Variations on the Shrutar tuning: three-stepsize-omintetrachordality; Graham Bre

ed's system; near 46-tET

Let's look at Dave Keenan's 1/8-diaschisma temperament again:

>fret name 3's 5's 7's cents step
>---------------------------------------------
>0 C 0 0 0 0.0 55.6
>1 C# -3 0 1 55.6 48.8
>2 Db -1 -1 0 104.4 55.6
>3 Cx 0 1 1 160.0 48.8
>4 D 2 0 0 208.8 55.6
>5 D# -1 0 1 264.4 48.8
>6 Eb 1 -1 0 313.2 78.0
>7 E 0 1 0 391.2 55.6
>8 E# -3 1 1 446.8 48.8
>9 F -1 0 0 495.6 55.6
>10 Ex 0 2 1 551.2 48.8
>11 F# 2 1 0 600.0 55.6
>12 Fx -1 1 1 655.6 48.8
>13 G 1 0 0 704.4 55.6
>14 G# -2 0 1 760.0 48.8
>15 Ab 0 -1 0 808.8 55.6
>16 Gx 1 1 1 864.4 48.8
>17 A 3 0 0 913.2 55.6
>18 A# 0 0 1 968.8 48.8
>19 Bb 2 -1 0 1017.6 78.0
>20 B 1 1 0 1095.6 55.6
>21 B# -2 1 1 1151.2 48.8
>22 C 1200.0

First notice that it has the step structure MSMSMSLMSMSMSMSMSMSLMS,
which is omnitetrachordal: All octave species can be split into two
identical "tetrachords", each spanning a 4/3 or 9 shrutis, plus
a "disjunction" of 4 shrutis:

MSMSMSMSLMSMSMSMSMSMSL = MSMSMSMSL + MSMS + MSMSMSMSL
SMSMSMSLMSMSMSMSMSMSLM = SMSMSMSLM + SMSM + SMSMSMSLM
MSMSMSLMSMSMSMSMSMSLMS = MSMSMSLMS + MSMS + MSMSMSLMS
SMSMSLMSMSMSMSMSMSLMSM = SMSMSLMSM + SMSM + SMSMSLMSM
MSMSLMSMSMSMSMSMSLMSMS = MSMSLMSMS + MSMS + MSMSLMSMS
SMSLMSMSMSMSMSMSLMSMSM = SMSLMSMSM + SMSM + SMSLMSMSM
MSLMSMSMSMSMSMSLMSMSMS = MSLMSMSMS + MSMS + MSLMSMSMS
SLMSMSMSMSMSMSLMSMSMSM = SLMSMSMSM + SMSM + SLMSMSMSM
LMSMSMSMSMSMSLMSMSMSMS = LMSMSMSMS + MSMS + LMSMSMSMS
MSMSMSMSMSMSLMSMSMSMSL = MSMS + MSMSMSMSL + MSMSMSMSL
SMSMSMSMSMSLMSMSMSMSLM = SMSM + SMSMSMSLM + SMSMSMSLM
MSMSMSMSMSLMSMSMSMSLMS = MSMS + MSMSMSLMS + MSMSMSLMS
SMSMSMSMSLMSMSMSMSLMSM = SMSM + SMSMSLMSM + SMSMSLMSM
MSMSMSMSLMSMSMSMSLMSMS = MSMS + MSMSLMSMS + MSMSLMSMS
or MSMSMSMSL + MSMSMSMSL + MSMS
SMSMSMSLMSMSMSMSLMSMSM = SMSM + SMSLMSMSM + SMSLMSMSM
or SMSMSMSLM + SMSMSMSLM + SMSM
MSMSMSLMSMSMSMSLMSMSMS = MSMS + MSLMSMSMS + MSLMSMSMS
or MSMSMSLMS + MSMSMSLMS + MSMS
SMSMSLMSMSMSMSLMSMSMSM = SMSM + SLMSMSMSM + SLMSMSMSM
or SMSMSLMSM + SMSMSLMSM + SMSM
MSMSLMSMSMSMSLMSMSMSMS = MSMS + LMSMSMSMS + LMSMSMSMS
or MSMSLMSMS + MSMSLMSMS + MSMS
SMSLMSMSMSMSLMSMSMSMSM = SMSLMSMSM + SMSLMSMSM + SMSM
MSLMSMSMSMSLMSMSMSMSMS = MSLMSMSMS + MSLMSMSMS + MSMS
SLMSMSMSMSLMSMSMSMSMSM = SLMSMSMSM + SLMSMSMSM + SMSM
LMSMSMSMSLMSMSMSMSMSMS = LMSMSMSMS + LMSMSMSMS + MSMS

This is the first example I've seen of an omnitetrachordal scale with
3 step sizes. As some readers may know, I view tetrachordality as a
preferable alternative to MOS as a determinant of "well-formedness"
so to speak.

Now rather than using the 7's in Dave's table, let's return those
shrutis to their traditional positions on the 5-limit lattice, with
3/2 in the center, but let's keep the diaschismic tempering in place:

shruti 3's 5's 7's cents step
---------------------------------------------
0 0 0 0 0.0 78.0
1 -5 0 0 78.0 26.4
2 -1 -1 0 104.4 78.0
3 -1 1 0 182.4 26.4
4 2 0 0 208.8 78.0
5 -3 0 0 286.8 26.4
6 1 -1 0 313.2 78.0
7 0 1 0 391.2 26.4
8 4 0 0 417.6 78.0
9 -1 0 0 495.6 26.4
10 3 -1 0 522.0 78.0
11 2 1 0 600.0 26.4
12 6 0 0 626.4 78.0
13 1 0 0 704.4 78.0
14 -4 0 0 782.4 26.4
15 0 -1 0 808.8 78.0
16 -1 1 0 886.8 26.4
17 3 0 0 913.2 78.0
18 -2 0 0 991.2 26.4
19 2 -1 0 1017.6 78.0
20 1 1 0 1095.6 26.4
21 5 0 0 1122.0 78.0
22 1200.0

Now there are only two step sizes, and this is Graham Breed's system.
It's also omnitetrachordal. There is a chain of fifths with 12 notes
and another with 10 notes; they alternate with one another as one
moves along an infinite diagonal strip in the 5-limit lattice. A
system such as this, entailing diaschismic equivalence, would help
explain "why 22" while conforming to the traditional lattice in
configuration. The schisma, though, is inflated to 26.4 cents --
clearly schisma-equivalence is antithetical to 22-ness. Explanations
invoking schisma-equivalency within 22 consecutive notes in a chain
of fifths must resort to a more arbitrary mechanism to explain why
the system stops at 22 -- and (2/1=22, 4/3=9, 3/2=13) fails to be a
usable measuring system for the intervals, since some of the fifths
in the chain are then reckoned as 12 shrutis.

Now back to Dave Keenan's system: as the traditional system
symmetrical around 3/2, we can invert the coordinates _and_ pitches
around that point to get an alternate 7-limit system:

shruti 3's 5's 7's cents step
---------------------------------------------
0 0 0 0 0.0 48.8
1 2 -1 -1 48.8 55.6
2 -1 -1 0 104.4 48.8
3 5 0 -1 153.2 55.6
4 2 0 0 208.8 48.8
5 4 -1 -1 257.6 55.6
6 1 -1 0 313.2 78.0
7 0 1 0 391.2 48.8
8 2 0 -1 440.0 55.6
9 -1 0 0 495.6 48.8
10 1 -1 -1 544.4 55.6
11 2 1 0 600.0 48.8
12 4 0 -1 648.8 55.6
13 1 0 0 704.4 48.8
14 3 -1 -1 753.2 55.6
15 0 -1 0 808.8 48.8
16 2 -2 -1 857.6 55.6
17 3 0 0 913.2 48.8
18 5 -1 -1 962.0 55.6
19 2 -1 0 1017.6 78.0
20 1 1 0 1095.6 48.8
21 3 0 -1 1144.4 55.6
22 1200.0

This is of course omitetrachordal too. Each of the pitches with a
factor of 7 in its ratio is thus displaced by 6.8 cents between this
version and the original. By tempering exactly halfway between the
two versions (3.4 cents from each), the near-46-tET version that Dave
Keenan reported results. But one might instead spread the tempering
unequally so that ratios closer (on the lattice) to the central 3/2
(such as 7/4 in the original and 9/7 above) are left alone (or even
pushed out to make for better 7:5 and 7:6 relationships) while other
shrutis are tempered in a weighted fashion depending on the relative
closenesses (on the lattice) of the two possible ratios to the
central 3/2. Any suggestions? I have a strong hunch that the result
will still be omnitetrachordal within the limits of human melodic
pitch discriminatation. (Furthermore, one could unequally distribute
the diaschisma to make for pure 1/1:5/4:3/2:7/4 and 9/7:3/2:9/5:9/8
chords at the center of the tuning -- and remember, everything is
reproduced a 3:2 lower on the other string [thus 4/3:5/3:1/1:7/6 and
12/7:1/1:6/5:3/2 chords] . . . how good a hybrid JI/omnitetrachordal
system can we work out?)

Paul Erlich wrote:

> But one might instead spread the tempering
> unequally so that ratios closer (on the lattice) to the central 3/2
> (such as 7/4 in the original and 9/7 above) are left alone (or even
> pushed out to make for better 7:5 and 7:6 relationships) while other
> shrutis are tempered in a weighted fashion depending on the relative
> closenesses (on the lattice) of the two possible ratios to the
> central 3/2. Any suggestions? I have a strong hunch that the result
> will still be omnitetrachordal within the limits of human melodic
> pitch discriminatation. (Furthermore, one could unequally distribute
> the diaschisma to make for pure 1/1:5/4:3/2:7/4 and 9/7:3/2:9/5:9/8
> chords at the center of the tuning -- and remember, everything is
> reproduced a 3:2 lower on the other string [thus 4/3:5/3:1/1:7/6 and
> 12/7:1/1:6/5:3/2 chords] . . . how good a hybrid JI/omnitetrachordal
> system can we work out?)

I love a challenge like this.

First, do we agree that all the perfect fifths must be the same size, which
must also be the same as the distance the two strings are tuned apart. If
they are not, we would need to split frets all over the place to give the
same note the same pitch on both strings, or alternatively we would have m
and s versions of a lot more notes apart from 16 17 18 19.

If we insist on keeping the 4 chains of fifths
m16 7 20 11 2 15 6 s19
m17 8 21 12 3 s16
m18 9 0 13 4 s17
m19 10 1 14 5 s18

Then we can't do better than we have already done.

It seems that, for reasons of tradition, we must keep the m16 and m18
chains intact and we must distribute the diaschisma so that srutis 2 and 11
retain their dual roles. For this purpose, fifths of 101/172 octave (704.7
cents) and major thirds of 56/172 octave (390.7 cents) seem a good choice.

What you seem to be saying is that we can break the other two chains. The
obvious places to break them are between 8 and 21 and between 14 and 5.
This means we will have m and s versions of two of these four notes in
addition to the others.

Try this.

/
m21--------12--------3--------s16 /
/ \`. ,'/ \`. ,'/ `. ,' `. /
/ \ m16-/---\--7--/------20--------11----
/ \/|\/ \/|\/ / \ / \
/ /\|/\ /\|/\ / \ / \
m14--------5--------s18 \ / \ / \
`. /,' `.\ /,' `.\ / \ / \ /
m18--------9---------0---------13--------4--------s17
/ \ / \ / \`. .'/ \`. ,'/ `. ,'
\ / \ / \ m17-/---\--8--/-----s21
\ / \ / \/|\/ \/|\/ /
\ / \ / /\|/\ /\|/\ /
-----2---------15--------6--------s19 \ /
/ `. ,' `. /,' `.\ /,' `.\ /
/ m19--------10--------1--------s14
/

We widen the 4:7's slightly, as you suggested, by making them 139/172
octave (969.8 cents).

On the "m" string:

704.7c 390.7c 969.8c pitch step step
sruti 3's 5's 7's (cents)(cents)(1/172 oct)
----------------------------------------------------
0 0 0 0 0.0 48.8 7
1 2 -1 -1 48.8 55.8 8
2 -1 -1 0 104.7 55.8 8
3 0 1 1 160.5 48.8 7
4 2 0 0 209.3 55.8 8
5 -1 0 1 265.1 48.8 7
6 1 -1 0 314.0 76.7 11
7 0 1 0 390.7 48.8 7
8 2 0 -1 439.5 55.8 8
9 -1 0 0 495.3 48.8 7
10 1 -1 -1 544.2 55.8 8
11 2 1 0 600.0 55.8 8
12 -1 1 1 655.8 48.8 7
13 1 0 0 704.7 48.8 7
14 3 -1 -1 753.5 55.8 8
15 0 -1 0 809.3 55.8 8
16 1 1 1 865.1 48.8 7
17 3 0 0 914.0 55.8 8
18 0 0 1 969.8 48.8 7
19 2 -1 0 1018.6 76.7 11
20 1 1 0 1095.3 48.8 7
21 3 0 -1 1144.2 55.8 8

This sure seems like what you are looking for. It also happens to be
strictly proper.

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

Paul,

My latest tempering of your scale is not omnitetrachordal. But I expect
it's still acceptable, given how close it is to 7-limit just.

s = 7/172 octave
m = 8/172 octave
L = 11/172 octave

smmsmsLsm smms smmsmsLsm
mmsmsLsms mmss mmsmsLsms
msmsLsmsm mssm msmsLsmsm
smsLsmsmm ssmm smsLsmsmm
msLsmsmms smms msLsmsmms
sLsmsmmss mmsm sLsmsmmsm failure
Lsmsmmssm msms Lsmsmmsms failure
smsm mssmmsmsL smsmmsmsL failure
msmm ssmmsmsLs msmmsmsLs failure
smms smmsmsLsm smmsmsLsm
mmss mmsmsLsms mmsmsLsms
mssm msmsLsmsm msmsLsmsm
ssmm smsLsmsmm smsLsmsmm
smms msLsmsmms msLsmsmms
mmsm sLsmsmmsm sLsmsmmss
msmsLsmsm msmsLsmsm mssm
smsLsmsmm smsLsmsmm ssmm
msLsmsmms msLsmsmms smms
sLsmsmmsm sLsmsmmss mmsm failure
Lsmsmmsms Lsmsmmssm msms failure
smsmmsmsL smsm mssmmsmsL failure
msmmsmsLs msmm ssmmsmsLs failure

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

/
m21--------12--------3--------s16 /
/ \`. ,'/ \`. ,'/ `. ,' `. /
/ \ m16-/---\--7--/------20--------11----
/ \/|\/ \/|\/ / \ / \
/ /\|/\ /\|/\ / \ / \
m14--------5--------s18 \ / \ / \
,' `. /,' `.\ /,' `.\ / \ / \ /
m18--------9---------0---------13--------4--------s17
/ \ / \ / \`. .'/ \`. ,'/ `. ,'
\ / \ / \ m17-/---\--8--/-----s21
\ / \ / \/|\/ \/|\/ /
\ / \ / /\|/\ /\|/\ /
-----2---------15--------6--------s19 \ /
/ `. ,' `. /,' `.\ /,' `.\ /
/ m19--------10--------1--------s14
/

Here's a possible notation. Do I have the "7" and "L" the right way 'round?
Does "7" usually sharpen by a septimal comma?

/
C\L-------G\L-------D\L--------A\L /
/ \`. ,'/ \`. ,'/ `. ,' `. /
/ \ A\-/---\--E\-/------B\--------F#\---
/ \/|\/ \/|\/ / \ / \
/ /\|/\ /\|/\ / \ / \
AbL-------EbL-------BbL\ / \ / \
,' `. /,' `.\ /,' `.\ / \ / \ /
Bb--------F---------C---------G---------D---------A
/ \ / \ / \`. .'/ \`. ,'/ `. ,'
\ / \ / \ A7-/---\--E7-/------B7
\ / \ / \/|\/ \/|\/ /
\ / \ / /\|/\ /\|/\ /
----Db/-------Ab/-------Eb/--------Bb/\ /
/ `. ,' `. /,' `.\ /,' `.\ /
/ Bb/7-------F/7-------C/7-------G/7
/

There's an 11-limit comma that is also well distributed by this tuning.

175:176 = 2^4 * 5^-2 * 7^-1 * 11^1 = 9.9 cents

There are eight 8:11's. As shruti pairs they are m19:7, 10:20, 1:11, s14:2,
11:m21, 2:12, 15:3, 6:s16. Probably mostly unusable.

In terms of small, medium and large steps
the approx 2:3 is 6s + 6m + L,
the approx 4:5 is 3s + 3m + L,
the approx 4:7 is 8s + 9m + L,
the approx 8:11 is 4s + 5m + L.

we get the following errors when embedding in the following ETs:

-tET 172 126 80
s = 7 5 3
m = 8 6 4
L = 11 8 5
Intvl Error (cents)
-------------------------
2:3 2.7 2.8 3.0
4:5 4.4 4.2 3.7
5:6 -1.7 -1.4 -0.6

4:7 0.9 2.6 6.2
5:7 -3.4 -1.6 2.5
6:7 -1.8 -0.2 3.1

4:9 5.4 5.6 6.1
5:9 1.0 1.5 2.4
7:9 4.5 3.0 -0.1

4:11 -0.2 1.1 3.7
5:11 -4.5 -3.1 0.0
6:11 -2.9 -1.7 0.6
7:11 -1.1 -1.5 -2.5
9:11 -5.5 -4.6 -2.4

Of course there's no reason for it to be embedded in any ET. The
distribution of the 2025:2048 diaschisma is completely independent of the
ratios of 7 or the 175:176 distribution. But the 126-tET version looks
pretty good.

Let's look at some versions of the tuning where 5-limit and 7-limit are
independently optimised.

Here's the tuning that gives minimax 7-limit errors. We have equal worst
errors of 3.3 cents (1/6-diaschisma) in 2:3 and 4:5 (5:6 is just). All
ratios of 7 have equal errors of 1.6 cents (1/12-diaschisma).
s = 50.4 c
m = 54.8 c
L = 73.9 c

Here's minimax 4:5:6:7 beat rates. At the 5 limit we have equal fastest
beats in 4:5, 5:6 and 4:6. At the 7-limit we have equal fastest beats in
4:7 and 6:7.
s = 51.8 c
m = 53.9 c
L = 71.4 c

Note that, melodically speaking, the above two tunings have only two step
sizes. The difference between s and m is insignificant.

If we force s and m to be equal, restoring omnitetrachordality, we get:

Minimax 7-limit errors (equal worst of 3.4 cents in 2:3 and 6:7, just 4:7's).
s = m = 52.7 c
L = 73.1 c

Minimax 4:5:6:7 beat rates (equal fastest beats in 5:6 and 6:7).
s = m = 53.0 c
L = 69.8 c

Let me know if you want any other optima. e.g. 9-limit or RMS.

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

Paul,

Here it is. The centre is just but the diaschisma and BP-comma are
distributed. I think this is awesome (if I do say so myself).

To maximise the number of chords that are playable, the lattice is based on
the open strings and the first 8 frets of the sa-grama strings and the
first 12 frets of the ma-grama strings.

s0 m9
s1 m10
s2 m11
s3 m12
s4 m13
s5 m14
s6 m15
s7 m16
s8 m16
m18
m19
m20
m21

==s8--------m21-------m12=======s3
/ \ / \`. ,'/ `. ,' `. /
/ \ / \ m16=/======s7--------m20-------m11=
/ \ / \/|\/ / \ / \ /
/ \ / /\|/\ / \ / \ /
--s1--------m14=======s5 \ / \ / \ /
`. ,' `. /,' `.\ / \ / \ /
m18=======m9========s0========m13=======s4
/ \ / \ / \`. .'/ `. ,' `.
/ \ / \ / \ m17=/======s8--------m21--
/ \ / \ / \/|\/ / \ /
/ \ / \ / /\|/\ / \ /
==m11=======s2--------m15=======s6 \ / \ /
/ `. ,' `. /,' `.\ / \ /
m19-------m10========s1--------m14==

=== are just fifths
--- are either 6.5 cents (1/3 diaschisma) or 7.1 cents (1/2 BP comma) wide.

Here are all the just intervals.

m12=======s3
,'/ `. ,'
m16=/======s7
/|\/ / \
/ |/\ / \
m14=======s5 \ / \
,' `. /,' `.\ / \
m18=======m9========s0========m13=======s4
\ / \`. .'/ `. ,'
\ / \ m17=/======s8
\ / \ | /
\ / \|/
m15=======s6
,'
m19 m10========s1

Here's the fret arrangement.

Fret Cents Step
0 0.0 55.9
1 55.9 49.3
2 105.2 49.9
3 155.1 48.8
4 203.9 63.0
5 266.9 48.8
6 315.6 70.7
7 386.3 48.8
8 435.1 63.0
9 498.0 48.8
10 546.8 49.9
11 596.7 49.3
12 646.1 55.9
13 702.0 55.9
14 757.8 49.3
15 807.2 49.9
16 857.1 48.8
17 905.9 63.0
18 968.8 48.8
19 1017.6 70.7
20 1088.3 48.8
21 1137.0 63.0
22 1200.0

Enjoy.

-- Dave Keenan
http://dkeenan.com

I wrote:

>>To maximise the number of chords that are playable, the lattice is based
>>on the open strings and the first 8 frets of the sa-grama strings and
>>the
first 12 frets of the ma-grama strings.

Paul Erlich replied:

>Can you elucidate your logic here?

Assumptions:
* The intervals that we most want to be just are those involving at least
one open string (s0 or m9).
* We make the intervals between open strings be just 3:4's and 2:3's.
* The place where finger-span is most limiting on what double-stop
intervals (and triple-stop chords) can be played, is near the nut.

Actually, the more I think about it, the less I think this actually had
much bearing on the final result. It just allowed me to get started, by
simplifying the problem sufficiently.

>Does it affect the logic of your
>derivation that I will by no means restrict myself to those frets?

No. It doesn't affect it. But the intervals outside these frets can be
taken as completely determined by those within them, so if any of them are
just it can be considered a bonus. And many bouses there are.

>Also, it
>looks like s4-s17 is just, but that's not in your lattice.

Agreed. And it is completely unplayable (except maybe way down the
fingerboard). However it turns out that m0 thru m8 can be the same as s0
thru s8 (and if they can be, they should be), so it is playable on a single
fret as m4-s17.

Now we redraw the lattice, removing the m and s qualifiers for those notes
that are the same pitch on both strings, and we include s16 thru s19. Note
that s16 thru s19 are flatter than the corresponding m notes by a septimal
comma (27.3 cents).

==s17
`. ,' `.
--m21-------m12========3========s16
/ \`. ,'/ \`. ,'/ `. ,' `. / /
/ \ m16=/===\==7--/-----m20-------m11========2--
/ \/|\/ \/|\/ / \ / \ /
\ / /\|/\ /\|/\ / \ / \ /
--m14========5========s18 \ / \ / \ /
`. ,' `. /,' `.\ /,' `.\ / \ / \ /
m18========9=========0=========13========4========s17
/ \ / \ / \`. .'/ \`. ,'/ `. ,' `.
/ \ / \ / \ m17=/===\==8--------m21--
/ \ / \ / \/|\/ \/|\/ / \
/ \ / \ / /\|/\ /\|/\ /
==m11========2--------m15========6========s19 \ /
/ / `. ,' `. /,' `.\ /,' `.\ /
m19-------m10========1--------m14==
`. ,'
m18==

To show the notes s10 s11 s12 s14 s15 s20 s21 we need to draw another
lattice replacing the corresponding m notes. Note that the s notes here are
flatter than the corresponding m notes by only 6.5 or 7.1 cents (1/3
diaschisma or 1/2 BP comma).

==s17
`. ,' `.
==s21-------s12--------3========s16
/ \`. ,'/ \`. ,'/ `. ,' `. / /
/ \ m16=/===\==7==/=====s20-------s11--------2--
/ \/|\/ \/|\/ / \ / \ /
\ / /\|/\ /\|/\ / \ / \ /
==s14--------5========s18 \ / \ / \ /
`. ,' `. /,' `.\ /,' `.\ / \ / \ /
m18========9=========0=========13========4========s17
/ \ / \ / \`. .'/ \`. ,'/ `. ,' `.
/ \ / \ / \ m17=/===\==8========s21--
/ \ / \ / \/|\/ \/|\/ / \
/ \ / \ / /\|/\ /\|/\ /
--s11--------2========s15--------6========s19 \ /
/ / `. ,' `. /,' `.\ /,' `.\ /
m19=======s10--------1========s14==
`. ,'
m18==

=== are just fifths
--- are either 6.5 cents (1/3 diaschisma) or 7.1 cents (1/2 BP comma) wide
(no worse than 22-tET).

Note that there are some remote major and minor thirds in this tuning with
errors of 13.0 cents or 14.2 cents (no worse than 12-tET), e.g. 4-m11,
8-m14. But I think they always have less tempered counterparts e.g. 4-s11,
8-s14.

Here's the fret arrangement (the scale on any string) expressed as ratios
(with errors when they exist).

Fret Ratio
0 1/1
1 28/27 -7.1 c or 36/35 +7.1 c
2 16/15 -6.5 c or 135/128 +13.0 c
3 35/32
4 9/8
5 7/6
6 6/5
7 5/4
8 9/7
9 4/3
10 48/35
11 45/32 +6.5 c or 64/45 -13.0 c
12 35/24 -7.1 c or 81/56 +7.1 c
13 3/2
14 14/9 -7.1 c or 54/35 +7.1 c
15 8/5 -6.5 c or 405/256 +13.0 c
16 105/64
17 27/16
18 7/4
19 9/5
20 15/8
21 27/14
22 2/1

Where 6.5 c is actually 1/3 diaschisma, 13.0 c is 2/3 diaschisma, 7.1 c is
1/2 BP comma.

I expect you will check this out thoroughly and let me know if there are
any remaining problems. Note that it is not merely 7-limit, but 9-limit.

Does it still have enough of the melodic properties you want?

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

Paul Erlich wrote:

>I don't think you have that right -- the 27/16 should only occur at fret
>4 on the 3/2 string, and not at all on the 1/1 string.

Some ratios switched strings with the aim of making more just chords
playable. Did that screw the melodic properties?

Here's another version. It should have things where you expect them, but
with no consideration given to playability. You could compare fingering
between the two, but maybe the previous one is too non-traditional and
asymmetric. This new one also distributes the errors better. The non-just
fifths only have 4.9 c or 4.7 c errors (1/4 diaschisma, 1/3 BP comma).

Here's the fret arrangement expressed as ratios.

Fret Ratio
0 1/1
1 28/27 -9.5 c or 36/35 +4.7 c
2 16/15 -4.9 c or 135/128 +14.7 c
3 35/32
4 9/8
5 7/6
6 6/5
7 5/4
8 9/7 +4.7 c or 35/27 -9.5 c
9 4/3
10 48/35
11 45/32 +9.8 c or 64/45 -9.8 c
12 35/24
13 3/2
14 14/9 -4.7 c or 54/35 +9.5 c
15 8/5
16 5/3
17 12/7
18 16/9
19 64/35
20 15/8 +4.9 c or 256/135 -14.7 c
21 27/14
22 2/1

Where 4.9 c is actually 1/4 diaschisma. 14.7 c is 3/4 diaschisma. 4.7 c is
1/3 BP comma. 9.5 c is 2/3 BP comma.

Here's the lattice for any one string (fret numbers).

===4
`. ,' `.
===8---------21--------12========3
/ \ / \`. ,'/ `. ,' `. /
/ \ / \ 16=/======7---------20--------11-
/ \ / \/|\/ / \ / \ /
/ \ / /\|/\ / \ / \ /
---1---------14--------5 \ / \ / \ /
`. ,' `. /,' `.\ / \ / \ /
18========9=========0=========13========4
/ \ / \ / \`. .'/ `. ,' `.
/ \ / \ / \ 17-/------8---------21--
/ \ / \ / \/|\/ / \ /
/ \ / \ / /\|/\ / \ /
---11--------2---------15=======s6 \ / \ /
/ `. ,' `. /,' `.\ / \ /
19========10--------1---------14==
`. ,' `.
18==
=== are just fifths
--- are either 4.9 cents (1/4 diaschisma) or 4.7 cents (1/3 BP comma) wide.

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

Paul,

Here are the cents and steps for my latest "well-tempering" of your shrutar
scale, given as ratios in the previos post. I don't think it is really a
well-temperament since we accept that there are still 4 wolves (729.2 c).
But then who's to say what "well temperament" means when applied to other
than 12 notes per octave.

Fret Cents Step
------------------
0 0.0 53.5
1 53.5 53.3
2 106.8 48.3
3 155.1 48.8
4 203.9 63.0
5 266.9 48.8
6 315.6 70.7
7 386.3 53.5
8 439.8 58.2
9 498.0 48.8
10 546.8 53.2
11 600.0 53.2
12 653.2 48.8
13 702.0 58.2
14 760.2 53.5
15 813.7 70.7
16 884.4 48.8
17 933.1 63.0
18 996.1 48.8
19 1044.9 48.3
20 1093.2 53.3
21 1146.5 53.5
22 1200.0

When you have time, would you work out fingering for the various complete
triads and tetrads with this tuning? At least the just ones. To make sure
you can actually play enough of them.

There are two 8:11s (only 0.2 c wide), namely 13-1 and 21-9. And two 8:13's
(0.5 c wide), namely 3-18 and 4-19.

I look forward to your first recording.

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

Paul Erlich wrote:
>The other two tetrads are largely unplayable, but we can't benefit from
>tempering them, since they're just by-products of the tetrads above,
>right?

Right.

>Now, how about including a just 1/1:3/2:5/4:11/8 chord?

Sure.

The maximum fifth error went up from 4.89 c to 4.93 c. eek! ;-)

Here's the core just lattice with the 11's shown without connecting lines.
You'll notice there's a complete otonal 11-limit hexad. 0-13-7-s18-4-m10
corresponds to 1:3:5:7:9:11.

m16========7========s20
/|\ /|\ / \
/ | \ / | \ / \
/ m5--------s18 \ / \
/,'m19`.\ /,'m10`.\ / m1 \
9=========0=========13========4
\ s21 / \ s12.'/ \`.s3 ,'/
\ / \ m17-/---\-s8 /
\ / \ | / \ | /
\ / \|/ \|/
m15========6========s19
Here's the tuning

Fret Cents
-------------
0 0.00
1 53.30
2 106.84
3 150.61
4 203.91
5 266.87
6 315.64
7 386.31
8 439.81
9 498.04
10 551.35
11 600.00
12 648.65
13 701.96
14 760.19
15 813.69
16 884.36
17 933.13
18 996.09
19 1049.39
20 1093.16
21 1146.70

One such 1:3:5:11 chord is 13-4-s20-m1.

At first I thought it would be unplayable because you wrote:

>note string fret
> 13 1/1 13
>s20 1/1 20
> 4 3/2 13

But you can't play s20 on the 1/1 string

The choices are:
13 1/1 13
or 3/2 0

s20 3/2 7

4 3/2 13
or 1/1 4

So I guess it has to be:
13 3/2 0
s20 3/2 7
4 1/1 4

Now we can add m1 to get the 8:11 from note 13.
13 3/2 0
s20 3/2 7
4 1/1 4
m1 1/1 1

That's quite a stretch from fret 1 to fret 7. It might barely be playable.
At least you can play it as a 6:8:11. 6:9:11 is possible too.
4 1/1 4
s17 3/2 4
m1 1/1 1

I'll leave it to you to figure out what subsets of that 11-limit hexad
0-13-7-s18-4-m10 are playable simultaneously.

If any of these 11's are unplayable, let me know so they can go back to
being tempered and let some other 7-limit intervals be just.

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