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On Fri, Apr 29, 2011 at 3:20 PM, Callahan White <cortaigne@...> wrote:
>
> I don't mean to spam the list, but I just started a blog and a lot of what I have in mind is definitely related to tuning and microtonal composition and performance. I would love to get any suggestions or feedback, and you are the go-to people for that. ;-)
>
> http://miskatonal.blogspot.com/

Nice! Blogger seems to have crashed, so I'll post my comments here:

You talk about equally dividing the tetrachord into 3 parts -
congrats, you have rediscovered porcupine temperament. The way to go
here is to make the fourth a little flat, so that what you were
calling a "neutral" third gets closer to a minor third. 22-equal is a
near-ideal tuning for this. 15-equal and 37-equal support it as well.
You might want to consider the mode with all generators going down -
i.e. 4333333 in 22-equal. As you can see, this has 11-limit
implications. Many people like porcupine[8], which is 13333333, but
ymmv. If instead you divide the tetrachord into 4 parts, you end up
with Negri temperament, which 19-equal is a good tuning for.

WRT 19-equal, you might still find that some pieces written in
12-equal just don't work - ones that subtly make use of the 128/125 or
648/625 unison vectors, which 19-equal doesn't vanish like 12-equal
does. This shouldn't pose too much of a problem, but you ought to be
aware of it - for example, if you ever use the altered scale over
dominant chords, things aren't going to line up the same. You should
also be aware that 19-equal has a lot more tonal systems in it than
12-equal - it supports magic, sensi, negri, semaphore, hanson, etc,
all in addition to the usual meantone. Also, 19-equal has been much
maligned for its lack of a strong 7 and 11, but it does have a pretty
good 13/10, so that you can turn minor 10:12:15 triads into
10:12:13:15 tetrads. Lastly, if you're trying to explore the 5-limit
for now, you might also want to check out 22-equal, which is of about
the same size as 19 but does NOT support meantone, so it really forces
you to think out of the box.

-Mike

🔗Callahan White <cortaigne@...>

Thank you! This is exactly the sort of feedback I was hoping for!

I have seen the name Porcupine on the list before, but to be honest I'm still pretty new at all this, so I'm not even sure what something like Porcupine[8] means, or what a generator is. In my own scribblings, something like 13333333 would be a step pattern for a subset of a tuning -- is that the case here?

Also, regarding "turn minor 10:12:15 triads into 10:12:13:15 tetrads", I need an explanation there as well. From what I gathered back when I read (parts of) Doty's "Just Intonation Primer," something like 6:5:4 would mean a 6:5 and a 5:4 -- is this the same idea?

Also also, what does it mean to vanish a unison vector?

Forgive me, but I've never been formally educated on anything beyond the difference between a half rest and a whole rest, or what a quarter note looks like. :-\

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Fri, Apr 29, 2011 at 3:20 PM, Callahan White <cortaigne@...> wrote:
> >
> > I don't mean to spam the list, but I just started a blog and a lot of what I have in mind is definitely related to tuning and microtonal composition and performance. I would love to get any suggestions or feedback, and you are the go-to people for that. ;-)
> >
> > http://miskatonal.blogspot.com/
>
> Nice! Blogger seems to have crashed, so I'll post my comments here:
>
> You talk about equally dividing the tetrachord into 3 parts -
> congrats, you have rediscovered porcupine temperament. The way to go
> here is to make the fourth a little flat, so that what you were
> calling a "neutral" third gets closer to a minor third. 22-equal is a
> near-ideal tuning for this. 15-equal and 37-equal support it as well.
> You might want to consider the mode with all generators going down -
> i.e. 4333333 in 22-equal. As you can see, this has 11-limit
> implications. Many people like porcupine[8], which is 13333333, but
> ymmv. If instead you divide the tetrachord into 4 parts, you end up
> with Negri temperament, which 19-equal is a good tuning for.
>
> WRT 19-equal, you might still find that some pieces written in
> 12-equal just don't work - ones that subtly make use of the 128/125 or
> 648/625 unison vectors, which 19-equal doesn't vanish like 12-equal
> does. This shouldn't pose too much of a problem, but you ought to be
> aware of it - for example, if you ever use the altered scale over
> dominant chords, things aren't going to line up the same. You should
> also be aware that 19-equal has a lot more tonal systems in it than
> 12-equal - it supports magic, sensi, negri, semaphore, hanson, etc,
> all in addition to the usual meantone. Also, 19-equal has been much
> maligned for its lack of a strong 7 and 11, but it does have a pretty
> good 13/10, so that you can turn minor 10:12:15 triads into
> 10:12:13:15 tetrads. Lastly, if you're trying to explore the 5-limit
> for now, you might also want to check out 22-equal, which is of about
> the same size as 19 but does NOT support meantone, so it really forces
> you to think out of the box.
>
> -Mike
>

🔗genewardsmith <genewardsmith@...>

🔗Mike Battaglia <battaglia01@...>

On Fri, Apr 29, 2011 at 4:21 PM, Callahan White <cortaigne@...> wrote:
>
> Thank you! This is exactly the sort of feedback I was hoping for!
>
> I have seen the name Porcupine on the list before, but to be honest I'm still pretty new at all this, so I'm not even sure what something like Porcupine[8] means, or what a generator is. In my own scribblings, something like 13333333 would be a step pattern for a subset of a tuning -- is that the case here?

Yes, it's a step pattern for 22-equal.

The basic idea is that we're trying to find patterns, as you did. One
famous pattern is that four 3/2's on top of one another gets you to
something close to a 5/1. So you can detune things slightly so that
four tempered 3/2's gets you to a tempered 5/1. This is meantone
temperament, which 12-equal and 19-equal supports.

Another famous pattern is that 3 5/4's on top of one another gets you
to something very close to an octave, so you can sharpen the major
thirds so that it does. This is augmented temperament, which 12-equal
and 15-equal supports.

Yet another famous pattern is that four 6/5's on top of one another
gets you to something very close to an octave, so you flatten the
minor thirds thirds so that it does. This is diminished temperament,
which 12-equal and 16-equal support.

A not so famous pattern is that 2 10/9's on top of one another gets
you to something close to a 6/5 minor third. If you do the math, this
also implies that three 10/9's gets you to a 4/3. This is what you
discovered, and it's called "porcupine temperament."

> Also, regarding "turn minor 10:12:15 triads into 10:12:13:15 tetrads", I need an explanation there as well. From what I gathered back when I read (parts of) Doty's "Just Intonation Primer," something like 6:5:4 would mean a 6:5 and a 5:4 -- is this the same idea?

Yes, except you'd notate it 4:5:6. That's a justly-tuned major chord.
Its inverse, a minor chord, is 1/(4:5:6), which is 1/4:1/5:1/6. If you
do the math and reduce this to lowest terms, you end up with 10:12:15.
It might be useful for you to play the harmonic series to see where
this chord lies out in it.

So 10:12:13:15 is just a minor chord with an extra note thrown in, a
13/10 (ultramajor third) over the root. So in 19-equal, that'd be C Eb
E# G. It's an interesting sound.

> Also also, what does it mean to vanish a unison vector?

You have to think about temperaments as "eliminating" a certain
interval. For example, 5/4, 3/2, and 6/5 are all fractions denoting
just frequency ratios, but they're also intervals. In the case of
meantone, we're making things so that 3/2 * 3/2 * 3/2 * 3/2 ~= 5/1.
However, 3/2 * 3/2 * 3/2 * 3/2 is actually equal to 81/16. The
difference between 81/16 and 5/1 is 81/80. So in meantone, this tiny
81/80 interval, which is called the "syntonic comma," vanishes and
gets equated with 1/1 - it becomes a unison.

> Forgive me, but I've never been formally educated on anything beyond the difference between a half rest and a whole rest, or what a quarter note looks like. :-\

Nobody's been formally educated on any of this here because most of
the theory's been developed on this list. Congratulations, you're now
on the bleeding edge of music theory. I think it's only a matter of
time before academia catches on though. Or perhaps popular music will
catch on first, with academia coming later.

-Mike

🔗genewardsmith <genewardsmith@...>

🔗Callahan White <cortaigne@...>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Fri, Apr 29, 2011 at 4:21 PM, Callahan White <cortaigne@...> wrote:
> >
> > Thank you! This is exactly the sort of feedback I was hoping for!
> >
> > I have seen the name Porcupine on the list before, but to be honest I'm still pretty new at all this, so I'm not even sure what something like Porcupine[8] means, or what a generator is. In my own scribblings, something like 13333333 would be a step pattern for a subset of a tuning -- is that the case here?
>
> Yes, it's a step pattern for 22-equal.

Okay, cool. So the [8] means there are eight steps? What about something like 22333333?

> The basic idea is that we're trying to find patterns, as you did. One
> famous pattern is that four 3/2's on top of one another gets you to
> something close to a 5/1. So you can detune things slightly so that
> four tempered 3/2's gets you to a tempered 5/1. This is meantone
> temperament, which 12-equal and 19-equal supports.
>
> Another famous pattern is that 3 5/4's on top of one another gets you
> to something very close to an octave, so you can sharpen the major
> thirds so that it does. This is augmented temperament, which 12-equal
> and 15-equal supports.
>
> Yet another famous pattern is that four 6/5's on top of one another
> gets you to something very close to an octave, so you flatten the
> minor thirds thirds so that it does. This is diminished temperament,
> which 12-equal and 16-equal support.

Excellent information! I've seen the explanation for meantone before, as I understand it's of historical significance, but if I've seen a good explanation for the different *kinds* of meantone it must have been long before I had any idea what it meant. The first preset tunings listed in Logic Pro are "1/2-comma meantone," "1/4-comma meantone with equal beating fifths," "1/4-comma meantone, 6/5 beats twice 3/2 (Pietro Aaron, 1523)," "1/6-comma meantone scale (Salinas tritonic)," and "1/6-comma meantone with equal beating fifths," and there are many other tunings later in the list with "meantone" in their name. Are you able to shed some light on these as well?

Augmented and diminished temperament are entirely new to me. Are they useful? :-)

> A not so famous pattern is that 2 10/9's on top of one another gets
> you to something close to a 6/5 minor third. If you do the math, this
> also implies that three 10/9's gets you to a 4/3. This is what you
> discovered, and it's called "porcupine temperament."

Cool. I'm looking at the xenharmonic wiki page on the Porcupine family, and it just reminds me how much more there is to learn. I've seen the terms monzo and wedgie and I have no clue what they mean, the math looks far beyond my experience. :-\

> > Also, regarding "turn minor 10:12:15 triads into 10:12:13:15 tetrads", I need an explanation there as well. From what I gathered back when I read (parts of) Doty's "Just Intonation Primer," something like 6:5:4 would mean a 6:5 and a 5:4 -- is this the same idea?
>
> Yes, except you'd notate it 4:5:6. That's a justly-tuned major chord.
> Its inverse, a minor chord, is 1/(4:5:6), which is 1/4:1/5:1/6. If you
> do the math and reduce this to lowest terms, you end up with 10:12:15.
> It might be useful for you to play the harmonic series to see where
> this chord lies out in it.
>
> So 10:12:13:15 is just a minor chord with an extra note thrown in, a
> 13/10 (ultramajor third) over the root. So in 19-equal, that'd be C Eb
> E# G. It's an interesting sound.

Very interesting! I will be studying this in detail. >:)

> > Also also, what does it mean to vanish a unison vector?
>
> You have to think about temperaments as "eliminating" a certain
> interval. For example, 5/4, 3/2, and 6/5 are all fractions denoting
> just frequency ratios, but they're also intervals. In the case of
> meantone, we're making things so that 3/2 * 3/2 * 3/2 * 3/2 ~= 5/1.
> However, 3/2 * 3/2 * 3/2 * 3/2 is actually equal to 81/16. The
> difference between 81/16 and 5/1 is 81/80. So in meantone, this tiny
> 81/80 interval, which is called the "syntonic comma," vanishes and
> gets equated with 1/1 - it becomes a unison.

Okay, I've definitely heard of tempering out a comma before, and had a vague idea it meant something like that. So ... is a "unison vector" any very small interval?

> > Forgive me, but I've never been formally educated on anything beyond the difference between a half rest and a whole rest, or what a quarter note looks like. :-\
>
> Nobody's been formally educated on any of this here because most of
> the theory's been developed on this list. Congratulations, you're now
> on the bleeding edge of music theory. I think it's only a matter of
> time before academia catches on though. Or perhaps popular music will
> catch on first, with academia coming later.

I think popular music could desperately use at least some of the basic ideas like this to start doing something besides the Sensitive Female Chord Progression, but I don't think industry sees a need. :-(

Thank you again for all the help!

🔗Mike Battaglia <battaglia01@...>

On Fri, Apr 29, 2011 at 6:08 PM, Callahan White <cortaigne@...> wrote:
>
> > Yes, it's a step pattern for 22-equal.
>
> Okay, cool. So the [8] means there are eight steps? What about something like 22333333?

The 8 means that you stack 8 porcupine "generators" on top of one
another. The generator for porcupine is a flat 10/9, and the period is
a 2/1. Rather than explain it formally, I'd just leave it at this:
meantone[7] is the major scale, and you get it by taking a chain of 7
meantone generators and smushing them within the meantone period. The
meantone generator is a fifth, and the meantone period is an octave.
So you can confirm that a stack of 7 fifths lands you the diatonic
scale. You get different modes depending on how many fifths go up from
the root and how many go down (major is one fifth down, the rest up).

Porcupine[8] is also an "MOS" of porcupine, which means it's a "moment
of symmetry" scale for that temperament. MOS scales are ones in which
every generic interval class comes in two sizes. For example,
meantone[7] is an MOS, because there are only two sizes of second -
major and minor - and two sizes of third - major and minor - and two
sizes of fourth - perfect and augmented - etc. Meantone[5] is also an
MOS, and so is meantone[12]. Meantone[6] doesn't form an MOS. These
scales are useful as starting points for the temperament you're trying
to explore, but you don't need to stick to the MOS's by any means
(sometimes it makes more sense not to).

Some ways to figure out the generator for a temperament

1) Magic
2) Voodoo
3) Figure out how the temperament you want creates a subspace of the
JI lattice and figure out what the set of basis vectors of this
subspace is
4) Ask on the list

I used to think that #4 was the main way to do it, but now I'm
exploring #3 as a viable option. Alternately, there's Graham's
temperament finder here:

http://x31eq.com/temper/

You'll probably need a bit more catching up before any of it makes
sense to you though.

> Excellent information! I've seen the explanation for meantone before, as I understand it's of historical significance, but if I've seen a good explanation for the different *kinds* of meantone it must have been long before I had any idea what it meant. The first preset tunings listed in Logic Pro are "1/2-comma meantone," "1/4-comma meantone with equal beating fifths," "1/4-comma meantone, 6/5 beats twice 3/2 (Pietro Aaron, 1523)," "1/6-comma meantone scale (Salinas tritonic)," and "1/6-comma meantone with equal beating fifths," and there are many other tunings later in the list with "meantone" in their name. Are you able to shed some light on these as well?

Those are specific tunings for the abstract meantone temperament in
general. 1/4 comma meantone means that every fifth is flattened by
1/4th of 81/80, so that four of them lands you a perfect 5, and so the
5/4's are perfect at the expense of the 3/2's (this is almost exactly
identical to 31-equal). 1/3 comma meantone does the same, but with
minor thirds (this is almost exactly identical to 19-equal). I
wouldn't worry about the equal beating stuff now. But they are
specific tunings of meantone temperament.

> Augmented and diminished temperament are entirely new to me. Are they useful? :-)

Sure - diminished is used all the time in jazz over dominant 7 chords.
Diminished[8] has a generator of a half step and a period of a minor
third. In 12-equal, the scale is

C Db D# E F# G A Bb C (dominant mode)
C D Eb F Gb Ab A B C (diminished mode)

Diminished[12] has a beautiful sound as well - in 12-equal, it's the
same as meantone[12], but you can kind of snap your brain around to
hear it differently.

> Okay, I've definitely heard of tempering out a comma before, and had a vague idea it meant something like that. So ... is a "unison vector" any very small interval?

No, it's just that the comma you're tempering out is called the unison
vector. If you aren't tempering it, it doesn't become a unison. We
call it a "vector" because some guy named Fokker worked all of this
out with something called the JI lattice, which is a vector space.

-Mike

🔗Chris Vaisvil <chrisvaisvil@...>

🔗Kalle Aho <kalleaho@...>

🔗genewardsmith <genewardsmith@...>

--- In tuning@yahoogroups.com, "Callahan White" <cortaigne@...> wrote:

If q is a positive rational number, then "q-comma meantone" should be meantone with the fifth flattened by (81/80)^q. However, all this equal beating stuff sometimes screws up how this simple idea is presented. 6/5 doesn't beat with 6/5 beating precisely twice 3/2 in 1/4 comma meantone (its 2.5 times, not 2 times), so requiring that it does actually leads to a different meantone, with a fifth of 696.296 cents, rather than the 696.578 cents of 1/4 comma meantone. That's awfully close to 81et meantone, actually.

> Cool. I'm looking at the xenharmonic wiki page on the Porcupine family, and it just reminds me how much more there is to learn. I've seen the terms monzo and wedgie and I have no clue what they mean, the math looks far beyond my experience. :-\

The math is for people who know math; you don't need to be a math whiz to be a musician and make use of the wide variety of xenharmonic resources available on the Xenwiki.

> Okay, I've definitely heard of tempering out a comma before, and had a vague idea it meant something like that. So ... is a "unison vector" any very small interval?

Here's the definition of a comma on the Xenwiki:

A comma is a small just interval, greater than 1 and not a power such as a square or cube of any other interval, generally in a low prime limit. The word is often used in reference to regular tempering when the comma is tempered out, which is to say reduced to a unison, by the temperament. For example the Didymos comma of 81/80 is tempered out by meantone temperament.

🔗Chris Vaisvil <chrisvaisvil@...>

🔗Graham Breed <gbreed@...>

On 29 April 2011 23:58, Mike Battaglia <battaglia01@...> wrote:

> You talk about equally dividing the tetrachord into 3 parts -
> congrats, you have rediscovered porcupine temperament. The way to go
> here is to make the fourth a little flat, so that what you were
> calling a "neutral" third gets closer to a minor third. 22-equal is a
> near-ideal tuning for this. 15-equal and 37-equal support it as well.
> You might want to consider the mode with all generators going down -
> i.e. 4333333 in 22-equal. As you can see, this has 11-limit
> implications. Many people like porcupine[8], which is 13333333, but
> ymmv. If instead you divide the tetrachord into 4 parts, you end up
> with Negri temperament, which 19-equal is a good tuning for.

Ah, so that's Porcupine! I've seen the numbers but I didn't have a
handle on the melodic pattern before. That tetrachord, then, is one
of three basic types that give 7 note tetrachordal scales with two
step sizes. The other two can be identified with Meantone and
Mohajira.

I knew about the melodic pattern a long time ago, but didn't know how
to find a regular temperament to fit it. It shows how far we've come
in the past 10 years that Porcupine's now a commonplace, sitting on
lists waiting for somebody to explore it further.

I think Myna has a 4&3 melodic pattern I couldn't reconcile before.
It works well in certain 13-limit subgroups.

Happy Miracle Day!

Graham

🔗Mike Battaglia <battaglia01@...>

On Sat, Apr 30, 2011 at 2:54 AM, Graham Breed <gbreed@...> wrote:
>
> Ah, so that's Porcupine! I've seen the numbers but I didn't have a
> handle on the melodic pattern before. That tetrachord, then, is one
> of three basic types that give 7 note tetrachordal scales with two
> step sizes. The other two can be identified with Meantone and
> Mohajira.

When you say tetrachordal, I assume you don't mean omnitetrachordal?
And what about enharmonic scales?

> I knew about the melodic pattern a long time ago, but didn't know how
> to find a regular temperament to fit it. It shows how far we've come
> in the past 10 years that Porcupine's now a commonplace, sitting on
> lists waiting for somebody to explore it further.

And then there's equal division of 4/3 into 4 parts, which is negri.

I really want to get my head wrapped around all of the 1L_s scales and
_L1s scales, so I can understand what they are. So far I have
1L5s - meantone 6-note whole tone scale
5L1s - machine[6], gamelismic[6]
1L6s - porcupine[7]
6L1s - meantone 7-note whole tone scale
1L7s - progression[8]
7L1s - porcupine[8]
1L8s - negri[9]
8L1s - progression[9]
1L9s - miracle[10]
9L1s - negri[10]
1L10s - ripple[11], passion[11]
10L1s - miracle[11]

That's all I got. Progression is another good one that people often
miss, probably because they're not a fan of sharp 5/4's. But I still
like it.

> Happy Miracle Day!

Oh happy day...

-Mike

🔗Callahan White <cortaigne@...>

What does "omnitetrachordal" mean?

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Sat, Apr 30, 2011 at 2:54 AM, Graham Breed <gbreed@...> wrote:
> >
> > Ah, so that's Porcupine! I've seen the numbers but I didn't have a
> > handle on the melodic pattern before. That tetrachord, then, is one
> > of three basic types that give 7 note tetrachordal scales with two
> > step sizes. The other two can be identified with Meantone and
> > Mohajira.
>
> When you say tetrachordal, I assume you don't mean omnitetrachordal?
> And what about enharmonic scales?
>
> > I knew about the melodic pattern a long time ago, but didn't know how
> > to find a regular temperament to fit it. It shows how far we've come
> > in the past 10 years that Porcupine's now a commonplace, sitting on
> > lists waiting for somebody to explore it further.
>
> And then there's equal division of 4/3 into 4 parts, which is negri.
>
> I really want to get my head wrapped around all of the 1L_s scales and
> _L1s scales, so I can understand what they are. So far I have
> 1L5s - meantone 6-note whole tone scale
> 5L1s - machine[6], gamelismic[6]
> 1L6s - porcupine[7]
> 6L1s - meantone 7-note whole tone scale
> 1L7s - progression[8]
> 7L1s - porcupine[8]
> 1L8s - negri[9]
> 8L1s - progression[9]
> 1L9s - miracle[10]
> 9L1s - negri[10]
> 1L10s - ripple[11], passion[11]
> 10L1s - miracle[11]
>
> That's all I got. Progression is another good one that people often
> miss, probably because they're not a fan of sharp 5/4's. But I still
> like it.
>
> > Happy Miracle Day!
>
> Oh happy day...
>
> -Mike
>

🔗Graham Breed <gbreed@...>

On 30 April 2011 15:24, Mike Battaglia <battaglia01@...> wrote:
> On Sat, Apr 30, 2011 at 2:54 AM, Graham Breed <gbreed@...> wrote:
>>
>> Ah, so that's Porcupine! I've seen the numbers but I didn't have a
>> handle on the melodic pattern before. That tetrachord, then, is one
>> of three basic types that give 7 note tetrachordal scales with two
>> step sizes. The other two can be identified with Meantone and
>> Mohajira.
>
> When you say tetrachordal, I assume you don't mean omnitetrachordal?
> And what about enharmonic scales?

I mean the same tetrachord is repeated, either conjuctly or
disjunctly. I explained it here a long time ago:

http://x31eq.com/7plus3.htm#tetra

7 notes.

>> I knew about the melodic pattern a long time ago, but didn't know how
>> to find a regular temperament to fit it. It shows how far we've come
>> in the past 10 years that Porcupine's now a commonplace, sitting on
>> lists waiting for somebody to explore it further.
>
> And then there's equal division of 4/3 into 4 parts, which is negri.

5 parts gives the classic pentatonic or a Semaphore pentatonic (every
other note of Negri). 9 note scales are interesting as complex
generalized diatonics. I've been looking at the tripod scale, the
Orwell MOS, and an Orwell non-MOS.

Graham