Yahoo Tuning Groups Ultimate Backup tuning Another temperament, with explanations for its commas

Here comes a classic Freivaldian overexplanation, cross-posted to the
tuning list and the Facebook XA. I apologize in advance to those of you
whom I bore.

So, if you use the perfect fifth of 80 EDO as a generator -- that's 47\80,
or exactly 705 cents -- then you get a 7- or 12-note MOS scales that temper
out the following commas:

-----
352/351, or "minthma", 4.93 cents, monzo | 5 -3 0 0 1 -1 >. This means that
going down three fifths gets you a "minor third" of about 13/11.

896/891, or "pentacircle", 9.69 cents, monzo | 7 -4 0 1 -1 >. This is the
product of gentle and minthma. It means that going up four fifths gets you
a "major third" of about 14/11.

364/363, or "gentle", 4.76 cents, monzo | 2 -1 0 1 -2 1 >. This means that
14/11 * 13/11 = 3/2: A major third + a minor third = a perfect fifth.

6656/6561, 24.89 cents, monzo | 9 -8 0 0 0 1 >. This means that going up
eight fifths gets you the thirteenth harmonic, 13/8. It also means that
going up six fifths gets you 13/9, and going up seven fifths gets you 13/12.

180224/177147, 29.81 cents, monzo | 14 -11 0 0 1 >. This means that going
up eleven fifths gets you the eleventh harmonic, 11/8. It also means that
going up nine fifths gets you 11/9, and going up ten fifths gets you 11/6.
-----

I'm not sure of the names of the last two commas.

Note that the effect of the first three commas is to do exactly what 81/80
does for meantone, except instead of 5/4 major thirds and 6/5 minor thirds,
you have 14/11 major thirds and 6/5 minor thirds. In other words, this is a
scale with meantone-like structures, and if you tuned a piano to it, the
modes would seem very familiar (black-key pentatonic, ionian, dorian,
phyrigian, etc.) but have different flavors because of the different
intervals.

Now, if you've played with meantone tunings before, you know that they can
get pretty good approximations to the seventh harmonic, 7/4, without really
"meaning to" (i.e., they weren't part of classical theory). With a good
7/4, some other intervals creep in, such as 7/5, 10/7, and 7/6. Those
intervals are there because the slightly flattened fifth drives them there:
As you repeat the generator again and again, the error accumulates to give
you (e.g.) those "flatter than minor" minor thirds and such.

Similarly, this slightly sharp generator gets you pretty good
approximations to the 11th and 13th harmonics (i.e., 13/8 and 11/8) without
really "meaning to", and pulls in other intervals (13/9, 13/12, 11/9, and
11/6) in the process. (The 11/7 was already pretty good, because the scale
was defined around a good 14/11, which is its inverse.) There's only one
11/8 in the whole scale, unfortunately, but it's there.

The result is the same in both cases: Using a meantone generator will get
you a 7-note MOS consisting of traditional diatonic scales, and going to
the full 12-note MOS will get you the 7/4, 7/6, etc. Using a 47\80
generator will get you a 7-note MOS consisting of traditional diatonic
scales, but tuned differently using 14/11 and 13/11 thirds (among other
differences); going to the full 12-note MOS will get you the 11/8, 13/8,
11/9, etc.

This is cool because you can actually use this same scale to play I-IV-V
chords with neutral thirds _or_ play I-IV-V chords with sharp-and-spicy
14/11 major thirds _or_ play i-iv-v chords with 13/11 soft-and-mournful
minor thirds. But it's also cool because you can play things like
1/1-13/12-11/9-3/2-13/8 with pretty good accuracy, which sounds completely
unlike anything remotely I-IV-V-ish. And you get it all within the same
twelve notes.

The following EDOs and generators temper out all of these commas, each with
slightly different tradeoffs. 46 is the least accurate of the bunch. 17 and
29 temper the commas out as well, but with a lot more damage.

46 704.34783
63 704.76190
80 705.00000
109 704.58716
126 704.76190
172 704.65116
189 704.76190

Interestingly, when I plug these commas into Graham's temperament finder, I
get 17 and 46 note scales, but no twelve-note scales. I haven't bothered to
compare them to see what the difference is.

Regards,
Jake

🔗Brofessor <kraiggrady@...>

If you look at the scale tree http://anaphoria.com/sctree.PDF page 11 it is easy to see many related scales in the series than runs up to it 12-17-29-46-63-80

--- In tuning@yahoogroups.com, Jake Freivald <jdfreivald@...> wrote:
>
> Here comes a classic Freivaldian overexplanation, cross-posted to the
> tuning list and the Facebook XA. I apologize in advance to those of you
> whom I bore.
>
> So, if you use the perfect fifth of 80 EDO as a generator -- that's 47\80,
> or exactly 705 cents -- then you get a 7- or 12-note MOS scales that temper
> out the following commas:
>
> -----
> 352/351, or "minthma", 4.93 cents, monzo | 5 -3 0 0 1 -1 >. This means that
> going down three fifths gets you a "minor third" of about 13/11.
>
> 896/891, or "pentacircle", 9.69 cents, monzo | 7 -4 0 1 -1 >. This is the
> product of gentle and minthma. It means that going up four fifths gets you
> a "major third" of about 14/11.
>
> 364/363, or "gentle", 4.76 cents, monzo | 2 -1 0 1 -2 1 >. This means that
> 14/11 * 13/11 = 3/2: A major third + a minor third = a perfect fifth.
>
> 6656/6561, 24.89 cents, monzo | 9 -8 0 0 0 1 >. This means that going up
> eight fifths gets you the thirteenth harmonic, 13/8. It also means that
> going up six fifths gets you 13/9, and going up seven fifths gets you 13/12.
>
> 180224/177147, 29.81 cents, monzo | 14 -11 0 0 1 >. This means that going
> up eleven fifths gets you the eleventh harmonic, 11/8. It also means that
> going up nine fifths gets you 11/9, and going up ten fifths gets you 11/6.
> -----
>
> I'm not sure of the names of the last two commas.
>
> Note that the effect of the first three commas is to do exactly what 81/80
> does for meantone, except instead of 5/4 major thirds and 6/5 minor thirds,
> you have 14/11 major thirds and 6/5 minor thirds. In other words, this is a
> scale with meantone-like structures, and if you tuned a piano to it, the
> modes would seem very familiar (black-key pentatonic, ionian, dorian,
> phyrigian, etc.) but have different flavors because of the different
> intervals.
>
> Now, if you've played with meantone tunings before, you know that they can
> get pretty good approximations to the seventh harmonic, 7/4, without really
> "meaning to" (i.e., they weren't part of classical theory). With a good
> 7/4, some other intervals creep in, such as 7/5, 10/7, and 7/6. Those
> intervals are there because the slightly flattened fifth drives them there:
> As you repeat the generator again and again, the error accumulates to give
> you (e.g.) those "flatter than minor" minor thirds and such.
>
> Similarly, this slightly sharp generator gets you pretty good
> approximations to the 11th and 13th harmonics (i.e., 13/8 and 11/8) without
> really "meaning to", and pulls in other intervals (13/9, 13/12, 11/9, and
> 11/6) in the process. (The 11/7 was already pretty good, because the scale
> was defined around a good 14/11, which is its inverse.) There's only one
> 11/8 in the whole scale, unfortunately, but it's there.
>
> The result is the same in both cases: Using a meantone generator will get
> you a 7-note MOS consisting of traditional diatonic scales, and going to
> the full 12-note MOS will get you the 7/4, 7/6, etc. Using a 47\80
> generator will get you a 7-note MOS consisting of traditional diatonic
> scales, but tuned differently using 14/11 and 13/11 thirds (among other
> differences); going to the full 12-note MOS will get you the 11/8, 13/8,
> 11/9, etc.
>
> This is cool because you can actually use this same scale to play I-IV-V
> chords with neutral thirds _or_ play I-IV-V chords with sharp-and-spicy
> 14/11 major thirds _or_ play i-iv-v chords with 13/11 soft-and-mournful
> minor thirds. But it's also cool because you can play things like
> 1/1-13/12-11/9-3/2-13/8 with pretty good accuracy, which sounds completely
> unlike anything remotely I-IV-V-ish. And you get it all within the same
> twelve notes.
>
> The following EDOs and generators temper out all of these commas, each with
> slightly different tradeoffs. 46 is the least accurate of the bunch. 17 and
> 29 temper the commas out as well, but with a lot more damage.
>
> 46 704.34783
> 63 704.76190
> 80 705.00000
> 109 704.58716
> 126 704.76190
> 172 704.65116
> 189 704.76190
>
> Interestingly, when I plug these commas into Graham's temperament finder, I
> get 17 and 46 note scales, but no twelve-note scales. I haven't bothered to
> compare them to see what the difference is.
>
> Regards,
> Jake
>

🔗Keenan Pepper <keenanpepper@...>

This sounds exactly like good old leapday temperament to me. I'm sure this is all very familiar to Margo Schulter since she's been a fan of exactly these kinds of scales for years and years.

Keenan

🔗Mike Battaglia <battaglia01@...>

🔗Jake Freivald <jdfreivald@...>

🔗jdfreivald@...

🔗Graham Breed <gbreed@...>

🔗Jake Freivald <jdfreivald@...>

🔗Keenan Pepper <keenanpepper@...>

--- In tuning@yahoogroups.com, Jake Freivald <jdfreivald@...> wrote:
>
> Thanks, Graham. Simpler does get me to a 12-note scale eventually, though
> not a 7-note one. I'll use more brute force on the "simpler" button in
> those situations from now on. :)

You can also just type in some vals that produce the temperament you want. Here's how I might do this:

First, go to "Temperament class from ETs" and type in some vals. For example if you type in "17 46" in the top field and "2.3.7.11.13" in the bottom field you get the temperament you're talking about (you can check to make sure it's the right one by looking at the mapping matrix, or the commas). If you enter "17 12" instead you get a choice between several different temperaments, and one of them, "17 & 12de", is the specific one you want. So now you know that "12de" is the correct val name. If you want a 7-note scale, try entering "7 12de" and seeing what comes up. Uh oh... now there is only one temperament shown, and it's not the right one. That means we need a different 7 val. If you look at the mapping matrix you see that 7, 11, and 13 are all mapped incorrectly, so try 7def. Hmmm, that isn't right either.

You could do trial and error at this point, or you could try to find the correct 7 val by doing a unison vector search for all your commas. If you go to the most "simple" side of the search, you get http://x31eq.com/cgi-bin/uv.cgi?uvs=169%3A168+352%3A351+364%3A363&page=13&limit=2_3_7_11_13 which shows 7dddeef as one of the rank-1 temperaments. Ah, so 7dddeef (or <7 11 18 23 25| in the 2.3.7.11.13 subgroup) is the val we're looking for.

So, if you type in "7dddeef 12de" into the "Temperament class from ETs" page, you get the temperament you're looking for, and it provides you with a 7-note Scala file.

(Alternatively, you could do it the easy way and let Scala make the MOS scale for you from the period and generator, which are always the same no matter what vals you use...)

> Thank you in general for the temperament finder, also. It's a great help in
> situations where I know what I'm looking for. I use the Regular Temperament
> Finder and the Unison Vector Search a fair amount.

I hope this post convinces you that "Temperament class from ETs" can also be useful. =)

Keenan

🔗monz <joemonz@...>

Hi all,

Just a reminder that you can use Tonescape to build
periodicity-blocks and their temperaments from unison-vectors,
and see the resulting lattice as well as compose in MIDI
with it.

http://tonalsoft.com

-monz

--- In tuning@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, Jake Freivald <jdfreivald@> wrote:
> >
> > Thanks, Graham. Simpler does get me to a 12-note scale eventually, though
> > not a 7-note one. I'll use more brute force on the "simpler" button in
> > those situations from now on. :)
>
> You can also just type in some vals that produce the temperament you want. Here's how I might do this:
>
> First, go to "Temperament class from ETs" and type in some vals. For example if you type in "17 46" in the top field and "2.3.7.11.13" in the bottom field you get the temperament you're talking about (you can check to make sure it's the right one by looking at the mapping matrix, or the commas). If you enter "17 12" instead you get a choice between several different temperaments, and one of them, "17 & 12de", is the specific one you want. So now you know that "12de" is the correct val name. If you want a 7-note scale, try entering "7 12de" and seeing what comes up. Uh oh... now there is only one temperament shown, and it's not the right one. That means we need a different 7 val. If you look at the mapping matrix you see that 7, 11, and 13 are all mapped incorrectly, so try 7def. Hmmm, that isn't right either.
>
> You could do trial and error at this point, or you could try to find the correct 7 val by doing a unison vector search for all your commas. If you go to the most "simple" side of the search, you get http://x31eq.com/cgi-bin/uv.cgi?uvs=169%3A168+352%3A351+364%3A363&page=13&limit=2_3_7_11_13 which shows 7dddeef as one of the rank-1 temperaments. Ah, so 7dddeef (or <7 11 18 23 25| in the 2.3.7.11.13 subgroup) is the val we're looking for.
>
> So, if you type in "7dddeef 12de" into the "Temperament class from ETs" page, you get the temperament you're looking for, and it provides you with a 7-note Scala file.
>
> (Alternatively, you could do it the easy way and let Scala make the MOS scale for you from the period and generator, which are always the same no matter what vals you use...)
>
> > Thank you in general for the temperament finder, also. It's a great help in
> > situations where I know what I'm looking for. I use the Regular Temperament
> > Finder and the Unison Vector Search a fair amount.
>
> I hope this post convinces you that "Temperament class from ETs" can also be useful. =)
>
> Keenan
>

🔗Margo Schulter <mschulter@...>

Hello, Jake and Keenan!

Keenan wrote:

> This sounds exactly like good old leapday temperament to
> me. I'm sure this is all very familiar to Margo Schulter since
> she's been a fan of exactly these kinds of scales for years
> and years.

Keenan, you're right, of course!

What you say has been true since the year 2000, when two of my
main interests were your proposed "Noble Fifth" tuning at 704.096
cents (the basis in 2002 for rank-3 Peppermint), and an e-based
temperament at 704.607 cents. The latter shares an important
element with a 705-cent tuning, Jake, which a comment you make
about meantone gives an opportunity to address.

Jake wrote:

>> Now, if you've played with meantone tunings before, you know
>> that they can get pretty good approximations to the seventh
>> harmonic, 7/4, without really "meaning to" (i.e., they weren't
>> part of classical theory). With a good 7/4, some other
>> intervals creep in, such as 7/5, 10/7, and 7/6.

Interestingly, this is also very much a characteristic of a
tuning with a fifth at 705.0 cents, or 2^(47/80). However, it
requires longer chains of fifths than in meantone, which is why
you wouldn't notice it in a 12-note set.

What happens at 705 cents is that 13 fourths up give us a
virtually just 9/7 at 435 cents (just would be 435.084 cents);
14 fifths, a near-7/6 at 270 cents; and 15 fifths, a near-7/4 at
975 cents. In a 24-note tuning, we get lots of these, as well as
the fine approximations you discussed of 14/13, 13/11, 11/9,
13/8, etc.

Since 3/2^14 would give 227.370 cents, narrow of 8/7 by the 3-7
schisma or 3.804 cents, while 14 fifths here approximate 7/6,
we might say that the comma tempered out is equal to the 3-7
schisma plus 49/48 (the difference between 8/7 and 7/6), or
14680064/14348907 (39.501 cents).

! reg705_24.scl
!
Regular 705-cent temperament, 24 of 80-tET
24
!
60.00000
135.00000
195.00000
210.00000
270.00000
285.00000
345.00000
420.00000
480.00000
495.00000
555.00000
630.00000
690.00000
705.00000
765.00000
840.00000
900.00000
915.00000
975.00000
990.00000
1050.00000
1125.00000
1185.00000
2/1

Best,

Margo

🔗Margo Schulter <mschulter@...>

Dear Jake, Keenan, and all,

In replying to your discussion on temperaments in the range
of 704-705 cents, and more specifically 705 cents (47/80 octave),
there's a very important about my own usage of such temperaments
that I should have made. This is a point that can get lost in
the theoretical and practical details that we love to explore
here.

What I should make clear is that in using something like the
e-based temperament at 704.607 cents -- part of the same family
as 705 cents -- I would not normally attempt major-major triadic
harmony, or for that matter the meantone modal styles of
16th-century Europe. I say "normally" meaning "in usual harmonic
timbres," since there are timbres which can make 420-cent major
thirds seem quite smooth and restful in the manner of something
much closer to 5/4.

Rather, I tend to use either 13th-14th century European
counterpoint, where major thirds around 14/11 fit quite
nicely, or some kind of Near Eastern style where Pythagorean
major third (sometimes a bit larger) are a typical
expectation.

However, I do understand that from a certain perspective,
using these large major thirds in styles where something
close to 5/4 might be expected could be an intonational
equivalent of being "cranked to 11," and might be heard
as "spiciness" rather than "out-of-tuneness."

There's a real hazard in tuning discussions of taking
things for granted that really aren't so obvious, for
example that fact that I would generally use meantone
for a meantone style, but something around 704-705 cents
for a medieval or neomedieval style, which seems to have
been my main interest over the last 12 years or so.

With many thanks,

Margo

🔗Jake Freivald <jdfreivald@...>

To Keenan, regarding the Temperament Classes from ETs: Um. Wow. :)

> (Alternatively, you could do it the easy way and let Scala make
> the MOS scale for you from the period and generator, which
> are always the same no matter what vals you use...)

That's more-or-less what I did, but I more-or-less stumbled into the scale.
I was actually building a comma pump using aprhowe. That's | 0 -3 0 -2 3
>, a.k.a. 1331/1323, a relatively simple no-fives 10.44-cent comma.

I wasn't looking at it for a musical goal, but just because it was a
relatively simple comma in the 11-limit. So I found an EDO that tempered
out aphrowe, 80 EDO, and started building a comma pump. I used a "down a
minor third, down a major third" pattern with 13/11 and 14/11 thirds; but
then I had to cancel out that pesky 13 because aphrowe is only 11-limit.
When I finished the pump, I tallied up the tones needed and found a
generator that would do the trick: ~3/2, or 705 cents.

When you use the fifth as a generator and 14/11 and 13/11 as thirds, you
get something with a meantone-ish structure (5L7S). That made me think of
other scales that I've thought to be similar, but this one had really nice
approximations to 13-ish intervals, which the others I'd played with didn't
have.

As it turns out, I didn't much care for the comma pump, but I liked the
scale.

> I don't think it has a trivial name yet, so there's no better name than
"2.3.7.11.13 leapday".

And I'm fine with not having Yet Another Name. It seems odd to me, though,
that it would be named after a comma that isn't in the basis of the
temperament. I get saying "2.3.5.13 porcupine", because 250/243 is a
5-limit comma and the five-limit is contained in the name: No matter what I
do with the 13, I get the same effect in the 5-limit because the porcupine
comma is tempered out. That's really not the case with a no-fives leapday.

To Margo:

I wrote:
>> Now, if you've played with meantone tunings before, you know
>> that they can get pretty good approximations to the seventh
>> harmonic, 7/4, without really "meaning to" (i.e., they weren't
>> part of classical theory). With a good 7/4, some other
>> intervals creep in, such as 7/5, 10/7, and 7/6.

Margo replied:
> Interestingly, this is also very much a characteristic of a
> tuning with a fifth at 705.0 cents, or 2^(47/80). However, it
> requires longer chains of fifths than in meantone, which is why
> you wouldn't notice it in a 12-note set.

Thank you very much for pointing that out. You're obviously right, but I
didn't notice because I rarely go beyond 12 notes, for technical reasons --
such as that I'm technically too lazy so far to figure out how to do it
easily with LilyPond and Timidity. :)

Also, it's interesting that you wouldn't use 14/11 and 13/11 for triadic
harmony. I understand why now, but when I started I didn't know the history
of Pythagorean tuning, or that 407 cents sounded out of tune to people back
in the day; I certainly didn't realize that 417 cents would be even more
extreme. I used Gene's Cantonpenta scale to do a lot of quartal and quintal
harmony (fancy words for the no-thirds power chords, sus2, and sus4 chords
that I grew up with on guitar) that finally settled down to major and minor
triads. I liked those sounds -- even using strings, which should be one of
the worst timbres to use. Here's an example of a "work in progress" (that
hasn't progressed for about a year):
https://soundcloud.com/jdfreivald/cantonpenta-chord-progression

Under your influence, I was thinking of trying to harmonize some Gregorian
Chant with one of these 14/11 / 13/11 scales, keeping in mind your idea
that the thirds would be a point of tension rather than one of rest.
There's a trove of chant and related music available for download here:
http://jeandelalande.org/HOME/index.htm ...and while I haven't had much
patience with traditional counterpoint, it seemed like this would be a
place to start trying more. I probably won't get to it for a little while,
though.

It's a little late here, and I'm not sure I'm coherent, but thank you both
again for your comments.

Regards,
Jake

🔗Margo Schulter <mschulter@...>

Hello, Jake and Keenan!

Jake wrote:

> I wasn't looking at it for a musical goal, but just because it
> was a relatively simple comma in the 11-limit. So I found an
> EDO that tempered out aphrowe, 80 EDO, and started building a
> comma pump. I used a "down a minor third, down a major third"
> pattern with 13/11 and 14/11 thirds; but then I had to cancel
> out that pesky 13 because aphrowe is only 11-limit. When I
> finished the pump, I tallied up the tones needed and found a
> generator that would do the trick: ~3/2, or 705 cents.

Jake, it's fascinating to compare notes on the different ways
that people seek out and find interesting tunings! The way I got
to 704.607 cents was a bit more random: I simply decided to set
Easley Blackwood's R, the ratio between whole tone and diatonic
semitone (the L and s in a 7-MOS), to Euler's e (2.71828...)!
My idea, I guess, was something about midway between 29-EDO
(R=2.5) and 17-EDO (R=3). But that e-based tuning and 705 cents
(R=2.8) are close neighbors, with 14/11 as a common theme.

[Keenan wrote:]

> I don't think it has a trivial name yet, so there's no better
> name than "2.3.7.11.13 leapday".

Keenan, in a 29-MOS of Herman Miller's leapday, we get 5/4 with
21 fifths up. In a 29-MOS at 705 cents, this gives 405 cents,
near 81/64 with 25 fourths at 375 cents closer to 5/4; and in a
46-MOS, 38 fifths up gives 390 cents, comparable to 21 fifths up
in 46-EDO as a good example of leapday. The name leapday refers
to the full 13-limit in a 29-MOS, thus February 29, so this would
be a different mapping, evidently 13-limit srutal (POTE 704.881
cents), with 80-EDO listed as an example on the xenwiki.

For a 12-MOS at 705 cents, why not "2.3.7.11.13 gentle"? We can
define gentle in a 12-MOS by the regular thirds around 14/11 and
13/11; augmented seconds (+9 fifths) as supraminor or small
neutral thirds somewhere from around 63/52 to 11/9; and
diminished fourths (-8 fifths) as submajor or large neutral
thirds somewhere from around 26/21 to 16/13.

A suggested range for gentle is from around 29-EDO to 80-EDO, so
here we're at the upper end, with 14/11 a bit wide, 13/11 narrow
at a near-just 33/28, and neutral thirds near 16/13 and 11/9.

Jake, you've addressed a number of the relevant commas, and
here's an often similar non-5 perspective on this region:

<http://launch.group.yahoo.com/group/tuning/message/11232>.

If we define gentle broadly based on a 12-MOS, this leaves open
the possibility of specific mappings within its region that seek
ratios like 7/4 from 15 fifths up; or 5/4 from 21 fifths up
(leapday), 25 fifths down (~704.547 cents), or 38 fifths up
(~704.903 cents).

Likewise, for 2.3.7 intervals, a rank-3 strategy is attractive in
the lower ranges of gentle, i.e. parapyth (George Secor's 29-HTT,
MET-24, parapyth POTE, Cantonpenta, Peppermint). By around
46-EDO, however, and even more so at 704.6-705.0 cents, a rank-2
strategy is quite accurate (109-EDO, e-based, 63-EDO, 80-EDO).

But for a 12-MOS, and a wonderful one at that anywhere in this
region, the name gentle may be most descriptive and flexible.
Maybe a 17-MOS at 705 cents could also be leapfrog, which tempers
out 169/168.

Best,

Margo

🔗gedankenwelt94 <gedankenwelt94@...>

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:
> [Keenan wrote:]
>
> > I don't think it has a trivial name yet, so there's no better
> > name than "2.3.7.11.13 leapday".
>
> Keenan, in a 29-MOS of Herman Miller's leapday, we get 5/4 with
> 21 fifths up. In a 29-MOS at 705 cents, this gives 405 cents,
> near 81/64 with 25 fourths at 375 cents closer to 5/4; and in a
> 46-MOS, 38 fifths up gives 390 cents, comparable to 21 fifths up
> in 46-EDO as a good example of leapday. The name leapday refers
> to the full 13-limit in a 29-MOS, thus February 29, so this would
> be a different mapping, evidently 13-limit srutal (POTE 704.881
> cents), with 80-EDO listed as an example on the xenwiki.

Thanks, that's interesting - I never thought about defining patent vals based on an MOS (instead of the underlying tuning)!

13-limit srutal should work if we add the diaschisma 2048/2025 to Jake's comma list, and apply a period of 1\2.

If we want an octave period, we could call it 2.3.7.11.13 srutal, or 2.3.25.7.11.13 srutal if we add 2048/2025, I guess, but that means no 5/4.

Please correct me if I'm wrong, I'm still not too confident with regular temperaments with a period other than the octave. ;)

> Jake, you've addressed a number of the relevant commas, and
> here's an often similar non-5 perspective on this region:
>
> <http://launch.group.yahoo.com/group/tuning/message/11232>.

I suppose you meant /tuning/topicId_11232.html#11232 ;)

Many thanks to all who contributed to this discussion, I found it very inspiring! :)

🔗Margo Schulter <mschulter@...>

Hello, Jake.

This is a reply to address a few points you raised in your recent
message to Keenan and me. On one of these points, the best answer
is that you've raised a question where I'd like to help in
figuring out the best answer.

> Thank you very much for pointing that out. You're obviously
> right, but I didn't notice because I rarely go beyond 12 notes,
> for technical reasons -- such as that I'm technically too lazy
> so far to figure out how to do it easily with LilyPond and
> Timidity. :)

I've used Timidity and know that I should learn LilyPond, so this
is a question where having a good solution available for handling
larger tuning sets could help lots of people. Of course, there
may be experts here who could suggest such a solution, but I'd be
glad to help if I can.

With Timidity, I know that either MIDI pitch bends or the more
elegant appraoch of a tuning table or .tbl file could support
these larger sets, but am not sure yet how this would tie in with
LilyPond. Scala can generate a .tbl file, although that might or
might not be relevant to the situation with LilyPond. For example,
for a 24-note regular tuning at 705 cents:

<http://www.bestII.com/~mschulter/reg705-24.scl>
<http://www.bestII.com/~mschulter/reg705-24.tbl>

> Also, it's interesting that you wouldn't use 14/11 and 13/11
> for triadic harmony. I understand why now, but when I started I
> didn't know the history of Pythagorean tuning, or that 407
> cents sounded out of tune to people back in the day; I
> certainly didn't realize that 417 cents would be even more
> extreme.

If "back in the day" means the 13th and 14th centuries, then
81/64 was fine as the everyday major third, relatively blending
but unstable, with yet wider thirds like 14/11 or 9/7 occurring
in expressive styles of intonation. But by the 16th century, with
the 4:5:6 and meantone now in vogue, the older 81/64 indeed
"sounded out of tune," with a meantone diminished fourth at
around 14/11, 32/25, or 9/7 "even more extreme."

So my hesitation would be to use something like 14/11 for
"triadic harmony" where major and minor thirds are expected to be
stable, and something close to 4:5:6 is expected, as in the
16th-17th centuries.

But from your perspective, my remark on styles like that of the
16th or early 17th century where "something close to 4:5:6 is
expected" might not necessarily apply to music with stable major
and minor triads in the 21st century, where influences such as
rock or extended jazz harmonies a la Mike Battaglia might open
things up to a much wider range of intonations. So what was
pleasantly spicy 700 years ago in an unstable context, a 14/11 or
whatever, might again be pleasantly spicy in some major/minor
triadic contexts as well!

> I used Gene's Cantonpenta scale to do a lot of quartal and
> quintal harmony (fancy words for the no-thirds power chords,
> sus2, and sus4 chords that I grew up with on guitar) that
> finally settled down to major and minor triads.

The interesting thing is that 2:3:4, 8:9:12, and 6:8:9 are also
very common in 13th-century European music, and I agree that
these tunings are great for quartal/quintal harmony! Around 705
cents or 3 cents wide isn't that much more tempering for fifths
and fourths than in 12-EDO, and in the region of 704 cents, as
with Gene's Cantonpenta, the impurity is almost identical or even
sometimes a bit less, albeit in the opposite direction.

> I liked those sounds -- even using strings, which should be one
> of the worst timbres to use. Here's an example of a "work in
> progress" (that hasn't progressed for about a year):
> <https://soundcloud.com/jdfreivald/cantonpenta-chord-progression>

Your link illustrates my technical problem in accessing music not
downloadable, if I'm right -- does this call for an application
like RealPlayer or the like, which I've not yet managed to
install? But I'd love to hear this!

> Under your influence, I was thinking of trying to harmonize
> some Gregorian Chant with one of these 14/11 / 13/11 scales,
> keeping in mind your idea that the thirds would be a point of
> tension rather than one of rest. There's a trove of chant and
> related music available for download here:
> [44]http://jeandelalande.org/HOME/index.htm ...and while I
> haven't had much patience with traditional counterpoint, it
> seemed like this would be a place to start trying more. I
> probably won't get to it for a little while, though.

Thank you for the link to the webpage on chant. One thing I
should clarify is that 13th-14th century medieval polyphony is
quite different from the "traditional counterpoint" usually
taught based on 16th-18th century European practice. They're
both beautiful traditions, but quite distinct.

To give some idea of the medieval traditon, here are two
pieces as recorded by Aaron Johnson in temperaments with thirds
near 14/11 and 13/11, the first piece from the 13th century and
the second from the late 14th century:

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

Here's a chant setting I did which might fit any of these
tunings:

<http://www.bestII.com/~mschulter/KyrieCunctipotens.pdf>

> It's a little late here, and I'm not sure I'm coherent, but
> thank you both again for your comments.

You're coherent, informative, and very readable -- very important
virtues in explaining these tunings!

Best,

Margo

🔗Margo Schulter <mschulter@...>

Hello, gendankenwelt94, and thank you both for some very thoughtful
comments and a helpful correction which I welcome even more than
I regret my failure to catch the erroneous link in the first
place!

>> Keenan, in a 29-MOS of Herman Miller's leapday, we get 5/4 with
>> 21 fifths up. In a 29-MOS at 705 cents, this gives 405 cents,
>> near 81/64 with 25 fourths at 375 cents closer to 5/4; and in a
>> 46-MOS, 38 fifths up gives 390 cents, comparable to 21 fifths up
>> in 46-EDO as a good example of leapday. The name leapday refers
>> to the full 13-limit in a 29-MOS, thus February 29, so this would
>> be a different mapping, evidently 13-limit srutal (POTE 704.881
>> cents), with 80-EDO listed as an example on the xenwiki.

> Thanks, that's interesting - I never thought about defining
> patent vals based on an MOS (instead of the underlying tuning)!
> 13-limit srutal should work if we add the diaschisma 2048/2025
> to Jake's comma list, and apply a period of 1\2.

Maybe my general approach would be to look at how primes or other
intervals are mapped in a given tuning size, which might
or might not be an MOS. For example, in a 24-note set with a
generator in the range of around 704.6-705.0 cents, the fact that
25 generators would be more or less close to 5/4 might be a bit
academic, and not so relevant to the purposes of the 24-note set,
for example 2.3.7.11.13.23. But that bit of information might be
less academic if someone has 5 as a priority and is ready to
expand the set! So various approaches might be relevant. With
leapday, I focused on a 29-MOS because that was the basis for
Herman Miller's name (i.e. February 29).

David Keenan, around 2000 or a bit earlier, focused on the
diaschismic (e.g. srutal) mapping with 1\2 as a period. This kind
of period other than the octave isn't so familiar to me either,
and it can lead to a different sense of classification. For
example, I would reason, "around 46-EDO, 21 fifths yield 5/4;
around 80-EDO, 38 fifths." As someone often involved with rank-2
or rank-3 tunings, I tend to focus on the chain of fifths. But
someone attuned to the diaschismic concept might say, "Ah, but
the two are really similar, in that each derives 5/4 from the
regular minor seventh less 600 cents." Like you, I'm not so
familiar with these periods other than 2/1.

Also, I tend to reason in terms of commas, although not
necessarily in an RMP manner (which I'll let others judge). For
example, I would note that around 704.0-704.4 cents, say, a close
5/4 results from 21 fifths, that is, a regular major third around
14/11 less a 17-comma on the order of 25-30 cents (in effect a
56/55). But by 705 cents, with the 17-comma at only 15 cents, we
need two of them to reduce a regular 420-cent major third to ~5/4 at 390 cents -- or +4 fifths for the 14/11, and +34 fifths for
the two 17-commas by which it is narrowed. These seem to me very
different situations, although from a diaschismic perspective
they might be synonymous (minor sevent less 600 cents gives 5/4).

> If we want an octave period, we could call it 2.3.7.11.13
> srutal, or 2.3.25.7.11.13 srutal if we add 2048/2025, I guess,
> but that means no 5/4. Please correct me if I'm wrong, I'm
> still not too confident with regular temperaments with a
> period other than the octave. ;)

Clearly you're far more acquainted with RMP theory than I am, so
others will need to judge your srutal analysis. Does your
notation refer to a 25/16 approximation? A curious thing I note
in Scala is that the two most common minor sixths, the regular
one at 780 cents and the septimal one at 765 cents, are almost
identically distant from 25/16 at 772.627 cents. Interesting how
either approximation is off by something a bit less than 225/224
with a 705-cent generator.

> I suppose you meant
> </tuning/topicId_11232.html#11232>

Thank you for catching and fixing this. My apologies for somehow
not testing the link before I posted it, but I'm so glad that you
repaired my oversight!

> Many thanks to all who contributed to this discussion, I found
> it very inspiring! :)

Indeed, most definitely including your contribution!

Best,

Margo

🔗Marcel de Velde <marcel@...>

Hello Margo,

I belief in some English music in this period the 81/64 major third quickly
became used in a "stable" way, not requiring resolution.

But by the 16th century, with
the 4:5:6 and meantone now in vogue, the older 81/64 indeed
"sounded out of tune," with a meantone diminished fourth at
around 14/11, 32/25, or 9/7 "even more extreme."

81/64 as "out of tune" is quite a strong statement.
I can show you any music from the 16th century on tuned to Pythagorean and
to most ears it will sound much more in tune than meantone.
Here 2 examples:
http://www.youtube.com/watch?v=XbGq43Ol0tk
http://www.youtube.com/watch?v=3-wq-yVGXU8

I do agree that for much music from a certain time period meantone is
historically correct, and that certain composers believed meantone to be the
best tuning (like Mozart's father). But that's a different thing than
calling 81/64 "out of tune".

As for the ~705 cents fifths range of temperaments.
I find these interesting as well for the following reason:
They turn the augmented prime into a small neutral second, and the
diminished third into a large neutral second (and the diminished fourth and
augmented second into neutral thirds).
So one can easily express maqams melodically, while also harmonizing them
with chromatic chord progressions following western harmony and counterpoint
rules.
I've done this and found that to me the additional deviation from
Pythagorean did not bring me anything, I prefer to tune pure, but for others
this may be different so I thought I'd mention it.

Kind regards,

Marcel

🔗Graham Breed <gbreed@...>

🔗Margo Schulter <mschulter@...>

Marcel wrote:

> Hello Margo,

Hello, Marcel.

[On 13th-14th century English music]

> I belief in some English music in this period the 81/64 major
> third quickly became used in a "stable" way, not requiring
> resolution.

We know, both from actual music and from treatises,
that major thirds were treated as stable and sometimes as
conclusive in some English styles -- but the question of their
tuning remains open. I am inclined to join Christopher Page, a
strong advocate of Pythagorean or wider major thirds for
Continental music of the era, in supposing that these stable
English thirds were "mollified" at something around 5/4.

Walter Odington, around 1300, tells us that 81/64 and 32/27 are
not far from the simpler ratios of 5/4 and 6/5, and that singers
make these intervals "fully consonant," which may suggest some
intonational nuances. In contrast, descriptions in Continental
treatises of the same era that the voices of a third "are heard
to differ greatly, and yet concord," or that thirds "are neither
perfectly concordant nor perfectly discordant" would fit nicely
with 81/64 and 32/27.

>> But by the 16th century, with the 4:5:6 and meantone now in
>> vogue, the older 81/64 indeed "sounded out of tune," with a
>> meantone diminished fourth at around 14/11, 32/25, or 9/7
>> "even more extreme."

> 81/64 as "out of tune" is quite a strong statement.

Since Jake, whom I was quoting on "out of tune," and I, very much
like large major thirds for various purposes, what we are both
addressing is the idea that 81/64 or 14/11 can sound "unstylish"
in a 16th-century context where some shading of meantone
(1/3-comma to 1/11 comma) would be the likely historical choice.
It's a beautiful major third in a less happy context.

This assumes harmonic timbres, and focuses specifically on the
16th and earlier 17th centuries, the periods which "music based
on stable major and minor thirds" mostly suggest to me in my own
musical activity. A main point which Jake clarified, with my
hearty agreement, is that 16th-century meantone norms do not
necessarily apply to 21st-century music also based on stable
major and minor thirds, but in what might be a quite different
stylistic context -- so that a 704-cent or 705-cent temperament
might be quite fine, as Jake finds it in practice.

> I can show you any music from the 16^th century on tuned to
> Pythagorean and to most ears it will sound much more in tune
> than meantone.
> Here 2 examples:
> [42]http://www.youtube.com/watch?v=XbGq43Ol0tk
> [43]http://www.youtube.com/watch?v=3-wq-yVGXU8

Since I can download and play audio files, but not video, I must
reserve judgment until I have heard these.

> I do agree that for much music from a certain time period
> meantone is historically correct, and that certain composers
> believed meantone to be the best tuning (like Mozart's
> father). But that's a different thing than calling 81/64 "out
> of tune".

The phrase "out of tune" can indeed be dangerous when people take
it as a statement about an interval in itself, rather than an
interval placed in a wrong or less apposite stylistic context.

As Mark Lindley has observed, there are some earlier 16th-century
pieces for lute that might fit a Pythagorean tuning; and in the
later 15th century, the traditional Pythagorean ideal and the
newer 5-limit or meantone outlook may have been in competition.
Pythagorean can nicely fit some of Ockeghem's textures and
contrapuntal passages, for example. With some German
music around 1500 in a traditional medieval style, of course,
Pythagorean could be beautiful!

But in the 16th century, there's a general consensus that what we
now term meantone is the standard keyboard tuning (whether
measured in fractional syntonic commas or otherwise), and that
tempering of fifths in the narrow direction is necessary for
such instruments. This fits the rules of counterpoint, which
strongly prefer and privilege thirds and sixths, with 4:5:6 as
the ideal. This is neither more nor less "natural" than
Pythagorean tuning or temperaments with wide fifths; it is simply
the general style and taste of that era.

And in a harmonic timbre, my ears agree that meantone wonderfully
agrees with a 16th-century style, and Pythagorean likewise with a
14th-century Continental style.

> As for the ~705 cents fifths range of temperaments.
> I find these interesting as well for the following reason:
> They turn the augmented prime into a small neutral second, and
> the diminished third into a large neutral second (and the
> diminished fourth and augmented second into neutral thirds).

This is precisely one of the main charms of these temperaments
from around 703.45 to 705 cents, let us say, and also heavier
temperaments out to 17-EDO (705.88 cents), where the diminished
fourth and augmented second are identical at around 353 cents.

> So one can easily express maqams melodically, while also
> harmonizing them with chromatic chord progressions following
> western harmony and counterpoint rules.

Certainly we can agree about these temperaments providing many of
the genera of maqam music, which indeed is one of my primary uses.
The question of whether, when, and why to develop hybrid forms
combining maqam with European polyphony, as I have done at times
in ways involving elements and patterns of 13th-14th century
European technique, is really a discussion in itself.

Best,

Margo

🔗Marcel de Velde <marcel@...>

Hi Margo,

Hello, Marcel.

[On 13th-14th century English music]

> I belief in some English music in this period the 81/64 major
> third quickly became used in a "stable" way, not requiring
> resolution.

Ah yes, agreed.
Also worth mentioning is perhaps that Ramis, who is accredited with making
the 5/4, and therefore ultimately meantone temperaments, popular,
based this 5/4 major third on his observations on how singers naturally sing
the major third.
Which also suggests that (at least some) practice of 5/4 major thirds
probably pre-dated the meantone era.

However, what I do not see is a strong link between differences in meantone
and Pythagorean leading to different music.
The music is all fifth based, and all music from the middle ages till now
functions perfectly well in Pythagorean.
And in my opinion would have just as well been written had the popular
tuning of the 16th century till the 20th century been Pythagorean instead of
meantone.

> 81/64 as "out of tune" is quite a strong statement.

Aah ok, sorry I missed that context of out of tune.
Yes agreed on unstylish. Not in an objective musical context (as if
meantone is implied by the music), but in a historically correct context
meantone is of course known to be correct.

Since I can download and play audio files, but not video, I must
reserve judgment until I have heard these.

I think you may have heard them before.
2 short piano pieces, one by Mozart and one by Mussorgsky, tuned to 12tet,
Pythagorean, ¼ comma meantone and 17tet (in that order).
I dont have them online in a different format. But you can download the
audio for the videos using an online converter like
http://www.listentoyoutube.com
If this doesnt work for you and you wish to hear them let me know and Ill
upload the audio somewhere or send it to you in a private email.

The phrase "out of tune" can indeed be dangerous when people take
it as a statement about an interval in itself, rather than an
interval placed in a wrong or less apposite stylistic context.

Well I dont really see it the same as you do.
I hear Pythagorean (or 12tet even) as fitting in all music.
The music itself does not determine a preference for a different temperament
to me.
The functioning of western music is fifth based. I see a major third as
de-constructible into 4 fifths (like a circle progression does for instance)
and so does western music.
I dont see any functional change by using 5/4 major thirds, nor does this
change the music. Its simply a different timbral color to me.
So why would some music fit meantone better than Pythagorean?

Style and taste yes.
But necessary, I dont see it. Meatone does give harsher sounds a more
pleasant timbre than Pythagorean.
But I do not see it as necessary for any instruments, including harpsichord.
Personal preferences do come in play here.

Cant agree with counterpoint rules though.
To me they imply the exact opposite.
The thirds and sixths are not perfect consonances in counterpoint, but
specifically imperfect consonances.
And the rules for imperfect consonances are very different from the perfect
consonances.
For instance: no parallel fifths or octaves allowed, but parallel thirds and
sixths are fine (within reason).

And in a harmonic timbre, my ears agree that meantone wonderfully
agrees with a 16th-century style, and Pythagorean likewise with a
14th-century Continental style.

> So one can easily express maqams melodically, while also
> harmonizing them with chromatic chord progressions following
> western harmony and counterpoint rules.

Perhaps a discussion well have somewhere in the future! :)
I find this one of the most interesting topics there are.
The neutral intervals and how they function in a harmonic context.

Kind regards,

Marcel