optimal fifth sizes for scales (in fixed-pitch circumstances)

With regard to the assumptions and findings of my paper (http://www-math.cudenver.edu/~jstarret/22ALL.html) and
subsequent posts:

Using total squared error (or equivalently, RMS),

The optimal "fifth" size for the pentatonic scale (with 3-limit harmony) is 701.955 cents, a perfect 3/2.

The optimal fifth size for the diatonic scale (with 5-limit harmony) is 696.165 cents, flat by 7/26 syntonic comma.

The optimal "fifth" size for the decatonic or double pentatonic scale (with 7-limit harmony) is 708.814 cents.

This is given in my paper as

(30.5+7*LOG(3)-5*LOG(35))/27

octaves (logs are base 2). I don't know if this can be interpreted as tempering by a fraction of some comma.

The optimal "fifth" size for the tetradecatonic (double diatonic) scale (with 7-limit harmony) is 693.651 cents and can be
derived as follows:

The 3/1 is constructed, of course, by one fifth plus an octave.

The 5/1 is constructed by four fifths.

The 5/3 is constructed by three fifths minus one octave.

The 7/1 is constructed by four fifths plus half an octave.

The 7/3 is constructed by three fifths minus half an octave.

The 7/5 is constructed by half an octave.

The total squared error given a fifth Q is

(log(3)-Q-1)^2+(log(5)-4*Q)^2+(log(5)-log(3)-3*Q+1)^2+(log(7)-4*Q-0.5)^2+(log(7)-log(3)-3*Q+0.5)^2+(log(7)-log(5)-0.5)^2

(logs are base 2)

Using the chain rule, half the derivative with respect to Q is

-(log(3)-Q-1)-4*(log(5)-4*Q)-3*(log(5)-log(3)-3*Q+1)-4*(log(7)-4*Q-0.5)-3*(log(7)-log(3)-3*Q+0.5)

Collecting terms:

-1.5+5*log(3)-7*log(35)+51*Q

And setting this equal to zero:

Q=(1.5-5*LOG(3)+7*LOG(35))/51

Which, multiplied by the conversion factor of 1200 cents per octave, gives 693.651 cents.

Although I originally found these tetradecatonic scales in 26-tET, 38-tET (=2*19-tET) would be better. 64-tET would be
much better still, and as I note in the paper, it can sort of do decatonics as well as diatonics. The problem with decatonics in
64-tET is that the resulting approximation of the 6:5 ratio is more than a syntonic comma sharp, even though an almost perfect
6:5 is available in 64-tET. But in the paper I give 76-tET as an all-around improvement over 64-tET, and it does
tetradecatonics well too since 76-tET=2*38-tET.

[Paul Erlich, TD 211.2]
>With regard to the assumptions and findings of my paper
>(http://www-math.cudenver.edu/~jstarret/22ALL.html) and
>subsequent posts:
>
>Using total squared error (or equivalently, RMS),
>
>The optimal "fifth" size for the pentatonic scale (with 3-limit harmony)
>is 701.955 cents, a perfect 3/2.
>
>The optimal fifth size for the diatonic scale (with 5-limit harmony)
>is 696.165 cents, flat by 7/26 syntonic comma.
>
>The optimal "fifth" size for the decatonic or double pentatonic scale
>(with 7-limit harmony) is 708.814 cents.
>...
>The optimal "fifth" size for the tetradecatonic (double diatonic)
>scale (with 7-limit harmony) is 693.651 cents ...

[Dave Keenan, now]
I agree with these values, although one needn't worry about 3 decimal
places! One decimal place would be more than enough. You can see how broad
the minima are in my charts.

diatonic (5-limit)
http://dkeenan.com/Music/1ch5limb.gif
double pentatonic (7-limit)
http://dkeenan.com/Music/2ch7limc.gif
double diatonic (7-limit)
http://dkeenan.com/Music/2ch7lima.gif
continued on
http://dkeenan.com/Music/2ch7limb.gif

[Paul Erlich]
>Although I originally found these tetradecatonic scales in 26-tET,
>38-tET (=2*19-tET) would be better. 64-tET would be
>much better still,

[Dave Keenan]
If RMS error is your measure, there's no significant difference between them:

Description Best RMS error for 14-atonic
fifth size 7-limit harmony
--------------------------------------------------------
26-tET 692.3c 11.9c
optimum 693.7c 11.2c
64-tET 693.8c 11.2c
38 or 76-tET 694.7c 11.6c

[Paul Erlich]
>and as I note in the paper, it [64-tET] can sort of do
>decatonics as well as diatonics. The problem with decatonics in
>64-tET is that the resulting approximation of the 6:5 ratio is more
>than a syntonic comma sharp, even though an almost perfect
>6:5 is available in 64-tET. But in the paper I give 76-tET as an
>all-around improvement over 64-tET, and it does
>tetradecatonics well too since 76-tET=2*38-tET.

[Dave Keenan]
With that many notes it seems like you ought to be able to do just about
anything since the difference between adjacent notes is less than the error
you're allowing in the 7:5. But I know consistency would say otherwise. It
_is_ kind of neat how the second-best fifth falls in the decatonic region.
I must show such ETs also against their second-best fifths on my charts in
future.

Description Second best RMS error for decatonic
fifth size 7-limit harmony
--------------------------------------------------------
64-tET 712.5c 13.4c
76-tET 710.5c 11.5c
optimum 708.8c 10.9c

Description Best RMS error for diatonic
fifth size 5-limit harmony
--------------------------------------------------------
64-tET 693.8c 8.1c
19,38,76-tET 694.7c 6.0c
optimum 696.2c 4.2c
12-tET 700.0c 12.1c

So yes, you can do all three (or four since pentatonic is a subset of
diatonic) in 76-tET but that's just too many notes for most purposes, so
what's the point?

Regards,
-- Dave Keenan
http://dkeenan.com

Dave Keenan wrote:

>[Paul Erlich, TD 211.2]
>>With regard to the assumptions and findings of my paper
>>(http://www-math.cudenver.edu/~jstarret/22ALL.html) and
>>subsequent posts:
>>
>>Using total squared error (or equivalently, RMS),
>>
>>The optimal "fifth" size for the pentatonic scale (with 3-limit harmony)
>>is 701.955 cents, a perfect 3/2.
>>
>>The optimal fifth size for the diatonic scale (with 5-limit harmony)
>>is 696.165 cents, flat by 7/26 syntonic comma.
>>
>>The optimal "fifth" size for the decatonic or double pentatonic scale
>>(with 7-limit harmony) is 708.814 cents.
>>...
>>The optimal "fifth" size for the tetradecatonic (double diatonic)
>>scale (with 7-limit harmony) is 693.651 cents ...
>
>[Dave Keenan, now]
>I agree with these values, although one needn't worry about 3 decimal
>places! One decimal place would be more than enough. You can see how broad
>the minima are in my charts.
>
>diatonic (5-limit)
> http://dkeenan.com/Music/1ch5limb.gif
>double pentatonic (7-limit)
> http://dkeenan.com/Music/2ch7limc.gif
>double diatonic (7-limit)
> http://dkeenan.com/Music/2ch7lima.gif
> continued on
> http://dkeenan.com/Music/2ch7limb.gif

Actually, the reason I went through the trouble of calculating the above is
because the last chart is sliced at 694 cents and I couldn't tell what side
of 694 the minimum was on.

>[Paul Erlich]
>>Although I originally found these tetradecatonic scales in 26-tET,
>>38-tET (=2*19-tET) would be better. 64-tET would be
>>much better still,
>
>[Dave Keenan]
>If RMS error is your measure, there's no significant difference between
them:
>
>Description Best RMS error for 14-atonic
> fifth size 7-limit harmony
>--------------------------------------------------------
>26-tET 692.3c 11.9c
>optimum 693.7c 11.2c
>64-tET 693.8c 11.2c
>38 or 76-tET 694.7c 11.6c

Well, the real reason is that, as a musician, I find that fifths flatter
than 695 cents become less than completely pleasing. Although fifths as flat
as 667 cents are musically useful (yes, as representations of the 3/2 ratio)
but harsh.

>[Paul Erlich]
>>and as I note in the paper, it [64-tET] can sort of do
>>decatonics as well as diatonics. The problem with decatonics in
>>64-tET is that the resulting approximation of the 6:5 ratio is more
>>than a syntonic comma sharp, even though an almost perfect
>>6:5 is available in 64-tET. But in the paper I give 76-tET as an
>>all-around improvement over 64-tET, and it does
>>tetradecatonics well too since 76-tET=2*38-tET.
>
>[Dave Keenan]
>With that many notes it seems like you ought to be able to do just about
>anything since the difference between adjacent notes is less than the error
>you're allowing in the 7:5. But I know consistency would say otherwise.

Huh? Consistency would disallow any of these except 26-tET for the double
diatonic scale.

> It
>_is_ kind of neat how the second-best fifth falls in the decatonic region.
>I must show such ETs also against their second-best fifths on my charts in
>future.
>
>Description Second best RMS error for decatonic
> fifth size 7-limit harmony
>--------------------------------------------------------
>64-tET 712.5c 13.4c
>76-tET 710.5c 11.5c
>optimum 708.8c 10.9c
>
>Description Best RMS error for diatonic
> fifth size 5-limit harmony
>--------------------------------------------------------
>64-tET 693.8c 8.1c
>19,38,76-tET 694.7c 6.0c
>optimum 696.2c 4.2c
>12-tET 700.0c 12.1c
>
>So yes, you can do all three (or four since pentatonic is a subset of
>diatonic) in 76-tET but that's just too many notes for most purposes, so
>what's the point?

Some of Ben Johnston's JI pieces have over 80 pitches. Several Boston
composers write extensively in 72-tET. James Tenney wrote a 72-tET piece for
six harps tuned 16.67 cents apart. One could write a piece in 76-tET for
four 19-tET guitars tuned 15.79 cents apart.

-Paul

[Paul Erlich TD 215.8]
>Actually, the reason I went through the trouble of calculating the above is
>because the last chart is sliced at 694 cents and I couldn't tell what side
>of 694 the minimum was on.

Yeah that is a bummer. Would it be better if I sliced the spectrum into 5
pieces (instead of 4) and provided a bit of overlap?

>>[Dave Keenan]
>>If RMS error is your measure, there's no significant difference between
>>[26, 64, and 38-tET and the optimum]

[Paul Erlich]
>Well, the real reason is that, as a musician, I find that fifths flatter
>than 695 cents become less than completely pleasing. Although fifths as flat
>as 667 cents are musically useful (yes, as representations of the 3/2 ratio)
>but harsh.

Agreed. Equal-weighted RMS leaves something to be desired. Maybe we need
something like Weight = Max( 6/Max(a,b), (a*b)/30) ), for a non
octave-equivalent ratio a:b in lowest terms? :-)

A reminder: The first component 6/Max(a,b) is intended to approximate equal
percentage increases in dissonance, while the second component (a*b)/30 is
intended to approximate equal beat rates. The crossover is made to occur at
the minor third 5:6.

For octave-equivalence one could use the inversion of each interval that
occurs in 4:5:6:7:9:11. Actually, the relative beat rate of an interval in
a chord (expressed in extended ratio form) is given by the LCM of the two
(unreduced) numbers. e.g. 12 for the 4:6 fifth and 18 for the 6:9 fifth,
but for all the others in the above chord the LCM is just the product.

>>[Dave Keenan]
>>With that many notes it seems like you ought to be able to do just about
>>anything since the difference between adjacent notes is less than the error
>>you're allowing in the 7:5. But I know consistency would say otherwise.

[Paul Erlich]
>Huh? Consistency would disallow any of these except 26-tET for the double
>diatonic scale.

I meant that when one enforces consistency it doesn't look so good as one
might expect from just looking at the scale-degree size. I think we're
agreeing.

[Paul Erlich]
>Some of Ben Johnston's JI pieces have over 80 pitches. Several Boston
>composers write extensively in 72-tET. James Tenney wrote a 72-tET piece for
>six harps tuned 16.67 cents apart. One could write a piece in 76-tET for
>four 19-tET guitars tuned 15.79 cents apart.

Good point.

In that case you might like to check out what is possible with two 34-tETs
tuned about 6 cents apart (not a half scale-degree), or the equivalent four
17-tETs. 5 cents apart if you favour RMS, 7 cents if you favour Max-Abs. I
think this does the same thing with fewer notes and better accuracy.

Regards,
-- Dave Keenan
http://dkeenan.com

I wrote in TD 217.12:

>In that case you might like to check out what is possible with two 34-tETs
>tuned about 6 cents apart (not a half scale-degree), or the equivalent four
>17-tETs. 5 cents apart if you favour RMS, 7 cents if you favour Max-Abs. I
>think this does the same thing with fewer notes and better accuracy.

Sorry. That's wrong. It does decatonic (to 9-limit) but not 14-atonic. It
does allow a second system to coexist with decatonic, but it's the (not
terribly interesting) one where a 4:7 is 8 fourths plus a half-octave (plus
6 cents in this case).

I like to visualise these systems as to how their complete otonalities look
on chain(s) of fifths.

diatonic
13..5

decatonic (double-pentatonic)
7.13
5...

14-atonic (double-diatonic)
13..5
....7

The not very interesting one near 46 and 58-tET (mentioned above)
7.......13
......5...

-- Dave Keenan
http://dkeenan.com

>In that case you might like to check out what is possible with two 34-tETs
>tuned about 6 cents apart (not a half scale-degree), or the equivalent four
>17-tETs. 5 cents apart if you favour RMS, 7 cents if you favour Max-Abs. I
>think this does the same thing with fewer notes and better accuracy.

I don't see how one would get a meantone diatonic scale there, let alone the
others. What are you calculating the RMS of? But it's a nice idea -- it
reminds me of one of my first proposals to this list (many years ago) of
approximating 5-limit JI my using two 12-tETs tuned about 15 cents apart,
which seems superior to Helmoltz's 24-tone schismatic system since 2 cents
from JI is JI to me. Since my first synth (a Casio) allowed for a split
keyboard with arbitrary transposition, I could actually play with that
tuning. It didn't thrill me (for the same reasons 5-limit JI doesn't usually
thrill me). Quartertones didn't thrill me either, except for the major chord
with added eleventh harmonic.

Dave Keenan wrote,

>>In that case you might like to check out what is possible with two 34-tETs
>>tuned about 6 cents apart (not a half scale-degree), or the equivalent
four
>>17-tETs. 5 cents apart if you favour RMS, 7 cents if you favour Max-Abs. I
>>think this does the same thing with fewer notes and better accuracy.

>Sorry. That's wrong. It does decatonic (to 9-limit)

This would be via the Keenan 9-limit tuning scheme with two sizes of fifth?
I don't like the way two tetrads are lost there, and the presence of more
than two steps sizes bothers me just a little too.

[Paul Erlich TD219.15]
>Dave Keenan wrote,
>
>>>In that case you might like to check out what is possible with two 34-tETs
>>>tuned about 6 cents apart (not a half scale-degree), or the equivalent
>four
>>>17-tETs. 5 cents apart if you favour RMS, 7 cents if you favour Max-Abs. I
>>>think this does the same thing with fewer notes and better accuracy.
>
>>Sorry. That's wrong. It does decatonic (to 9-limit)
>
>This would be via the Keenan 9-limit tuning scheme with two sizes of fifth?

I wondered if you'd realise. You didn't disappoint me. :-)

>I don't like the way two tetrads are lost there, and the presence of more
>than two steps sizes bothers me just a little too.

Yes. For some dumb reason I thought that extending it in this way would
give us back those two tetrads with the broken 4:7 (they weren't entirely
lost). So this scheme doesn't do your decatonics at all. Sorry.

In checking it out I came up with an alternative projection of the 7-limit
triangular lattice that has fewer crossings and is more useful when
extended to 9-limit. Here are some examples comparing the two projections.

Erlich's symmetrical decatonic scale

A/ -------- E/ -------- B/
,' /|\`. ,' /|\ `. ,' /|
B/ -/-|-\- F#/ -/-|-\- C#/ / |
| / | \ | / | \ | / |
| / Eb -------- Bb -------- F
|/,' /|\ `.\|/,' /|\ `.\|/,' /|
F --/-|-\-- C --/-|-\-- G / |
| / | \ | / | \ | / |
| / A/ -------- E/ -------- B/
|/,' `.\|/,' `.\|/,'
B/ ------- F#/ ------- C#/

B/ ------- F#/ ------- C#/
| `. /| `. /|
| `. / | `. / |
| A/ -------- E/ -------- B/
| ,' ^ | ,' ^ | ,' ^
| ,' |\ | ,' |\ | ,' |
| ,' | \| ,' | \| ,' |
F -------|- C -------|- G |
| `. | /| `. | /| `. |
| `. |/ | `. |/ | `. |
| Eb -------- Bb -------- F
| ,' ^ | ,' ^ | ,' ^
| ,' |\ | ,' |\ | ,' |
| ,' | \| ,' | \| ,' |
B/ ------| F#/ ------| C#/ |
| / `. | / `. |
|/ `. |/ `. |
A/ -------- E/ -------- B/

Erlich's pentachordal decatonic scale

E/ -------- B/ ------- F#/
,' /|\ `. ,' /| `. ,' |
B/ ------- F#/ -/-|-\- C#/ -/-|- - G#/ |
| | / | \ | / | |
| Eb -------- Bb -------- F --------- C
| ,' / \ `. |/,' /|\ `.\|/,' /|\ /|
F --/---\-- C --/-|-\-- G / | \ / |
| / \ | / | \ | / | \ / |
| / \ | / E/ -------- B/ -\---/- F#/
|/ `.\|/,' `.\|/,' `.\ /,'
B/ ------- F#/ ------- C#/ ------- G#/

B/ ------- F#/ ------- C#/ ------- G#/
| | `. /| `. / `.
| | `. / | `. / `.
| | E/ -------- B/ ------- F#/
| | ,' ^ | ,' ^ |
| | ,' |\ | ,' | |
| | ,' | \| ,' | |
F --------- C -------|- G | |
| `. /| `. | /| `. | |
| `. / | `. |/ | `. | |
| Eb -------- Bb -------- F --------- C
| ,' ^ | ,' ^ | ,' ^ ,' |
| ,' \ | ,' |\ | ,' |\ ,' |
| ,' \| ,' | \| ,' | \ ,' |
B/ ------- F#/ ------| C#/ ------| G#/ |
| / `. | / `. |
|/ `. |/ `. |
E/ -------- B/ ------- F#/

Keenan's strange 9-limit temperament (12-notes)

A/ ======== E/ ======== B/ ------- F#/
/|\ `. ,' /|\ `. ,' /|\ `. /|
B/ -/-|-\- F#/ =/=|=\= C#/ =/=|=\= G#/ / |
| / | \ | / | \ | / | \ | / |
| / Eb ======== Bb ======== F --------- C
|/ /|\ `.\|/,' /|\ `.\|/,' /|\ `.\|/ /|
F --/-|-\-- C ==/=|=\== G ==/=|=\== D / |
| / | \ | / | \ | / | \ | / |
| / A/ ======== E/ ======== B/ -\-|-/- F#/
|/ `.\|/,' `.\|/,' `.\|/
B/ ------- F#/ ======= C#/ ======= G#/

Too hard to show 5:9 and 7:9 above.

Note that I use C ========= G ========= D to mean: Not only are C-G
and G-D usable 2:3's but C-D is also a usable 4:9.

B/ ------- F#/ ======= C#/ ======= G#/
| /| `. , '/| `. , '/|
| /,| ' `. /,| ' `. / |
| A/ ======== E/ ======== B/ ------- F#/
| ,' ^ `| ,' ^ `| ,' ^ | ,' |
| ,' |\ | `,' |\ | `,' |\ | ,' |
| ,' | \| ,' ` |.\| ,' ` |.\| ,' |
F -------|- C =======|= G =======|= D |
| | /| `. , |'/| `. , |'/| |
| |/,| ' `. |/,| ' `. |/ | |
| Eb ======== Bb ======== F --------- C
| ,' ^ `| ,' ^ `| ,' ^ | ,' |
| ,' |\ | `,' |\ | `,' |\ | ,' |
| ,' | \| ,' ` |.\| ,' ` |.\| ,' |
B/ ------| F#/ ======| C#/ ======| G#/ |
| / `. , |'/ `. , |'/ |
|/, ' `. |/, ' `. |/ |
A/ ======== E/ ======== B/ ------- F#/

All available 9-limit intervals shown.

Regards,
-- Dave Keenan
http://dkeenan.com

Dave Keenan wrote,

>In checking it out I came up with an alternative projection of the 7-limit
>triangular lattice that has fewer crossings and is more useful when
>extended to 9-limit. Here are some examples comparing the two projections.

>Erlich's symmetrical decatonic scale

> A/ -------- E/ -------- B/
> ,' /|\`. ,' /|\ `. ,' /|
>B/ -/-|-\- F#/ -/-|-\- C#/ / |
>| / | \ | / | \ | / |
>| / Eb -------- Bb -------- F
>|/,' /|\ `.\|/,' /|\ `.\|/,' /|
>F --/-|-\-- C --/-|-\-- G / |
>| / | \ | / | \ | / |
>| / A/ -------- E/ -------- B/
>|/,' `.\|/,' `.\|/,'
>B/ ------- F#/ ------- C#/
>
>
>B/ ------- F#/ ------- C#/
>| `. /| `. /|
>| `. / | `. / |
>| A/ -------- E/ -------- B/
>| ,' ^ | ,' ^ | ,' ^
>| ,' |\ | ,' |\ | ,' |
>| ,' | \| ,' | \| ,' |
>F -------|- C -------|- G |
>| `. | /| `. | /| `. |
>| `. |/ | `. |/ | `. |
>| Eb -------- Bb -------- F
>| ,' ^ | ,' ^ | ,' ^
>| ,' |\ | ,' |\ | ,' |
>| ,' | \| ,' | \| ,' |
>B/ ------| F#/ ------| C#/ |
> | / `. | / `. |
> |/ `. |/ `. |
> A/ -------- E/ -------- B/

>Erlich's pentachordal decatonic scale
>
> E/ -------- B/ ------- F#/
> ,' /|\ `. ,' /| `. ,' |
>B/ ------- F#/ -/-|-\- C#/ -/-|- - G#/ |
>| | / | \ | / | |
>| Eb -------- Bb -------- F --------- C
>| ,' / \ `. |/,' /|\ `.\|/,' /|\ /|
>F --/---\-- C --/-|-\-- G / | \ / |
>| / \ | / | \ | / | \ / |
>| / \ | / E/ -------- B/ -\---/- F#/
>|/ `.\|/,' `.\|/,' `.\ /,'
>B/ ------- F#/ ------- C#/ ------- G#/
>
>
>B/ ------- F#/ ------- C#/ ------- G#/
>| | `. /| `. / `.
>| | `. / | `. / `.
>| | E/ -------- B/ ------- F#/
>| | ,' ^ | ,' ^ |
>| | ,' |\ | ,' | |
>| | ,' | \| ,' | |
>F --------- C -------|- G | |
>| `. /| `. | /| `. | |
>| `. / | `. |/ | `. | |
>| Eb -------- Bb -------- F --------- C
>| ,' ^ | ,' ^ | ,' ^ ,' |
>| ,' \ | ,' |\ | ,' |\ ,' |
>| ,' \| ,' | \| ,' | \ ,' |
>B/ ------- F#/ ------| C#/ ------| G#/ |
> | / `. | / `. |
> |/ `. |/ `. |
> E/ -------- B/ ------- F#/
>
>
>Keenan's strange 9-limit temperament (12-notes)
>
> A/ ======== E/ ======== B/ ------- F#/
> /|\ `. ,' /|\ `. ,' /|\ `. /|
>B/ -/-|-\- F#/ =/=|=\= C#/ =/=|=\= G#/ / |
>| / | \ | / | \ | / | \ | / |
>| / Eb ======== Bb ======== F --------- C
>|/ /|\ `.\|/,' /|\ `.\|/,' /|\ `.\|/ /|
>F --/-|-\-- C ==/=|=\== G ==/=|=\== D / |
>| / | \ | / | \ | / | \ | / |
>| / A/ ======== E/ ======== B/ -\-|-/- F#/
>|/ `.\|/,' `.\|/,' `.\|/
>B/ ------- F#/ ======= C#/ ======= G#/
>
>Too hard to show 5:9 and 7:9 above.
>
>Note that I use C ========= G ========= D to mean: Not only are C-G
>and G-D usable 2:3's but C-D is also a usable 4:9.
>
>B/ ------- F#/ ======= C#/ ======= G#/
>| /| `. , '/| `. , '/|
>| /,| ' `. /,| ' `. / |
>| A/ ======== E/ ======== B/ ------- F#/
>| ,' ^ `| ,' ^ `| ,' ^ | ,' |
>| ,' |\ | `,' |\ | `,' |\ | ,' |
>| ,' | \| ,' ` |.\| ,' ` |.\| ,' |
>F -------|- C =======|= G =======|= D |
>| | /| `. , |'/| `. , |'/| |
>| |/,| ' `. |/,| ' `. |/ | |
>| Eb ======== Bb ======== F --------- C
>| ,' ^ `| ,' ^ `| ,' ^ | ,' |
>| ,' |\ | `,' |\ | `,' |\ | ,' |
>| ,' | \| ,' ` |.\| ,' ` |.\| ,' |
>B/ ------| F#/ ======| C#/ ======| G#/ |
> | / `. , |'/ `. , |'/ |
> |/, ' `. |/, ' `. |/ |
> A/ ======== E/ ======== B/ ------- F#/

>All available 9-limit intervals shown.

That's fantastic, Dave! What happens to the Fokker-Lumma scale?