like to do my part to
> de-escalate any tensions that may have arisen in this thread. This
> should be a friendly cultural exchange, in which we seek to understand
> our different worldviews and approaches to tuning design and
> optimization. It should be at least mutually amusing, and perhaps
> mutually edifying.

I'm sorry if I've been excessively brusque; I wasn't aware of any tensions and did not mean to convey any. There is a difficulty which one could call a tension I suppose arising from the fact that we have such different actual and conceptual vocabularies. If I say I don't understand something, please take that literally.

> Let's look at your numbers above as one way to understand why I find
> it very curious that MET-24 would be described as an instance of pele,
> although I would agree that someone looking for a pele pathway will
> find it. The simple fact, as I see it, is that MET-24 uses a way
> larger linear generator or 703.711 cents, way beyond POTE pele and
> somewhat beyond George Secor's HTT at 703.579 cents.

But pretty close to the 13/9, 7/6 eigenmonzo tuning of 703.782. In any case, of course, I don't define temperaments by tunings, but according to what JI intervals they temper and what they temper out.

> One indication of this is that in 1978, George _didn't_ rely on pele
> to get 5/4: instead, he used an ingenious irregular 29-note structure
> to make 5/4 just!

Was that irregular structure basically a modification of pele, however? I presume you mean the scale secor29htt.scl; I think it might be worthwhile to analyze that in terms of closeness to a regular temperament.

Wow, it looks like some really productive discussion has been going on here! I just want to chime in and try to clarify a few things.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Margo Schulter <mschulter@> wrote:
> > In your worldview, yes; but in mine, your reasonable point about
> > 11/10 gives rise to this: 2.3.11/10.7.11.13. I'd call it a
> > variant of parapyth, because the object is isn't 5-limit thirds,
> > but a tetrachord like al-Farabi's 1/1-9/8-99/80-4/3 or
> > 9:8-11:10-320:297.
>
> In my lexicon, something like "2.3.11/10.7.11.13" is on a different level of abstraction than it seems to be in yours, and involves group theory. The result is that 2.3.11/10.7.11.13 and 2.3.5.7.11.13 mean exactly the same thing.

Yes, as soon as I read Margo's comment I had an urge to say exactly this. In the lexicon of regular temperament theory, "2.3.11/10.7.11.13" means an infinite set of JI intervals, consisting of all possible *combinations* of those listed. Therefore "2.3.11/10.7.11.13" contains 5/4 in it, because 5/4 = (2)^(-3) * (11/10)^(-1) * (11)^(1). Similarly, 2.3.5.7.11.13 contains 11/10 already, so the two subgroups are one and the same.

> In Graham's temperament finder they don't, but someone else will have to unravel the mysteries of what, exactly, it does and why.

I believe I understand what exactly it's doing. The only reason why the results are different for the two different bases is that, for optimization purposes, Graham's finder always uses the L2 norm of the weighted error vector components *in the given basis*. So, even though the two bases are different expressions of the same vector space, the *norm* Graham's finder uses on this vector space depends on the basis.

Let me give numerical examples of this to avoid any confusion:

If you put in Pele in the standard 2.3.5.7.11.13 basis ( http://x31eq.com/cgi-bin/rt.cgi?ets=46p+17c+41p&limit=2.3.5.7.11.13 ) you get this as the TE tuning map:

<1199.498, 1902.618, 2787.030, 3366.302, 4152.316, 4441.951]

so the weighted error vector is

[-0.000418, 0.000349, 0.000257, -0.000749, 0.000240, 0.000321]

and the L2 norm of that is 0.00104. (Note that Graham's finder gives it in cents per octave, and it's also root *mean* square instead of root *sum* square, so that's an overall factor of 1200/sqrt(6) in this case.)

If you took that same tuning map but expressed it in the 2.3.11/10.7.11.13 basis, you would get

Tuning map: <1199.498, 1902.618, 165.788, 3366.302, 4152.316, 4441.951]
Weighted error: [-0.000418, 0.000349, 0.004750, -0.000749, 0.000240, 0.000321]
L2 norm: 0.00486

Note that the component corresponding to 11/10 is much larger than the others, so it dominates the L2 norm. In other words, because 11/10 was given in the basis, the algorithm isn't treating it as a complex interval, but as a "prime" interval that's much smaller than 2, and hence more important to optimize.

So, if you put in Pele in the 2.3.11/10.7.11.13 basis ( http://x31eq.com/cgi-bin/rt.cgi?ets=46p+17p+41p&limit=2.3.11/10.7.11.13 ), you don't get the above tuning map, but instead this other one:

Tuning map: <1199.434, 1902.594, 165.002, 3367.030, 4152.691, 4442.078]
Weighted error: [-0.000472, 0.000336, -0.000014, -0.000533, 0.000331, 0.000349]
L2 norm: 0.000923

Note that in this tuning map, 11/10 is tuned much more accurately, which decreases the L2 norm in this other basis because it's much more sensitive to the mistuning of 11/10.

AFAIK, this kind of norm doesn't have much theoretical justification at all unless it coincides with the T2 norm, which only happens when every element in the basis is a prime power. Note that some subgroups, for example 2.5/3.7/3, cannot be expressed in this way; it would be nice to have something that optimized these using the actual T2 norm rather than some arbitrary L2 norm related to a particular basis.

Keenan

--- In tuning@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:
> AFAIK, this kind of norm doesn't have much theoretical justification at all unless it coincides with the T2 norm, which only happens when every element in the basis is a prime power.

Sorry, this isn't quite right. I think the correct statement is that the L2 and T2 norms only coincide when the basis consists of pairwise coprime integers. (Things like 2.7/5 still can't be expressed in this way.)

Keenan

> I'm sorry if I've been excessively brusque; I wasn't aware of
> any tensions and did not mean to convey any. There is a
> difficulty which one could call a tension I suppose arising
> from the fact that we have such different actual and
> conceptual vocabularies. If I say I don't understand
> something, please take that literally.

Dear Gene,

Thank you for pointing to the joy as well as challenge of these
dialogues: seeking mutual understanding when the musical and
mathematical focuses can be a bit different. Your last sentence
is especially good advice, which will apply to me also if I
express difficulty in understanding some point.

Likewise, if I say "if I understand correctly," please take that
literally as an acknowledgement that I might be mistaken in
understanding your concept or viewpoint, and welcome correction.

And I'll apologize for the length of what follows, but emphasize
that my purpose is to express my worldview in a bit more detail,
something the RMP documents and posts written by you and Graham
and others already do for your perspective, together of course
what you're sharing in this dialogue.

So at times I'm making an effort not to take concepts for
granted which might not be so obvious to others, but to explain
myself in a more step-by-step fashion, which sometimes can mean
a longer fashion. But I don't want it to interfere with the
give and take of free conversation -- an "optimization" dilemma
of its own kind, maybe.

Thus I would ask your patience, even while seeking the same
spirit of inquiry that your words so beautifully express.

[On MET-24 (2/1, 703.711, 57.422) and Pele]

> But pretty close to the 13/9, 7/6 eigenmonzo tuning of
> 703.782. In any case, of course, I don't define temperaments
> by tunings, but according to what JI intervals they temper and
> what they temper out.

This may be a real difference in concepts and viewpoint, but not
one, I think, that can't be bridged by a bit of patient dialogue,
so that we come to understand each other's perspectives. And one
of the things we may learn is that our viewpoints may differ a
bit in how we "lump and split" different kinds of tunings, seeing
them as shadings of the same category, or representatives of
different categories.

The "lumping and splitting" thing may come from linguistics. Some linguistics like to "split," looking for lots of different
language families; other like to "lump," proposing various groups
or supergroups that go so deep (e.g. Nostratic) that it's hard to
verify or falsify them using standard comparative methods.

Here, we've identified an interesting difference in how we lump
or split tunings with our respective paradigms.

Clearly you see POTE Parapyth at (2/1, 703.856, 58.339) as in a
different category than the 13/9, 7/6 eigenmonzo tuning at what I
believe from the linear generator and eigenmonzos must be a
tuning of (2/1, 703.782, 59.308). This is a temperament of the
fifth by precisely (28672/28431)^(1/8), or 1/8-Secorian comma,
resulting in a just 26/21 or 21/13, the difference between the
other eigenmonzos of 7/6 and 13/9.

More specifically, if I understand correctly, you see MET-24 as
falling more within Pele because it is closer to your 13/9, 7/6
tuning than to POTE Parapyth.

This is a point I would like to understand more clearly, but
already know must have something to do with the commas tempered
out in each of these tunings. The best thing I can do is to
invite you to explain how RMP distinguishes between these
tunings, and then carefully read your reply and ask any questions
I might have.

* * *

At the same time, in my next message, I'll explain why I see
MET-24 and O3 (the latter with a generator very close to POTE
Parapyth) as slightly different shadings of the same basic
scheme. An important point I'll develop is that I see both, as
well as POTE Parapyth itself, as having 11:10 as a factor,
whether intentionally (as for with O3 and MET-24 alike) or
otherwise, as with POTE Parapyth if it is purely 2.3.7.11.13.

I'll explain that an important part in my optimizations of O3 and
more recently MET-24 is what I might term ZEO, or Zalzalian
Epimore Optimization: getting 14:13, 13:12, 12:11, and in these
tunings also 11:10 as close to just as possible, within the
overall structure and other constraints of the tuning scheme.
Your tuning with eigenmonzos of 7/6, 13/9, and 26/21 also
provides an interesting example of how, with the same generator
of 703.782 cents, somewhere between MET-24 and POTE Parapyth or
O3, I might usually optimize a bit differently, aiming for a just
13/12 and 7/4 rather than 7/6 and 13/9.

Whereas POTE gives more weight to simpler primes, if I understand
correctly from Mike, ZEO gives lots of weight to ratios like
13:12 and 12:11 -- and, often, about equal weight to 7/6 and
13/8, for example. As I'll explain, this means around the region
we're exploring, say 703.6-703.9 cents, using ZEO will promote a
slightly smaller spacing generator than POTE.

This is a difference you noted between MET-24 (2/1, 703.711,
57.422) and POTE Parapyth (2/1, 703.856, 58.339). What I owe you
is a coherent account of how and why I would set the spacing at
57.422 cents, which happens to be 49 tuning units of 1024-EDO.
And that will be a theme of my next post explaining ZEO.

>> One indication of this is that in 1978, George _didn't_ rely on pele
>> to get 5/4: instead, he used an ingenious irregular 29-note structure
>> to make 5/4 just!

> Was that irregular structure basically a modification of pele,
> however? I presume you mean the scale secor29htt.scl; I think
> it might be worthwhile to analyze that in terms of closeness
> to a regular temperament.

A good question, and worth pursuing as I shall in a moment. But
first, I should apologize for maybe being too defensive about my
own tuning when simply explaining what I was doing when I
designed MET-24 in 2011, or O3 in 2010, might be more helpful in
giving you insight into my optimization concepts than looking at
what George Secor was doing in 1978. The latter question,
however, is highly interesting in itself!

You're absolutely right: let's see what happens to 5/4 in the
regular temperament. We know that 63/52 is just, and that the
spacing for George's just 7/4 is 58.090 cents, a tiny amount
greater than 91/88 (0.054 cents). So in a regular rank-3 scheme,
we have a third of 390.298 cents, the same tiny amount greater
than 441/352. The impurity of 5/4 is 3.930 cents.

And in my more mathematically semiliterate approach, if we may
call it that, I would simply say to myself, "Well, 5/4 is about
1/6 Pythagorean comma wide (3.910 cents), and a bit less than the
1/5 syntonic comma (4.301 cents) we'd get in 1/5-comma meantone.
So there's little question that 5 is present here, however
intentionally or otherwise."

So to me, it's not quantum leaps from one comma to another which
make one tuning 2.3.7.11.13 and another 13-limit Pele in this
region, but the pervasive reality that we're going to get rather
nice meantone-quality thirds wherever we go in Parapyth, through
what RMP terms either the Pele and/or the Leapday pathway -- but
maybe those are wrong as the general terms I want. What I want to
say is "either the pathway of 9-generators-plus-spacing in
rank-3, or 21-generators-up in rank-2."

The one landmark that stands out to me is the point where, if we
space for a just 7/4, for example (as I routinely do around
703.7-703.9 cents), these pathways are equally accurate at
approximating 5/4. We would really a 58-note system to take full
advantage of each pathway (2x29-MOS), but the relevant tuning
would be (2/1, 703.8445, 57.292). At this point, either pathway
yields a 5/4 impure by 5.579 cents -- somewhere, in my analysis,
between 1/4 syntonic and 1/4 Pythagorean comma. The sizes are
380.734 and 391.893 cents.

! patheq58.scl
!
Aug2-plus-spacing and 21-fifths pathways to 5/4 equally (in)accurate
58
!
22.64890
46.13400
57.29241
80.77750
103.42640
126.91150
138.06991
173.04550
184.20390
207.68900
230.33790
253.82300
264.98141
288.46650
311.11540
334.60050
345.75891
380.73450
391.89290
415.37800
438.02690
461.51200
472.67041
496.15550
518.80440
542.28950
553.44791
576.93300
599.58190
623.06700
634.22541
669.20100
680.35940
703.84450
726.49340
749.97850
761.13691
784.62200
807.27090
830.75600
841.91441
876.89000
888.04840
911.53350
934.18240
957.66750
968.82591
992.31100
1014.95990
1038.44500
1049.60341
1084.57900
1095.73740
1119.22250
1141.87140
1165.35650
1176.51490
2/1

So 703.8445 cents is the point of equipoise. If we go lower,
keeping the spacing for a 7/4 eigenmonzo, then Pele gets more
accurate; if we go higher, then Leapday gets more accurate, with
a 5/4 eigenmonzo at 704.110 cents, and still quite reasonable
accuracy around 46-EDO or a pure 14/11 tuning (704.377 cents).

Thus Secor's tuning is in the region where Pele would give a 5/4
wide by 3.930 cents, just a bit outside his tolerance of twice
the temperament of the fifth, or 3.248 cents. You're probably the
one -- maybe together with him -- to look at secor29htt.scl and
appreciate all the subtleties and commas.

But my understanding is that balancing out the circle of 29
fifths, where one is about 715.477 cents (the same kind thing we
find in a regular rank-3 Parapyth for 2x12-MOS where the top of
the lower chain connects with the bottom of the upper chain),
calls for some meantone-like fifths, one or more of which
corrects for the 441/440 and results in seven pure 5/4 thirds.

However, from my point of view, the brute reality is that any
Parapyth scheme will tend to have 5-limit approximations lurking
about: even Peppermint-24, at (2/1, 704.096, 58.680), has what I
would call a "9-generators-up-plus-spacing" third at 395.540
cents -- as it happens, just about identical to 1/7-comma
meantone (395.531 cents)! Now we can call this a near-just 49/39,
and indeed it is, since 14/13 is virtually just and 7/6 an
eigenmonzo -- but lots of people lots of the time would call it a
recognizable approximation of 5/4. And if we expand to
Peppermint-58 with a 29-MOS, then Leapday is very close to a 5/4
eigenmonzo!

From my viewpoint, these approximations of 5 simply come with the
territory: and my stance is neither to disown them nor to make
them a part of my optimization process itself, which focuses on
2.3.7.11.13 and ZEO. The unintended remote intervals we get in
our tunings are special gifts and surprises.

Also, I tend to decide whether a given ratio or interval is
represented in a tuning by intuitive methods not much depending
on commas, although I admire the sophistication of RMP and find
that a commatic analysis is fascinating in revealing more of the
fine structure.

For example, I would say that POTE Parapyth does represent 11/10,
which at 161.440 cents is about 3.564 cents narrow. The 9/8 is
wide by 3.802 cents, so 11/10 is actually better!

And your 5/4 approximation is 393.043 cents, 6.729 cents wide, or
a bit less than 1/3 syntonic comma -- in other words, a tad
better than 1/6-comma meantone. So if 5 is present in MET-24,
which it is as a musical option (however sparse), I'd say that
it's present in POTE Parapyth: we are simply looking further into
the exact shadings, and the commas to which they may relate.

That commatic analysis might yield lots of musical insights, but
is unlikely to change the simple reality, as I see it, that all
of Parapyth has meantone-quality approximations of 5 through what
I will call the pathway of "9-generators-up-plus-spacing."

Note that I'm resisting the temptation to say "Pele" as a
nickname for this pathway, since that name would imply some
specific comma or commas. To me, it's a simple reality on
keyboard, in MET-24, O3, or Peppermint: take an augmented second
like F-G# at around 63/52, for example, and add that spacing,
thus F-G#* in my typical notation, and we get something a bit
wide of 5/4. Whether it's around 390 or 392 or 395 cents is just
a matter of shading: to me, the phenomenon is basically the
same.

It's not a part of the design, but a predictable result of it.
So here I do a bit of "lumping," and don't usually think in terms
of specific commas: "An augmented second or supraminor third plus
spacing gives a meantone-like shading of 5/4."

So in my understanding, Parapyth is a 2.3.7.11.13 system where
we're going to get some ratios of 5 in any 2x12-MOS system via
the 9-generators-up-plus-spacing pathway. Someone could use that
pathway now and then as a variation on the main 2.3.7.11.13
theme, my approach; or seek to make it an integral part of the
design and maybe expand to 2x17-MOS or indeed 2x29-MOS, etc.

Taking what to me seems a purely descriptive perspective, quite
apart from the intent of any tuning designer, in describing a
given Parapyth tuning we can simply note any pathways to 5 that
might be available in a given tuning size, e.g. 24-note rank-3
(2x12-MOS) or 29-note rank-2, etc. It's like a strait in
geography: you might or might not choose to sail there, but it's
there if and when you want to use it.

Obviously George Secor was very specifically out to optimize 5,
and he used an irregular tuning for that specific purpose. It
would be very valuable to understand the exact commas involved,
and on that question my main role might be to listen to you and
George unfolding the fine points, and possibly offer an odd
comment now and get about the kinds of commas like 28672/28431
that seem to attract my special affection.

With many thanks,

Margo

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:

> Clearly you see POTE Parapyth at (2/1, 703.856, 58.339) as in a
> different category than the 13/9, 7/6 eigenmonzo tuning at what I
> believe from the linear generator and eigenmonzos must be a
> tuning of (2/1, 703.782, 59.308).

Actually, I don't. I see a two-parameter family of tuning possibilities, and these are two points in that space of tunings.

> More specifically, if I understand correctly, you see MET-24 as
> falling more within Pele because it is closer to your 13/9, 7/6
> tuning than to POTE Parapyth.

Analyzing MET-24 on the basis of your met24.scl file shows it makes the most sense interpreted as a no-fives parapyth scale, though of course you can also "extend" it to a pele usage. You had another file with 5-limit intervals explicitly in it; I wouldn't equate the two.

> At the same time, in my next message, I'll explain why I see
> MET-24 and O3 (the latter with a generator very close to POTE
> Parapyth) as slightly different shadings of the same basic
> scheme.

I'm afraid I don't know what O3 is.

Dear Gene.

As promised, this is my explanation as to the optimizations I
used for O3 in 2010 and MET-24 in 2011, both of which, of course,
draw on the basic parapyth 2.3.7.11.13 subset of George Secor's
29-HTT (1978), secor29htt.scl.

The parameters for O3, actually designed in 1024-EDO, are
theoretically (2/1, 703.893, 57.148 cents), with the generator
very close to that of POTE Parapyth (2/1, 703.856, 58.339).
However, the logic may be a bit different, and it's my purpose
here to explain that, rather than get into possible RMP questions
or interpretations, which we can do in our dialogue.

The parameters for MET-24, mainly designed to improve 3/2 and 9/8
a bit as well as to strike as equal a balance between 14/13 and
11/10, are a theoretical (2/1, 703.711, 57.422). This is an
interesting example where the structure of 1024-EDO itself
pointed to the choice of generator (actually the average of two
generators in the real-world tuning), but there's a theoretical
way of reaching an almost identical result not tied to any
specific EDO. We might speak of these as the "bitmap" and
"vectorized" versions, if an analogy from computer graphics might
be helpful.

I've decided to reserve the question of an alternative
optimization with the generator of the 7/6-13/9-26/21 eigenmonzo
tuning you called to my attention (2/1, 703.782, 59.308) for
another post, because this question deserves more consideration
on my part. Thus this post will stick with O3 and MET-24.

What I should emphasize at the outset, before explaining the
optimization of these tunings, is one point about O3 as well as
the later MET-24.

Both tunings are very specifically designed to optimize the 11/10
neutral second, as well as 12/11, 13/12, and 14/13. That was a
major focus of O3 as well as MET-24, as I made clear in my
announcement of O3:

</tuning/topicId_91499.html#91499>

In that announcement, I noted a specific desire for a system
"optimizing the large Zalzalian or neutral third at 26/21 (370
cents)" for an historical flavor of Ottoman Rast. "More
generally," I continued, "it was my goal to optimize all four of
the [superparticular] Zalzalian or middle second steps discussed
by medieval Islamic theorists such as Safi al-Din, and also Ozan,
the 14:13, 13:12, 12:11, and 11:10."

One interesting cultural point here is that 26/21 is a basic
interval for me, and might go far to define a tuning system, once
I make clear my implicit assumption that this interval will be
the diminished fourth from 8 generators up.

At the same time, I was drawn to idea of a just 22/21 limma,
another ratio I really like. And, of course, 14/13 and 13/11
would be important, as well as 7/6, 7/4, 9/7, and 21/16, etc.

----------------------------------------------------------------
1. Optimizing O3 in 1024-EDO: Some help from the limma for 11/10 ----------------------------------------------------------------

For O3 in 1024-EDO, my goal within a single 12-MOS chain was to
get 26/21 close to just and at the same time have 11/10 with
roughly Secor's HTT tolerance of just, 3.247 cents, which since
2001, when he shared that tuning with me, has become a kind of
emblem of "near-just" quality.

This brings me to what may be a basic difference between the POTE
worldview and my optimization strategy.

The POTE, I assume, looks at all relevant primes at the same
time, and gives them weights by which to arrive, say, at a rank-3
Parapyth system with the parameters simultaneously determined.

In my approach, it's more of a layered process: first optimizing
a 12-MOS or its slightly irregular equivalent in 1024-EDO; and
then deciding on the spacing which will complete a rank-3,
24-note tuning.

My approach is somewhat analogous to the layered or successive
composition of 13th-14th century European polyphony, where often
one voice is added at a time. Indeed, one might often start with
an existing melody, and add one or more new melodic lines to it.
This is what happened in Peppermint-24: Keenan Pepper had already
defined the 12-MOS, and I simply picked the distance at which to
set the second identical chain (58.680 cents) for a pure 7/6.
There neither the 12-MOS of Keenan Pepper nor the spacing was
conceived in 1024-EDO, but O3 was designed from the ground up
specifically for 1024-EDO.

This means that for O3, I first designed and optimized the basic
rank-2 12-MOS, and then decided on the spacing. Both stages of
the process were based in 1024-EDO, as I'll describe.

Here I realized that a just 22/21 and 26/21 were highly
compatible ideas; if asked to explain, I might note that the
difference between these intervals is 13/11, which in O3 is
always within a cent of just.

Another element of the 12-MOS equation for O3, and also for
MET-24 as we'll see, is the subtle balance between 14/13 and
11/10, the two epimoric neutral seconds represented within a
single chain of generators. I knew that 14/13 would be a bit
narrow at a just 22/21 tuning, but still well within the HTT
tolerance: 11/10 was more the issue.

Of course, I knew that the theoretical generator was
(84/11)^(1/5) or 3/2 tempered wide by (896/891)^(1/5), at 703.893
cents. Rather humorously, this is 1.938 cents wide, or almost
precisely as impure as in 12n-EDO (1.955 cents), but in the
opposite direction!

In practice, however, I was designing and tuning in 1024-EDO,
where there is no fifth of this size, but available nearby sizes
of 600 or 601 steps: 703.125 or 704.297 cents. My purpose was
develop a bearing plan so that, if possible, all limmas or
diatonic semitones from 5 fifths down would be at the best
possible approximation of 22/21.

This approximation was 69 steps, or 80.859 cents -- wide of 22/21
by 0.322 cents. The reason for this slight difference from just
(80.537 cents) is that while 599 steps is almost exactly 3/2, the
representation of 896/891 at 8 steps (9.375 cents) falls slightly
short of the just size at 9.688 cents.

My strategy was to lay out a digital "bearing plan" so that each
five generators in a chain would be wide by 8 tuning steps in all
for the best 22/21. I'll first give the scheme, and then consider
the consequences for 11/10.

704.3 703.1 704.3 703.1 704.3 704.3 703.1 704.3 703.1 704.3 704.3
Eb Bb F C G D A E B F# C# G#
601 600 601 600 601 601 600 601 600 601 601
+2 +1 +2 +1 +2 +2 +1 +2 +1 +2 +2

One compromise of this scheme, by the way, is that although the
average temperament might be a bit less in than 12n-EDO, 7 or our
11 fifths in a chain are a bit more impure than in 12n-EDO. But
that's an aside: let's get back to 11/10.

Because of the slight inaccuracy of 1024-EDO in the wide direction
for 22/21, and our consistent use of this closest approximation
at 69 tuning steps or 80.859 cents, it follows that 11/10 from
two such limmas, or a chain of ten fifths down, will be a bit
larger than (22/21)^2 or 484/441 at 161.074 cents -- i.e. 138
tuning steps or 161.719 cents, about 3.285 cents narrow of 11/10.
This is very close to the emblematic HTT standard of 3.247
cents, and so I deemed it a happy optimization.

Another feature of this optimization is that 13/11 is always
within a cent of just (289.210 cents), at either 4 or 5 tuning
steps narrow of the near-just 1024-EDO version of 32/27 at 251
tuning steps: 288.281 cents or 289.453 cents.

How about our 26/21 with a just size of 369.747 cents (thinking
of airlines helps to remember the decimal value!)? We get it at
369.141 or 370.312 cents, again always within a cent of just.

There is a humorous touch here, as I realized: in a certain
Pickwickian way, O3 might be said to observe the 2080/2079
distinguishing 99/80 or 9/8 plus 11/10 (368.914 cents), an
interval occurring in al-Farabi's 9:8-11:10-320:297, from 26/21 in
a permutation of Ibn Sina's 9:8-14:13-208:189, for example, as
well as his exquisite 8:7-13:12-14:13.

It is possible to view the smaller tempering of the submajor or
large neutral second at 369.141 cents as representing 99/80
(0.226 cents wide), and the larger at 370.312 cents as
representing 26/21. This kind of thing is a big part of my
culture of optimization, which should be fun!

The 14/11 major third is at 414.844 or 416.016 cents, the first
value actually leaning a bit toward 33/26 (412.745 cents), and
the second more strongly toward 14/11 (417.508 cents). These
results are fine, and subtly distinct from those of Peppermint
based on Keenan Pepper's Noble Fifth tuning (2/1, 704.096), where
in 1024-EDO, 14/11 is at 416.016 cents (as sometimes here) or the
virtually just 417.187 cents.

---------------------------------------------------------
2. Optimizing O3 in 1024-EDO: Setting the spacing and ZEO
---------------------------------------------------------

In the process of successive or layered optimization, we proceed
from the now-defined 12-note chain to the final parameter
defining O3 as a rank-3 tuning: the size of the spacing generator
between our established chain and the duplicate we are about to
add. In other words, what distance shall we place between the
chains.

Here two approaches happily coincide: eigenmonzos or ZEO
(Zalzalian Epimore Optimization). Note that 2010 was not yet
familiar with the meaning of the term eigenmonzo, although I seem
to recall hearing it various times on the tuning lists. However,
since "going for a just interval" is pretty much synonymous, I'll
use the RMP term, which I find very comfortable.

Apart from 22/21 and 26/21 -- and thus 13/11 -- my intended
eigenmonzos were 7/4, and also 11/6 (7/4 plus 22/21). Here
1024-EDO introduces a charming ambiguity and leeway: the "within
a cent or tuning unit" test for "virtual justness."

One form of virtual justness we have seen within a single chain,
specifically with 22/21, is always using the nearest
approximation (here 80.859 cents). A different form, which we
have seen with 13/11 and 26/21, where there are two tempered
sizes because of the inevitably irregular chain of fifths in a
1024-EDO "bitmap" or "halftoning pattern" to get the desired
shading of temperament, is having each of these sizes no further
than a 1024-EDO tuning unit (1.171875 cents) from just.