13-limit and continued fractions

Hey folks, I know I've been quiet lately; I'll be in lurk mode for a while and won't be posting much except if/when I get music written.

I have a couple questions:

First, I've been Googling and searching the archives to see if there's a linear or equal tempermant like MIRACLE, but extended to 13-limit. I'm seeing 175-EDO in the tuning-math archives, and 224 and 270 also work well, but I'd like something simpler. 53-tone will do for now despite its weakness in the prime 11 (the only mismapped intervals in the 15-limit square are 14/11 and 11/7), and the kind of music I'm into relies on strong fifths and fourths.

Second, what do you call this method of JI scale derivation:

1) Take a non-JI scale: equal temperament, well-temperament, meantone, etc.
2) Find the continued fractions of the frequency ratios.
3) Replace the pitches with the simplest unique fraction of each pitch.

So 12-tET comes out to [1/1, 17/16, 9/8, 6/5, 5/4, 4/3, 7/5, 3/2, 8/5, 5/3, 7/4, 15/8, 2/1]. A basic JI scale, more suited for monophony (6/5 to 7/4 is not a good fifth). This method would be more useful for larger scales.

~D.

--- In tuning@yahoogroups.com, "Danny Wier" <dawiertx@...> wrote:

> First, I've been Googling and searching the archives to see if
there's a
> linear or equal tempermant like MIRACLE, but extended to 13-limit. I'm
> seeing 175-EDO in the tuning-math archives, and 224 and 270 also
work well,
> but I'd like something simpler.

The 224&270 linear temperament, with period a half-octave and
generator a 44/39, is a standout among ultra high precision 13-limit
linear temperaments, but of course it is very complex. If you want
simpler temperaments which can beat its logflat badness figure, I'm
afraid there don't seem to be any. However 58 notes of the "mystery"
temperament, the 29&58 temperament, sounds like what you might be
looking for. This would be two 29-et cycles stacked unevenly,
separated by 16 cents.

Gene Ward Smith wrote:

> --- In tuning@yahoogroups.com, "Danny Wier" <dawiertx@...> wrote:
>
>> First, I've been Googling and searching the archives to see if
> there's a
>> linear or equal tempermant like MIRACLE, but extended to 13-limit. I'm
>> seeing 175-EDO in the tuning-math archives, and 224 and 270 also
> work well,
>> but I'd like something simpler.
>
> The 224&270 linear temperament, with period a half-octave and
> generator a 44/39, is a standout among ultra high precision 13-limit
> linear temperaments, but of course it is very complex. If you want
> simpler temperaments which can beat its logflat badness figure, I'm
> afraid there don't seem to be any. However 58 notes of the "mystery"
> temperament, the 29&58 temperament, sounds like what you might be
> looking for. This would be two 29-et cycles stacked unevenly,
> separated by 16 cents.

I think I asked the same question months ago and you gave me the same answer. Thanks again then.

~D.

> I have a couple questions:
>
> First, I've been Googling and searching the archives to see
> if there's a linear or equal tempermant like MIRACLE, but
> extended to 13-limit. I'm seeing 175-EDO in the tuning-math
> archives, and 224 and 270 also work well, but I'd like
> something simpler. 53-tone will do for now despite its
> weakness in the prime 11 (the only mismapped intervals in
> the 15-limit square are 14/11 and 11/7), and the kind of
> music I'm into relies on strong fifths and fourths.

Territory beyond the 11-limit isn't well explored.
Nevertheless, Gene may have something to say. Have
you looked at 612?

> Second, what do you call this method of JI scale derivation:
>
> 1) Take a non-JI scale: equal temperament, well-temperament,
> meantone, etc.
> 2) Find the continued fractions of the frequency ratios.
> 3) Replace the pitches with the simplest unique fraction of
> each pitch.

I'd call that "rational approximation of an irrational
scale". Could you explain step 3) a little more? Come
come every scale of n tones/octave doesn't come out the
same?

-Carl

That's not such a bad scale for harmony:

input 17/16 9/8 6/5 5/4 4/3 7/5 3/2 8/5 5/3 7/4 15/8 2/1
set attrib notation
chords/match/constrained "major triad"

0-5-9: Major Triad 2nd inversion 3:4:5
C F A
0-4-7: Major Triad 4:5:6
C E G
0-3-8: Neapolitan Sixth, Major Triad 1st inversion 5:6:8
C Eb G#
2-7-11: Major Triad 2nd inversion 3:4:5
D G B
3-8-12: Major Triad 2nd inversion 3:4:5
Eb G# C
4-7-12: Neapolitan Sixth, Major Triad 1st inversion 5:6:8
E G C
5-9-12: Major Triad 4:5:6
F A C
7-12-16: Major Triad 2nd inversion 3:4:5
G C E
7-11-14: Major Triad 4:5:6
G B D
8-12-15: Major Triad 4:5:6
G# C Eb
9-12-17: Neapolitan Sixth, Major Triad 1st inversion 5:6:8
A C F
11-14-19: Neapolitan Sixth, Major Triad 1st inversion 5:6:8
B D G
Total of 12

chords/match/constrained "minor triad"

0-3-7: Minor Triad 10:12:15
C Eb G
0-4-9: Minor Triad 1st inversion 12:15:20
C E A
0-5-8: Minor Triad 2nd inversion 15:20:24
C F G#
3-7-12: Minor Triad 1st inversion 12:15:20
Eb G C
4-7-11: Minor Triad 10:12:15
E G B
4-9-12: Minor Triad 2nd inversion 15:20:24
E A C
5-8-12: Minor Triad 10:12:15
F G# C
7-11-16: Minor Triad 1st inversion 12:15:20
G B E
7-12-15: Minor Triad 2nd inversion 15:20:24
G C Eb
8-12-17: Minor Triad 1st inversion 12:15:20
G# C F
9-12-16: Minor Triad 10:12:15
A C E
11-16-19: Minor Triad 2nd inversion 15:20:24
B E G
Total of 12

Actually, for the life of me I couldn't figure out how to enter the
scale on one line in Scala. Notice the frequency ratios on the
right. They can appear in the chromatic clavier too, now. :)

Joe

--- In tuning@yahoogroups.com, "Danny Wier" <dawiertx@...> wrote:
>
> Hey folks, I know I've been quiet lately; I'll be in lurk mode for
a while
> and won't be posting much except if/when I get music written.
>
> I have a couple questions:
>
> First, I've been Googling and searching the archives to see if
there's a
> linear or equal tempermant like MIRACLE, but extended to 13-limit.
I'm
> seeing 175-EDO in the tuning-math archives, and 224 and 270 also
work well,
> but I'd like something simpler. 53-tone will do for now despite
its weakness
> in the prime 11 (the only mismapped intervals in the 15-limit
square are
> 14/11 and 11/7), and the kind of music I'm into relies on strong
fifths and
> fourths.
>
> Second, what do you call this method of JI scale derivation:
>
> 1) Take a non-JI scale: equal temperament, well-temperament,
meantone, etc.
> 2) Find the continued fractions of the frequency ratios.
> 3) Replace the pitches with the simplest unique fraction of each
pitch.
>
> So 12-tET comes out to [1/1, 17/16, 9/8, 6/5, 5/4, 4/3, 7/5, 3/2,
8/5, 5/3,
> 7/4, 15/8, 2/1]. A basic JI scale, more suited for monophony (6/5
to 7/4 is
> not a good fifth). This method would be more useful for larger
scales.
>
> ~D.
>

--- In tuning@yahoogroups.com, "Danny Wier" <dawiertx@...> wrote:
>
> Hey folks, I know I've been quiet lately; I'll be in lurk mode for
a while
> and won't be posting much except if/when I get music written.
>
> I have a couple questions:
>
> First, I've been Googling and searching the archives to see if
there's a
> linear or equal tempermant like MIRACLE, but extended to 13-limit.
I'm
> seeing 175-EDO in the tuning-math archives, and 224 and 270 also
work well,
> but I'd like something simpler. 53-tone will do for now despite its
weakness
> in the prime 11 (the only mismapped intervals in the 15-limit
square are
> 14/11 and 11/7), and the kind of music I'm into relies on strong
fifths and
> fourths.

130-ET is 15-limit consistent and has 15-limit error slightly better
than the 11-limit error for Miracle, but I image that's also too
complicated for you. I don't imagine that there's any generating
interval in 130 that will give all 5 primes in a sequence of tones
reasonably small in number. (Perhaps Gene might have an idea for
something involving more than one chain of generators.)

--George

How about 79/80 MOS 159-tET? It functions in 13-limit too.

----- Original Message -----
From: "George D. Secor" <gdsecor@yahoo.com>
To: <tuning@yahoogroups.com>
Sent: 28 Eyl�l 2006 Per�embe 17:59
Subject: [tuning] Re: 13-limit and continued fractions

> --- In tuning@yahoogroups.com, "Danny Wier" <dawiertx@...> wrote:
> >
> > Hey folks, I know I've been quiet lately; I'll be in lurk mode for
> a while
> > and won't be posting much except if/when I get music written.
> >
> > I have a couple questions:
> >
> > First, I've been Googling and searching the archives to see if
> there's a
> > linear or equal tempermant like MIRACLE, but extended to 13-limit.
> I'm
> > seeing 175-EDO in the tuning-math archives, and 224 and 270 also
> work well,
> > but I'd like something simpler. 53-tone will do for now despite its
> weakness
> > in the prime 11 (the only mismapped intervals in the 15-limit
> square are
> > 14/11 and 11/7), and the kind of music I'm into relies on strong
> fifths and
> > fourths.
>
> 130-ET is 15-limit consistent and has 15-limit error slightly better
> than the 11-limit error for Miracle, but I image that's also too
> complicated for you. I don't imagine that there's any generating
> interval in 130 that will give all 5 primes in a sequence of tones
> reasonably small in number. (Perhaps Gene might have an idea for
> something involving more than one chain of generators.)
>
> --George
>
>

Carl Lumma wrote:

> Territory beyond the 11-limit isn't well explored.
> Nevertheless, Gene may have something to say. Have
> you looked at 612?

I wish Partch hadn't have stopped at 11. (I still need to finish reading "Genesis" and then read .)

I've used both 612 and 665 as as schisma/twelfth-comma measurements. Realistically, I could get by with quarter-comma EDOs (212, 217 and 224), since if I'm playing fretless bass in realtime, I don't think of ET, I think of Pythagorean tones and semitones, commas, half-commas and quarter-commas. And third-commas if I feel the need to go to 17-limit or higher.

> I'd call that "rational approximation of an irrational
> scale". Could you explain step 3) a little more? Come
> come every scale of n tones/octave doesn't come out the
> same?

It's definitely a rational approximation, but there's no prime limit; nor is there a set limit in the denominator like there is with Farey.

I did need to explain better. I'll demonstrate using 5-tET fake slendro to be brief.

0. 0 cents
1. 240
2. 480
3. 720
4. 960
5. 1200

Then I convert to continued fractions (Scala has this function):

0. [1;]
1. [1;6,1,2,1,1,1,3,25,1,4,3,3,7,54,2,44,7,4,1,...]
2. [1;3,7,1,2,2,1,2,4,56,1,14,2,1,1,3,3,2,4,18,...]
3. [1;1,1,15,2,2,5,2,113,1,6,1,2,3,1,1,2,2,1,8,...]
4. [1;1,2,1,6,3,1,1,1,12,2,2,6,1,1,1,3,106,1,1,...]
5. [2;]

I'll use a continued fraction calculator like this one, http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/cfCALC.html, and come up with lists of fractions for each pitch. (I'll limit denominators to 100.)

0. 1/1
1. 1/1, 7/6, 8/7, 23/20, 31/27, 54/47, 85/74
2. 1/1, 4/3, 29/22, 33/25, 95/72
3. 1/1, 2/1, 3/2, 47/31, 97/64
4. 1/1, 2/1, 5/3, 7/4, 47/27, 148/85
5. 2/1

When I said "unique fractions", I meant that two intervals can't be used more than once. Obviously 1/1 has to map to the first note and 2/1 to the last, so I take out the extra 1/1s and 2/1s. (In larger scales, you'll get multiple cases of 3/2, 4/3 and so on, so in an equal temperament, find which pitch the ratio maps to and remove all instances of the ratio outside of that pitch.

After that, use only the simplest fractions left over.

0. 1/1
1. 7/6
2. 4/3
3. 3/2
4. 5/3
5. 2/1

Another variation of this method, used only for equal and symmetrical temperaments, is to use the simplest fractions for pitches that are the inverse (2/p) of their octave complements. 7/6 would be replaced by 8/7 and 5/3 by 7/4. Also, if only 5/3 is replaced, again by 7/4, you have two disjunct identical trichords, forming a septimal blues pentatonic scale.

~D.

> > I'd call that "rational approximation of an irrational
> > scale". Could you explain step 3) a little more? Come
> > come every scale of n tones/octave doesn't come out the
> > same?
>
> It's definitely a rational approximation, but there's no prime
> limit; nor is there a set limit in the denominator like there
> is with Farey.

The technique of using continued fractions to find
rational approximations is widely used. Routines to do
it are included in many programming languages by default.

> When I said "unique fractions", I meant that two intervals
> can't be used more than once. Obviously 1/1 has to map to
> the first note and 2/1 to the last, so I take out the extra
> 1/1s and 2/1s. (In larger scales, you'll get multiple cases
> of 3/2, 4/3 and so on, so in an equal temperament, find which
> pitch the ratio maps to and remove all instances of the ratio
> outside of that pitch.
>
> After that, use only the simplest fractions left over.
>
> 0. 1/1
> 1. 7/6
> 2. 4/3
> 3. 3/2
> 4. 5/3
> 5. 2/1

How do you resolve the contensions? Order matters. It
seems like you take the available ratios in increasing order
of complexity and find their best approximations in the
scale. But then it's unclear what purpose approximating the
scale members seperately serves in the first place -- just
start with 1/1 and go up in Tenney limit (num * den) until
all scale tones are mapped. (?)

> Another variation of this method, used only for equal and
> symmetrical temperaments, is to use the simplest fractions for
> pitches that are the inverse (2/p) of their octave complements.

What's p?

-Carl

Gene:
>>The 224&270 linear temperament, with period a half-octave and
>>generator a 44/39, is a standout among ultra high precision 13-limit
>>linear temperaments, but of course it is very complex. If you want
>>simpler temperaments which can beat its logflat badness figure, I'm
>>afraid there don't seem to be any. However 58 notes of the "mystery"
>>temperament, the 29&58 temperament, sounds like what you might be
>>looking for. This would be two 29-et cycles stacked unevenly,
>>separated by 16 cents.

Danny:
> I think I asked the same question months ago and you gave me the same > answer. Thanks again then.

I think the stand-out equal temperaments go 29, 46, 58, 72, ... Technically 46 is inconsistent but there's only one obvious way of mapping it. 26 might have some significance as well. Anyway, you can get 2-D temperaments by pairing those off.

Mystery58 is indeed good. Every 15-limit interval has either a complexity of 0 or 1, meaning it either uses one 29-et cycle or both of them. But it comes out a lot worst in average complexity. I think some diaschismic extensions come out better there, two of which Erv looked at, so try 46&58. Some 15-limit intervals will be more complex but the average consonance is simpler. Or the simpler consonances tend to be more simple. I forget what the accuracy is.

Maybe 58&72 has something going for it as well. If you want to check, I've still got the searches online, something like

http://microtonal.co.uk/temper

and if you run the code locally you can do weighted searches as well.

If you want a Partchian JI superscale, 58 notes is a good place to
start.

In practical terms, it may well be that 13 is too complicated. There's still a notable absence of 13-limit music out there. It might be more fruitful to look at different subsets. In that case, you can choose which primes to leave out, and run the searches.

Graham

Carl Lumma wrote (about my method of crude rational approximation):

>> When I said "unique fractions", I meant that two intervals
>> can't be used more than once. Obviously 1/1 has to map to
>> the first note and 2/1 to the last, so I take out the extra
>> 1/1s and 2/1s. (In larger scales, you'll get multiple cases
>> of 3/2, 4/3 and so on, so in an equal temperament, find which
>> pitch the ratio maps to and remove all instances of the ratio
>> outside of that pitch.
>>
>> After that, use only the simplest fractions left over.

> How do you resolve the contensions? Order matters. It
> seems like you take the available ratios in increasing order
> of complexity and find their best approximations in the
> scale. But then it's unclear what purpose approximating the
> scale members seperately serves in the first place -- just
> start with 1/1 and go up in Tenney limit (num * den) until
> all scale tones are mapped. (?)

I took the available ratios in order and used the simplest ones I could without having the same ratio map to more than one pitch. It would be a good idea when working with equal temperaments to take the simplest fraction in each pitch, calculate which degree of ET the note is closest too, and if the numbers don't match, move on to the next ratio in each set. In 12-TET, 2/1 appears as the first ratio for all the pitches from perfect fourth to perfect octave, but since it properly maps only to the octave, you cross out 2/1 on all the other pitches and go on to the next one. And several pitches have 3/4 in their lists, but you only use 3/4 in the perfect fifth, and so on.

Ratios that are not too much more complex may be used as alternates, such as 18/17 in place of 17/16 for the minor second in 12-TET.

The Tenney limit method might work better though.

>> Another variation of this method, used only for equal and
>> symmetrical temperaments, is to use the simplest fractions for
>> pitches that are the inverse (2/p) of their octave complements.
>
> What's p?

The frequency of the particular note in relation to the first pitch, but I could've just as easily used 'r' for ratio.

~D.

Graham Breed wrote (about my 13-limit question):

> I think the stand-out equal temperaments go 29, 46, 58, 72, ...
> Technically 46 is inconsistent but there's only one obvious way of
> mapping it. 26 might have some significance as well. Anyway, you can
> get 2-D temperaments by pairing those off.

I forgot about 72 being viable for 13-limit.

But based on my own calculations, I'd say 224-EDO, as high a degree an ET as it is, is probably a best bet. Four steps is the 81/80 comma, five steps (1� comma) is 64/63, ten steps (2� commas) is 33/32, and twelve steps (3 commas) is 27/26. All the intervals in the 15-limit square are approximated within a standard 33.33% tolerance.

And Athenian Sagittal, which is essentially 665-EDO, will also probably do. (The four above commas are then 12, 15, 30 and 36 steps; the Pythagorean comma is 13 steps.)

> If you want a Partchian JI superscale, 58 notes is a good place to
> start.

That's 58 notes including the octave right? I was thinking that number, or 57 without the octave. I have come up with sets of up to 72 pitches including the octave, but some of these are alternatives, like 64/63 for 81/80 and 15/14 for 16/15.

I've also called it Partch Plus, SuperPartch, Neo-Partch, Partch-13, or Triskaidekaphilia.

> In practical terms, it may well be that 13 is too complicated. There's
> still a notable absence of 13-limit music out there. It might be more
> fruitful to look at different subsets. In that case, you can choose
> which primes to leave out, and run the searches.

The reason I went to 13-limit because 11-limit leaves gaps in approximating 53-tone Pythagorean, which not only has pairs of major and minor intervals, but pairs of neutral intervals as well, and 11-limit only handles one of each. There was no easy way to represent the eleventh degree (semiaugmented second or semidiminished third) in 11-limit, for example, but 13/11 approximates it almost perfectly. And 16/13 was proposed by ibn Sina for the neutral third in the Arabic-Persian tetrachord anyway.

Ozan said he goes to 17-limit, and that would include a couple of al-Farabi's ratios (18/17 for the minor second, for example). I still have to master 13 first.

~D.

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> I think the stand-out equal temperaments go 29, 46, 58, 72, ...
> Technically 46 is inconsistent but there's only one obvious way of
> mapping it. 26 might have some significance as well. Anyway, you can
> get 2-D temperaments by pairing those off.

The 13-limit temperaments to 1000 with logflat minimax badness figure
under 1 are 26, 29, 46, 58, 111, 130, 224, 270, 311, 494 and 684.

> The Tenney limit method might work better though.

I just meant it might be the same, since if you include
all the semiconvergeants for each note in the target scale,
the simplest ratios will (all?) be there somewhere, and
will get mapped to the closest pitch in the scale. So you
might as well just list the simplest ratios in one go and
map them. Then you could say, 'This is 5-et in JI with
a Tenney limit of 10.' Or with a Tenney limit of 20.
Another way to go is to say how many cents error you want,
and get the simplest ratios that don't exceed it. Either
way, you can be precise about what you're getting.

-Carl

> > I think the stand-out equal temperaments go 29, 46, 58, 72, ...
> > Technically 46 is inconsistent but there's only one obvious way
> > of mapping it. 26 might have some significance as well. Anyway,
> > you can get 2-D temperaments by pairing those off.
>
> The 13-limit temperaments to 1000 with logflat minimax badness
> figure under 1 are 26, 29, 46, 58, 111, 130, 224, 270, 311, 494
> and 684.

Here's the difference between JI, 12, 46, and 58:

http://lumma.org/tuning/46vs58.zip

46 is clear winner on bang/buck I think, given that 26 and 29
have very large errors.

-Carl

That's right Danny. 17 is a good practical limit for exotic harmony in my
opinion. It is the furthest I would choose to go anyway. Still, 79/80 MOS
159-tET can approximate even higher primes up to our satisfaction.

Oz.

----- Original Message -----
From: "Danny Wier" <dawiertx@sbcglobal.net>
To: <tuning@yahoogroups.com>
Sent: 02 Ekim 2006 Pazartesi 1:29
Subject: Re: [tuning] Re: 13-limit and continued fractions

> Graham Breed wrote (about my 13-limit question):
>
> > I think the stand-out equal temperaments go 29, 46, 58, 72, ...
> > Technically 46 is inconsistent but there's only one obvious way of
> > mapping it. 26 might have some significance as well. Anyway, you can
> > get 2-D temperaments by pairing those off.
>
> I forgot about 72 being viable for 13-limit.
>
> But based on my own calculations, I'd say 224-EDO, as high a degree an ET
as
> it is, is probably a best bet. Four steps is the 81/80 comma, five steps
(1�
> comma) is 64/63, ten steps (2� commas) is 33/32, and twelve steps (3
commas)
> is 27/26. All the intervals in the 15-limit square are approximated within
a
> standard 33.33% tolerance.
>
> And Athenian Sagittal, which is essentially 665-EDO, will also probably
do.
> (The four above commas are then 12, 15, 30 and 36 steps; the Pythagorean
> comma is 13 steps.)
>
> > If you want a Partchian JI superscale, 58 notes is a good place to
> > start.
>
> That's 58 notes including the octave right? I was thinking that number, or
> 57 without the octave. I have come up with sets of up to 72 pitches
> including the octave, but some of these are alternatives, like 64/63 for
> 81/80 and 15/14 for 16/15.
>
> I've also called it Partch Plus, SuperPartch, Neo-Partch, Partch-13, or
> Triskaidekaphilia.
>
> > In practical terms, it may well be that 13 is too complicated. There's
> > still a notable absence of 13-limit music out there. It might be more
> > fruitful to look at different subsets. In that case, you can choose
> > which primes to leave out, and run the searches.
>
> The reason I went to 13-limit because 11-limit leaves gaps in
approximating
> 53-tone Pythagorean, which not only has pairs of major and minor
intervals,
> but pairs of neutral intervals as well, and 11-limit only handles one of
> each. There was no easy way to represent the eleventh degree
(semiaugmented
> second or semidiminished third) in 11-limit, for example, but 13/11
> approximates it almost perfectly. And 16/13 was proposed by ibn Sina for
the
> neutral third in the Arabic-Persian tetrachord anyway.
>
> Ozan said he goes to 17-limit, and that would include a couple of
> al-Farabi's ratios (18/17 for the minor second, for example). I still have
> to master 13 first.
>
> ~D.
>
>

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
> That's right Danny. 17 is a good practical limit for exotic
> harmony in my opinion.

I agree. -C.

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:

> Here's the difference between JI, 12, 46, and 58:
>
> http://lumma.org/tuning/46vs58.zip
>
> 46 is clear winner on bang/buck I think, given that 26 and 29
> have very large errors.

What was the chord?

> > Here's the difference between JI, 12, 46, and 58:
> >
> > http://lumma.org/tuning/46vs58.zip
> >
> > 46 is clear winner on bang/buck I think, given that 26 and 29
> > have very large errors.
>
> What was the chord?

2:3:5:6:7:9:11:13

-C.

--- In tuning@yahoogroups.com, "Danny Wier" <dawiertx@...> wrote:
>
> Graham Breed wrote (about my 13-limit question):
>
> > I think the stand-out equal temperaments go 29, 46, 58, 72, ...
> > Technically 46 is inconsistent but there's only one obvious way of
> > mapping it. 26 might have some significance as well. Anyway,
you can
> > get 2-D temperaments by pairing those off.
>
> I forgot about 72 being viable for 13-limit.
>
> But based on my own calculations, I'd say 224-EDO, as high a degree
an ET as
> it is, is probably a best bet. Four steps is the 81/80 comma, five
steps (1¼
> comma) is 64/63, ten steps (2½ commas) is 33/32, and twelve steps
(3 commas)
> is 27/26. All the intervals in the 15-limit square are approximated
within a
> standard 33.33% tolerance.
>
> And Athenian Sagittal, which is essentially 665-EDO, will also
probably do.
> (The four above commas are then 12, 15, 30 and 36 steps; the
Pythagorean
> comma is 13 steps.)

Athenian-level (medium-resolution) Sagittal JI is comparable to 224-
EDO, not 665.

If you're considering something as complex as 224-EDO, then why not
give serious consideration to 130-EDO? Besides being a lot simpler,
it can be notated quite easily with the Spartan symbol set.

BTW, you and/or Gene might be interested in this: I've found a
compact way to arrange the tones of 130 in a 4x4 lattice that
contains a 9-limit pentad:

105 34 93 22
88 17 76 5
71 0 59 118
54 113 42 101

If you expand this, you'll find prime 11 (60deg) 7 positions up and 1
left from tone 0, and prime 13 (91deg) is 3 positions down and 2 left.

The above also applies to 46-EDO, with the added benefit that prime
11 (21deg46) also maps to the 4x4 lattice:

37 12 33 8
31 6 27 2
25 0 21 42
19 40 15 36

--George

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

> BTW, you and/or Gene might be interested in this: I've found a
> compact way to arrange the tones of 130 in a 4x4 lattice that
> contains a 9-limit pentad:
>
> 105 34 93 22
> 88 17 76 5
> 71 0 59 118
> 54 113 42 101

This appears to be a 6144/6125 planar temperament formation. Here are
16 notes obtained using generators of 48/35 and 35/32, reduced modulo
6144/6125:

[36/35, 35/32, 9/8, 6/5, 5/4, 4/3, 48/35, 35/24, 3/2, 8/5, 105/64,
12/7, 7/4, 64/35, 15/8, 2]

Here are some "tictactoe" diagrams. Perhaps it would be interesting to
look at these from the point of view of finding generators which
minimize area, as yours seem to be more efficient.

/tuning-math/message/12297

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@> wrote:
>
> > BTW, you and/or Gene might be interested in this: I've found a
> > compact way to arrange the tones of 130 in a 4x4 lattice that
> > contains a 9-limit pentad:
> >
> > 105 34 93 22
> > 88 17 76 5
> > 71 0 59 118
> > 54 113 42 101
>
> This appears to be a 6144/6125 planar temperament formation. Here
are
> 16 notes obtained using generators of 48/35 and 35/32, reduced
modulo
> 6144/6125:
>
> [36/35, 35/32, 9/8, 6/5, 5/4, 4/3, 48/35, 35/24, 3/2, 8/5, 105/64,
> 12/7, 7/4, 64/35, 15/8, 2]
>
> Here are some "tictactoe" diagrams. Perhaps it would be interesting
to
> look at these from the point of view of finding generators which
> minimize area, as yours seem to be more efficient.
>
> /tuning-math/message/12297

Sure, go for it!

--George

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

> Sure, go for it!
>
> --George

Followup at

/tuning-math/message/15785