Yahoo Tuning Groups Ultimate Backup tuning a JI 7-limit "Indian" tuning and its application to the Shrutar

I'll take the usual JI Modern Indian Gamut as a requirement:
1/1 16/15 9/8 6/5 5/4 4/3 45/32 3/2 8/5 27/16 9/5 15/8 2/1

A 22-tone periodicity block I found which contains this is based on the
unison vectors

2025/2048 (diaschisma)
245/243 (BP comma)
64/63 (septimal comma)

It looks like this:

shruti ratio cents successive successive
ratio cents

0 1 0 28/27 63
1 28/27 63 36/35 49
2 16/15 112 525/512 43
3 35/32 155 36/35 49
4 9/8 204 28/27 63
5 7/6 267 36/35 49
6 6/5 316 25/24 71
7 5/4 386 28/27 63
8 35/27 449 36/35 49
9 4/3 498 525/512 43
10 175/128 541 36/35 49
11 45/32 590 28/27 63
12 35/24 653 36/35 49
13 3/2 702 28/27 63
14 14/9 765 36/35 49
15 8/5 814 525/512 43
16 105/64 857 36/35 49
17 27/16 906 28/27 63
18 7/4 969 36/35 49
19 9/5 1018 25/24 71
20 15/8 1088 28/27 63
21 35/18 1151 36/35 49

10
/ \
/ \
/ \
/ \
8---------21--------12--------3--------16
/ \ / \ / \`. ,'/ `. ,' `.
/ \ / \ / \ 7--/------20--------11
/ \ / \ / \/|\/ / \ / \
/ \ / \ / /\|/\ / \ / \
1---------14--------5---------18 \ / \ / \
`. ,' `. /,' `.\ / \ / \
9---------0---------13--------4---------17
/ \ / \ / \ /
/ \ / \ / \ /
/ \ / \ / \ /
/ \ / \ / \ /
2---------15--------6---------19

This, is, of course, a CS, and not too different from Wilson's tuning NO. 3
in http://www.anaphoria.com/trans22.PDF -- but the step sizes here are more
uniform, Wilson's smallest step being a syntonic comma and my smallest being
more than twice as large.

This tuning also has the property that the interval structure from 1/1 to
4/3 is repeated exactly between 3/2 and 1/1.

If this is the fretting of a Shrutar, you have a very manageable set of
frets. In particular, if this is the scale on the 3/2 strung, on the whole
instrument you have:

fret# pitch on pitch on
3/2 string 1/1 string
ratio cents ratio cents

0 1/1 0 4/3 498
1 28/27 63 112/81 561
2 16/15 112 64/45 610
3 35/32 155 35/24 653
4 9/8 204 3/2 702
5 7/6 267 14/9 765
6 6/5 316 8/5 814
7 5/4 386 5/3 884
8 35/27 449 140/81 947
9 4/3 498 16/9 996
10 175/128 541 175/96 1039
11 45/32 590 15/8 1088
12 35/24 653 35/18 1151
13 3/2 702 1/1 0
14 14/9 765 28/27 63
15 8/5 814 16/15 112
16 105/64 857 35/32 155
17 27/16 906 9/8 204
18 7/4 969 7/6 267
19 9/5 1018 6/5 316
20 15/8 1088 5/4 386
21 35/18 1151 35/27 449

or, in other words:

shruti pitch on pitch on
# 3/2 string 1/1 string
ratio cents ratio cents

0 1/1 0 1/1 0
1 28/27 63 28/27 63
2 16/15 112 16/15 112
3 35/32 155 35/32 155
4 9/8 204 9/8 204
5 7/6 267 7/6 267
6 6/5 316 6/5 316
7 5/4 386 5/4 386
8 35/27 449 35/27 449
9 4/3 498 4/3 498
10 175/128 541 112/81 561
11 45/32 590 64/45 610
12 35/24 653 35/24 653
13 3/2 702 3/2 702
14 14/9 765 14/9 765
15 8/5 814 8/5 814
16 105/64 857 5/3 884
17 27/16 906 140/81 947
18 7/4 969 16/9 996
19 9/5 1018 175/96 1039
20 15/8 1088 15/8 1088
21 35/18 1151 35/18 1151

Using m (for ma-grama) to mean the 1/1 string, s (for sa-grama) to mean the
3/2 string, and neither letter to mean either string, the full lattice is:

m19-------s10
/ \ / \
/ \ / \
/ \ / \
/ \ / \
m17--------8---------21--------12--------3-------s16
/ \ / \ / \`. ,'/ \`. ,'/ `. ,' `.
/ \ / \ / \ m16--------7--/------20-------s11
/ \ / \ / \/|\/ \/|\/ / \ / \
/ \ / \ / /\|/\ /\|/\ / \ / \
m10-------1---------14-----/--5--\-----s18 \ / \ / \
`. ,' `. /,' `.\ /,' `.\ / \ / \
m18--------9---------0---------13--------4--------s17
/ \ / \ / \ / \ /
/ \ / \ / \ / \ /
/ \ / \ / \ / \ /
/ \ / \ / \ / \ /
m11--------2---------15--------6---------19

There are a lot of new, easily played triads here: any m---s (or m--- or
---s) fifth falls on a single fret. And with some stretching we've got two
otonal and two utonal tetrads -- and even a hexany!

Unless Dave Keenan tells me why I should temper it, I might take this just
tuning as the tuning of my Shrutar!

🔗David C Keenan <D.KEENAN@UQ.NET.AU>

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

🔗David C Keenan <D.KEENAN@UQ.NET.AU>

Here's what the lattice looks like when microtempered (approx ratios of 11
not shown). This does not consider what happens when you make all frets
continuous. This is just one string.

10
/ \
1 14 / \ 5
/ \
/ \
8---------21--------12--------3---------16
/ \ / \ / \ /
/ \ / \ / \ 7 / 20 11
/ \ / 10 \ / \ /
/ \ / \ / \ /
1---------14--------5---------18
\ / \ / \ /
\ / \ 9 / \ 0 / 13
12 \ / 3 \ / 16 \ /
\ / \ / \ /
7---------20--------11
/ \ / \ / \
/ \ 2 / \ 15 / \ 6
14 5 / 18 \ / \ / \
/ \ / \ / \
9---------0---------13--------4---------17
/ \ / \ / \ /
/ \ / \ / \ /
7 / 20 \ / 11 \ / \ /
/ \ / \ / \ /
2---------15--------6---------19

13 4 17

We have 6 otonal and 5 utonal tetrads and one hexany (3,5,10,14,16,20).
Most of the tetrads extend to 9-limit pentads.

As opposed to mapping it to 72-tET, my favourite 11-limit microtempering
distributes the errors as follows:

Intvl Error
(cents)
2:3 -1.05
4:5 -3.18
5:6 2.13
4:7 -0.74
5:7 2.44
6:7 0.31
4:9 -2.10
5:9 1.08
7:9 -1.35
4:11 -1.63
5:11 1.55
6:11 -0.58
7:11 -0.89
9:11 0.47

I'd be interested to hear what effect this has on the step sizes.

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

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

🔗Dave Keenan <D.KEENAN@UQ.NET.AU>

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

Dave Keenan wrote,

>So try widening the 2:3's and 4:5's by 1/6-diaschisma (3.3 cents) or
>widen the 2:3's by 1/8-diaschisma (2.4 cents) and widen the the 4:5's
>by 1/4-diaschisma (4.9 cents).

. . . which would temper the 3:5s by 1/8-diaschisma. Cool! I'll have to try
it later . . .

>It probably smooths out the scale a bit, but I don't think it does
>much for numbers of 7 limit harmonies.

It's really for 5 limit harmonies, and melodically where it will help --
note that the first 9 frets on the 1/1 string are currently the same as
frets 9-18 on the 3/2 string -- I want to extend that similarity out four
more frets, so that I can extend whatever scale I'm playing on the highest
3/2 string downward over the full range of the instrument, using only the
first 12 frets of the 1/1 string and the open 1/1 string.

I gotta go. Can someone help Mango?

🔗David C Keenan <D.KEENAN@UQ.NET.AU>

>David -- "microtempering" here means tempering out how many unison vectors?
>Which ones?

224:225 septimal kleisma and 384:385 11-limit whatever.

>>I'd be interested to hear what effect this has on the step sizes.
>
>Perhaps you could work this out?

Three sizes. s = 35.2 cents, M = 50.6 cents, L = 65.4 cents.
L/s = 1.86. M/s = 1.44, L/M = 1.29

3's 5's cents step
--------------------------------
C 0 0 0.0 65.4
C# -1 2 65.4 50.6
Db -1 -1 116.0 35.2
Cx 2 3 151.2 50.6
D 2 0 201.8 65.4
D# 1 2 267.2 50.6
Eb 1 -1 317.8 65.4
E 0 1 383.1 65.4
E# -1 3 448.5 50.6
F -1 0 499.1 35.2
Ex 2 4 534.3 50.6
F# 2 1 584.9 65.4
Fx 1 3 650.3 50.6
G 1 0 700.9 65.4
G# 0 2 766.3 50.6
Ab 0 -1 816.9 35.2
Gx 3 3 852.1 50.6
A 3 0 902.7 65.4
A# 2 2 968.1 50.6
Bb 2 -1 1018.7 65.4
B 1 1 1084.0 65.4
B# 0 3 1149.4 50.6
C 1200.0

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

🔗David C Keenan <D.KEENAN@UQ.NET.AU>

Here's the 1/8-diaschisma tempering of it.

Three step sizes. s = 48.8 cents, M = 55.6 cents, L = 78.0 cents.
L/s = 1.60. M/s = 1.14, L/M = 1.40

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

It's 2.8 c away from being proper. The smallest ET this could be embedded
in would be 172-tET, which has about 1/7-diaschisma (2.7 c) wide fifths.
It would be proper if it was in 172-tET. 172-tET is only 3-limit consistent
so I guess we'd be using second-best 4:5's.

Here are the step sizes in 1/172 nds of an octave.

8 7 8 7 8 7 11 8 7 8 7 8 7 8 7 8 7 8 7 11 8 7

We can distribute the BP comma too (243:245, 14.2 cents) by narrowing the
4:7's by 3.43 cents (in conjunction with the above 1/8-diaschisma).

This is essentially the following mode of 46-tET

2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 2 2 2 2 3 2 2

46-tET is 13 limit consistent with the following unsigned errors
ratio 2:3 4:5 4:7 4:9 4:11 4:13
cents 2.4 5.0 8.6 8.6 8.6 10.7

A long way from Just now, but it's interesting to see the progression of
related temperaments.

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

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

Dave Keenan wrote,

>Here's the 1/8-diaschisma tempering of it.

>Three step sizes. s = 48.8 cents, M = 55.6 cents, L = 78.0 cents.
>L/s = 1.60. M/s = 1.14, L/M = 1.40

>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

That's awesome! Now the entire step structure from C up to G is the same as
that from G up to D, which is what I wanted. The A is a little high though
-- perhaps distibuting the diaschisma _unequally_ would satisfy me better .
. .

>We can distribute the BP comma too (243:245, 14.2 cents) by narrowing the
>4:7's by 3.43 cents (in conjunction with the above 1/8-diaschisma).

What extra chords do we get this way? What do the step sizes look like?

>This is essentially the following mode of 46-tET

>2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 2 2 2 2 3 2 2

You mean the result of distributing the BP comma as well as the diaschisma
is essentially this "tetrachordal" 22-out-of-46? Wild!

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

🔗Dave Keenan <D.KEENAN@UQ.NET.AU>

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:
> Dave Keenan wrote,
>
> >This is essentially the following mode of 46-tET
>
> >2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 2 2 2 2 3 2 2
>
> That may or may not be identical to what Graham Breed refers to
here:
>
> http://x31eq.com/diaschis.htm
>
> "46-equal is fairly close to just in the 7-limit. The third is the
worst
> interval. 46 is a lot of notes, but you can use it with a 22 note
keyboard
> mapping. The problem is that the 7-limit approximation isn't
consistent with
> 22-equal. So, you won't be putting your fingers where you would if
you were
> playing 22-equal."
>
> That last statement surprises me greatly. Is Graham right about
that?

Absolutely.

You'll also see it in my
http://dkeenan.com/Music/2ChainOfFifthsTunings.htm

In both cases the 4:5 is two fourths plus a half-octave.

In 22-tET and related double chains of fifths/fourths, the best 4:7
approximation is a chain of two fourths.

But in 46-tET and related, the best 4:7 is eight fourths plus a
half-octave. That's why it has so few 7-limit ratios in 22 notes.

Here's an otonal tetrad shown on twin chains of 11 fifths for 46-tET.

- - - - - - - - 4 6 -
7 - - - - - 5 - - - -

So you can see that we only get 4 otonal and 4 utonal tetrads for our
22 notes.

Damn! Now I've answered a previous question you asked, for which I had
said "your turn". :-)

-- Dave Keenan

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

I wrote,

>>So, you won't be putting your fingers where you would if
>>you were
>> playing 22-equal."
>
>> That last statement surprises me greatly. Is Graham right about
>>that?

Dave Keenan wrote,

>Absolutely.

>In both cases the 4:5 is two fourths plus a half-octave.

>In 22-tET and related double chains of fifths/fourths, the best 4:7
>approximation is a chain of two fourths.

>But in 46-tET and related, the best 4:7 is eight fourths plus a
>half-octave. That's why it has so few 7-limit ratios in 22 notes.

>Here's an otonal tetrad shown on twin chains of 11 fifths for 46-tET.

>- - - - - - - - 4 6 -
>7 - - - - - 5 - - - -

But you'd still be putting your fingers where you would for a 22-tET tetrad,
wouldn't you? I mean, it's the same generic pattern out of 22 notes, though
it only

>So you can see that we only get 4 otonal and 4 utonal tetrads for our
>22 notes.

But there are more than 22 notes -- 17, 18, and 19 still come in m and s
varieties. I'm working on the full lattice.

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

>> >We can distribute the BP comma too (243:245, 14.2 cents) by
narrowing the
>> >4:7's by 3.43 cents (in conjunction with the above 1/8-diaschisma).
>
>> What extra chords do we get this way?

>Your turn. :-)

Well, starting with this lattice:

m19-------s10
/ \ / \
/ \ / \
/ \ / \
/ \ / \
m17--------8---------21--------12--------3--------s16
/ \ / \ / \`. ,'/ \`. ,'/ `. ,' `.
/ \ / \ / \ m16-/---\--7--/------20-------s11
/ \ / \ / \/|\/ \/|\/ / \ / \
/ \ / \ / /\|/\ /\|/\ / \ / \
m10-------1---------14-----/--5--------s18 \ / \ / \
`. ,' `. /,' `.\ /,' `.\ / \ / \
m18--------9---------0---------13--------4--------s17
/ \ / \ / \ / \ /
/ \ / \ / \ / \ /
/ \ / \ / \ / \ /
/ \ / \ / \ / \ /
m11--------2---------15--------6--------s19

and first tempering out the diaschisma, we get
m17--------8------...
/ \ / \ .
/ \ / \ .
/ \ / \ .
/ \ / \ /
m19--------10--------1---------14...
/ \ / \ / `. ,'
/ \ / \ / m18-----...
/ \ / \ / / \ .
/ \ / \ / / \ .
m17--------8---------21--------12--------3--------s16 / \ .
/ \ / \ / \`. ,'/ \`. ,'/ `. ,' `. / \ /
/ \ / \ / \ m16-/---\--7--/------20--------11--------2-...
/ \ / \ / \/|\/ \/|\/ / \ / \ /
/ \ / \ / /\|/\ /\|/\ / \ / \ /
10--------1---------14-----/--5--------s18 \ / \ / \ /
\ / `. ,' `. /,' `.\ /,' `.\ / \ / \ /
\ / m18--------9---------0---------13--------4--------s17
\ / / \ / \ / \ / \ /
\ / / \ / \ / \ / \ /
...-s16 / \ / \ / \ / \ /
,' `. / \ / \ / \ / \ /
20--------11--------2---------15--------6--------s19
\ / \ /
\ / \ /
\ / \ /
\ / \ /
...--4--------s17
/
/
/
/
s19

So, like you said, not a lot of new chords. But then tempering out the
245:243 gives:

/ \ / \ /. . . . . . . m17--------8------...
12--------3--------s16 . . . . . . ./|\. ./|\. .
\`. ,'/ `. ,' \ / \ / \ / /\|/\ /\|/\ .
..\.-7--/------20--------11--------2---------15--------6--------s19 \ .
\/ \/ / \ / \ / `. ,' `. /,' `.\ /,' `.\ /
/\ /\ / \ / \ / m19--------10--------1---------14...
../-s18 \ / \ / \ / / \ / \ / `. ,'
/,' \ / \ / \ / / \ / \ / m18-----...
0---------13--------4--------s17 / \ / \ / / \ .
\`. ,'/ \`. ,'/ `. ,' `. / \ / \ / / \ .
\ m17-/---\--8--/------21--------12--------3--------s16 / \ .
\/|\/ \/|\/ / \`. ,'/ \`. ,'/ `. ,' `. / \ /
/\|/\ /\|/\ / \ m16-/---\--7--/------20--------11--------2-...
...--6--------s19 \ / \/|\/ \/|\/ / \ / \ / `.
/ `.\ /,' `.\ / /\|/\ /\|/\ / \ / \ /
10--------1---------14-----/--5--------s18 \ / \ / \ / .
\ / `. ,' `. /,' `.\ /,' `.\ / \ / \ / .
\ / m18--------9---------0---------13--------4--------s17 .
\ / / \ / \ / \`. .'/ \`. ,'/ `. ,' `. /
\ / / \ / \ / \ m17-/---\--8--/------21--------12...
...-s16 / \ / \ / \/|\/ \/|\/ / \ ,'/ \`.
,' `. / \ / \ / /\|/\ /\|/\ / \ m16-/---...
20--------11--------2---------15--------6--------s19 \ / \/|\/ ..
\ / \ / `. ,' `. /,' `.\ /,' `.\ / /\|/\ ..
\ / \ / m19--------10--------1---------14--------5--\---...
\ / \ / / \ / \ / `. ,' `. /,' `.\ /,'
\ / \ / / \ / \ / m18--------9---------0
...--4--------s17 / \ / \ / / \ / \ / \
,'/ `. ,' `. / \ / \ / / \ / \ / .
8--/------21--------12--------3--------s16 / \ / \ / .
| / / \`. ,'/ \`. ,'/ `. ,' `. / \ / \ / .
|/ . . m16-.---.--7--.-----20---------11--------2---------15-----...
s19 . ./ \. ./ \. / \ / \ / `. ,' `.

So you were right, 4 otonal tetrads and 4 utonal tetrads . . . what's nice
is that half of them involve open strings, so they can actually be played .
. . and half of those each involve _both_ open string pitches, which I would
consider the essential tetrads you'd want in "dronality" such as this. There
are are 2 hexanies too.

I wrote,

>17, 18, and 19 still come in m and s varieties.

16 too.

🔗graham@microtonal.co.uk

Paul Erlich wrote:

> >>So, you won't be putting your fingers where you would if
> >>you were
> >> playing 22-equal."
> >
> >> That last statement surprises me greatly. Is Graham right about
> >>that?

> But you'd still be putting your fingers where you would for a 22-tET
> tetrad,
> wouldn't you? I mean, it's the same generic pattern out of 22 notes,
> though
> it only

No it isn't. You remember we had this discussion way back about 7-limit
approximations for these kinds of scales. You favor a mapping that's
consistent with 22-equal so that your decatonic scales work (or
vice-versa). I came up with two different mappings, which can give better
7(9, etc)-limit approximations. But they aren't consistent with decatonic
scales, and this one in particular isn't consistent with 22-equal.

There's another mapping that covers the ground between 34 and 22. That
will have the right number of steps in the 7-limit intervals, but they'll
be in the "wrong" places, like the 5-limit schismic intervals in a
Pythagorean tuning are in the "wrong" places if you're expecting meantone.
the 46-equal mapping is more different than this.

Note: this also means that scales defined on shrutis can have different
7-limit interpretations depending on whether you consider the scale to be
equal tempered-like or Pythagorean-like.

My "shruti" tuning has a 12-note tritone like this, starting on C:

r p r p r-p p r p r p r p

where r is the larger and p the smaller interval

Removing the E key to match Paul's mapping, that gives the 22 note octave

r p r r p r p r p r p r p r r p r p r p r p

The equivalent of the traditional shruti scale starts on Eb, according to
the comment in my tuning file.

S r R g G M m P d D n N S
r p r p r p r p r p r r p r p r p r p r p r
F G A B C D F G A B C D

Top row is Hindustani name, middle row is intervals, bottom row shows you
the notes the keys would normally play.

That still looks wrong to me, because I thought there should be an r
either side of P. But that can be fixed by using the F key instead of E
for that bit:

S r R g G M m P d D n N S
r p r p r p r p r p r p r r p r p r p r p r
F G A B C D E G A B C D

Anyway, take the thing Dave Keenan came up with

2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 2 2 2 2 3 2 2

Oh, it isn't the same, because my "p" is one step from 46. So 2 would be
the r-p above.

The diatonic semitone, r+p, will match up both scales. So we'd get

S r R g G M m P d D n N S
2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 2 2 2 2 3 2 2
F G A B C D E G A B C D

The top row gives the same pitches as me. This is such a stunning
coincidence, you'd almost thing it was a fix ;)

It may well be that this scale *does* use the same fingerings for some
intervals as 22-equal. But I'd guess some of the 5-limit must break ...?

I'm going to be offline for around a week now, and so can't follow it up
properly. My scale is more "shruti-like" in that the tuning given by B.
C. Deva and others (one on the Net, try a search) definitely has commas,
and Daves scale doesn't. Then again, the original shrutis used by Bharata
would probably have been more like 22-equal, as suggested by Paul in his
paper. (I think he falls short of asserting that the historical scale
matched his ideas, but does say it would make sense if it did. Or
something.)

Very interesting. Manuel's mode list gives historical ragas defined on
shrutis. I went through these, and found that some made sense with an
unequal scale like that above, and others were more like 22 equal shrutis.
You may like to revisit this in the light of your new scale. Also, can
anybody find more ragas defined against shrutis?

Graham

🔗Monz <MONZ@JUNO.COM>

--- In tuning@egroups.com, graham@m... wrote:

> http://www.egroups.com/message/tuning/16821
> ...

Graham,

You display a great breadth of technical scale knowledge
in this post!

> Note: this also means that scales defined on shrutis can
> have different 7-limit interpretations depending on whether
> you consider the scale to be equal tempered-like or
> Pythagorean-like.

This made me think of the lexicographical question of spelling
for the classical Indian term for a small interval: "sruti"
vs. "shruti".

Can anyone with information on other names for this term
- i.e., citations from the literature as it filtered into
Persian or possibly Arab or other cultures - bring them forward,
so I may include them in my Dictionary?

-monz
http://www.ixpres.com/interval/monzo/homepage.html
'All roads lead to n^0'

🔗David C Keenan <D.KEENAN@UQ.NET.AU>

>So you were right, 4 otonal tetrads and 4 utonal tetrads . . . what's nice
>is that half of them involve open strings, so they can actually be played.
>. . and half of those each involve _both_ open string pitches, which I would
>consider the essential tetrads you'd want in "dronality" such as this. There
>are are 2 hexanies too.

I was only right by accident. My premises were wrong. I was talking about
one string (22 notes not 26) and it isn't twin chains of fifths a half
octave apart. There are of course 4 (inequal length) chains as you showed.

>But you'd still be putting your fingers where you would for a 22-tET tetrad,
>wouldn't you? I mean, it's the same generic pattern out of 22 notes, though
>it only

Yes. You are right. I was wrong. Where the 7 limit intervals exist, they
span the same number of scale steps as in 22-tET.

With this 22-tET notation:

0 C
1 C/ or Db
2 Db/ or C#\
3 D\ or C#
4 D
5 D/ or Eb
6 Eb/ or D#\
7 E\ or D#
8 E
9 F
10 F/ or Gb
11 Gb/ or F#\
etc.

a 4:5:6:7 approximation looks like C:E\:G:Bb.

Here's the corresponding notation for 46-tET:

0 C
1 C/
2 Db\
3 Db
4 Db/ or C#\
5 C#
6 C#/
7 D\
8 D
9 D/
10 Eb\
11 Eb
12 Eb/ or D#\
13 D#
14 D#/
15 E\
16 E
17 E/
18 F\
19 F
20 F/
21 Gb\
22 Gb
23 Gb/ or F#\
etc.

Now the 4:5:6:7 approximation is C:E\:G:Bb\ (note Bb\ not Bb).

8 tetrads can be acheived with only 20 notes from 46-tET, and with 26 notes
one can get up to 16 tetrads, but of course these scales have comma-sized
steps and are not "tetrachordal".

You have done well to get 8 given those constraints.

-- Dave Keenan
http://dkeenan.com

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

🔗Dave Keenan <D.KEENAN@UQ.NET.AU>

Paul,

To be fair, Graham did write:

>It may well be that this scale *does* use the same fingerings for
>some intervals as 22-equal. But I'd guess some of the 5-limit must
>break ...?

Graham,

Indeed many intervals break, relative to 22-tET, and not merely
5-limit ones. This is of course what allows the remaining intervals to
be rendered more accurately. Paul's "tetrachordal" 22-of-46 may be
considered a detempering of 22-tET. But where good 7-limit
approximations _do_ occur they have the same pattern of scale steps in
Paul's tuning as they do in 22-tET.

However, the ratios of 7 do not retain the same pairs of note-names in
any consistent naming scheme (across the two tunings), while the
5-limit ones do.

Regards,
-- Dave Keenan

🔗Bill Alves <ALVES@ORION.AC.HMC.EDU>

Monz:
>This made me think of the lexicographical question of spelling
>for the classical Indian term for a small interval: "sruti"
>vs. "shruti".
>
>Can anyone with information on other names for this term
>- i.e., citations from the literature as it filtered into
>Persian or possibly Arab or other cultures - bring them forward,
>so I may include them in my Dictionary?
>
Sruti is the usual transliteration from Sanskrit. Shruti is possibly a
Hindi version. To my knowledge, the term first appears in the Natyasastra,
the early Indian treatise on dramaturgy and music. The date of this work is
highly disputed, and it has been variously dated at 300 BCE to 500 CE.
Probably a date of c. 300 is reasonable. Anyway, this work is the source of
the idea that the Indian scale consists of 22 srutis per octave. They may
be grouped according to two scale types, or grama:

Sa-grama: 4 3 2 4 4 3 2
Ma-grama: 4 3 4 2 4 3 2

Bharata, the supposed author of the Natyasastra, also defines intervals
through srutis: consonant intervals (samvadi) have 9 or 13 sruti between
them; 2 or 20 sruti makes an interval dissonant (vivadi); other intervals
are "assonant" or neither consonant nor dissonant. However, he never
associates any precise tuning with the sruti.

Nevertheless, the tuning of sruti has been the source of endless debate
since the time of the Natyasastra. Some regard them as equal, others
non-equal, some just, others tempered. Complicating the picture is that
fact that what was, at least theoretically, a 22-tone system at the time of
Bharata, became by modern times, at least practically speaking, a 12-tone
system. There have been many attempts to reconcile the 22 with the 12.
There is enough intonational flexibility in Indian unfretted strings and
voice that pinning down a single particular tuning, especially a fixed
22-tone tuning, is probably pointless.

Today sruti is also used in an informal way as the equivalent of our word
"microtone." Thus a singer will speak of ornamenting a tone by sliding up
"a sruti." "Sruti" is also used to mean simply "interval" or "step."

For more on this debate and various definitions, I recommend:

Mark Levy's _Intonation in North Indian Music_
Nazir Jairazbhoy's _The Rags of North Indian Music_ (he has also written a
couple of articles on the subject)
and the article on India in _The New Groves Dictionary_.

Bill

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
^ Bill Alves email: alves@hmc.edu ^
^ Harvey Mudd College URL: http://www2.hmc.edu/~alves/ ^
^ 301 E. Twelfth St. (909)607-4170 (office) ^
^ Claremont CA 91711 USA (909)607-7600 (fax) ^
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

🔗Monz <MONZ@JUNO.COM>

I wrote:

> http://www.egroups.com/message/tuning/16833
>
> This made me think of the lexicographical question of spelling
> for the classical Indian term for a small interval: "sruti"
> vs. "shruti".
>
> Can anyone with information on other names for this term
> - i.e., citations from the literature as it filtered into
> Persian or possibly Arab or other cultures - bring them forward,
> so I may include them in my Dictionary?

Many thanks to Bill Alves for an in-depth reply:

http://www.egroups.com/message/tuning/16856

I also got one from a Hindi list reader, who will soon be
subscribing. He asked me to forward his reply to the list:

(Go easy on him, Paul, till he gets used to you... ;-)

-monz

-------- forwarded message -----------------------

Date: Sat, Dec. 23, 2000
From: Haresh BAKSHI <hareshbakshi@hotmail.com>
Subject: Shruti

The spelling: sruti (the first letter "s" as the diacritical
Roman symbol) has been used by Sir Monier-Williams, Walter Kaufmann,
Ashok Da Ranade, L. Subramaniam; shruti, by Alain Danielou, Swami
Prajnananda and several others. It is a question of how we prefer
to represent the Sanskrit sound, as in the English word "shrewd",
for transliteration into the Roman alphabet.

Definitions: Lack of standardisation is the hallmark of Sanskrit
terms. Shruti has been variously translated as: microtone,
microtonic interval, interval, step etc. It is mainly determined
through fine auditory perception or grasp. The number of sounds
falling within a scale is infinite; but the number which can be
differntiated (or perceived, or grasped) as musically useful,
is 22. These 22 shruti-s are further grouped into five classes,
bringing in the aesthetic concept that a shruti is not only an
auditory perception, but also a distinct expression to the
listener's mind. There is a broad agreement that shruti-s are
not equal; that the seven notes are the sounds selected from
among these 22 shruti-s; that each note is established on its
first shruti; that shruti-s can be reproduced on a string
instrument, in terms of various vibrating lengths; that shruti-s
can, therefore, be stated in terms of frequencies, though Indian
music, being truly modal, is built on the relationship of sounds
with the tonic (also called the reference note, key etc.).

References: The earliest mention appears to have been made in
Bharat muni's Natyashastra (about 500 B.C.). Later references
include Narada of Shiksha (1st century A.D.), Kohala (quoted in
Brihaddeshi) Dattila's Brihaddeshi, and several later works.

Remarks: Shruti is a part of tonal concept. Talented singers use
shruti-s, regularly and unknowingly, as part of their musical
repertoire. Quite often, even the raga-s have their notes
established on shruit-s, rather than the usual notes. For example,
the "komal Re" (Db, if C is the tonic) of the raga Todi, is really
a microtone, lower than the usual location of "komal Re" on the
Indian scale. Even the singers of a capella, use the notes of
the harmonic scale, not the notes of the tempered scale; this
is because of absence of accompaniment on piano or any instrument
with fixed, tempered scale. So, singing correct microtones comes
naturally to the voice of a talented singer.

Haresh BAKSHI

---------------------------

-monz
http://www.ixpres.com/interval/monzo/homepage.html
'All roads lead to n^0'

🔗graham@microtonal.co.uk

Paul H. Erlich wrote:

> I wrote,
>
> >> But you'd still be putting your fingers where you would for a 22-tET
> >> tetrad,
> >> wouldn't you? I mean, it's the same generic pattern out of 22 notes,
>
> Graham wrote,
>
> >No it isn't.
>
> Well, Graham, you must be misunderstanding something, as Dave Keenan
> confirmed that it is.

Yes, later in the post I discovered that Dave's scale is not the same as
mine. The 7-limit intervals aren't 22-equal consistent in my tuning.

Look at the tritones. Here's my original scale:

S r R g G M m P d D n N S
r p r p r p r p r p r r p r p r p r p r p r
F G A B C D F G A B C D

As the scale repeats every 11 notes, each interval of 11 notes will be an
exact half-octave of 23 steps. If it doesn't look that way, there's a
mistake. However, 46-equal's best approximation to 7/5 is 22 steps, a p
flat. Such an interval exists on my mapping, between the F and Eb keys.
but that's only 10 notes on the keyboard. The best approximation in
22-equal is 11 notes, so there's a discrepancy.

My adapted, more sruti-like scale, has different intervals for 11 notes:

S r R g G M m P d D n N S
r p r p r p r p r p r p r r p r p r p r p r
F G A B C D E G A B C D

Between the F and E keys the interval is 5r+6p. That's 5*3+6*1=21 steps.
That's too flat for a 7/5. So the 22-equal fingering never matches the
best 7-limit one. QED.

Now take Dave's scale:

S r R g G M m P d D n N S
2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 2 2 2 2 3 2 2
F G A B C D E G A B C D

This is interesting in that it's more like 22-equal, but still keeps the
accuracy of 46-equal. The notes on the top row are unchanged, so the
5-limit matches the other scales. Between B and B or C and C is 2*11=22
steps. So yes, it's 7-limit 22-equal consistent.

For 14- and 26-equal, I've dabbled with this tuning:

C D E F G A B C
r r r r s r r r r r r s

This time, the way it fits the keyboard matches the way it sounds. The
white notes are in meantone, and each black note is midway between the
neighbouring white notes. So this is a kind of temperament where the
5-limit matches meantone but the 7-limit diverges.

Dave's "shruti" scale is the 12/22 equivalent. The base 12 notes are
identical to a regular diaschismic/whatever scale, and the extra 10 notes
split the original 12. That means, again, the 7-limit mappings diverge.

This deserves more attention, but not from me right now as I want to get
to bed.

Graham

🔗graham@microtonal.co.uk

I wrote:

> For 14- and 26-equal, I've dabbled with this tuning:
>
> C D E F G A B C
> r r r r s r r r r r r s
>
> This time, the way it fits the keyboard matches the way it sounds. The
> white notes are in meantone, and each black note is midway between the
> neighbouring white notes. So this is a kind of temperament where the
> 5-limit matches meantone but the 7-limit diverges.

This mapping can be described as "double negative" as it is fully
consistent with Bosanquet's terminology.

It would have been more relevant if I'd mentioned the double positive
mapping that fits 22 and 46-equal:

C D E F G A B C
s s s s r s s s s s s r

This isn't so useful as a keyboard tuning, but makes some sense as a basis
for notation.

> Dave's "shruti" scale is the 12/22 equivalent. The base 12 notes are
> identical to a regular diaschismic/whatever scale, and the extra 10
> notes split the original 12. That means, again, the 7-limit mappings
> diverge.
>
> This deserves more attention, but not from me right now as I want to
> get to bed.

Well, I've done so now. It's some kind of 22-based scale. So probably
22-ly double negative in Wilson's terminology. I haven't worked it out.
Whatever, it can be generalized:

S r R g G M m P d D n N S
q q q q q q r q q q q q q q q q q q q r q q
F G A B C D E G A B C D

A 2:3 is always r+12q, a 4:5 is r+6q and a 4:7 is always r+17q. So any
7-prime limit interval can be defined on this scale. The minimax 7-odd
limit temperament is with 7/4 just, which means r is 73.1 cents and q is
52.7 cents. The worst intervals are 3/2 and 7/6 which are 3.4 cents out.
The same tuning was given by Dave Keenan on the 26th.

So, Dave, isn't this also the minimax 9-limit tuning? That'll make 9/8
and 9/7 each 6.8 cents off.

The 22-ly double-negative(?) scales are 24, 46, 68, 90, etc. In 24-equal,
q really is a quartertone. 68-equal is a good 7-limit approximation, and
you might get it to work with a clever tuning of a guitar fretted for
34-equal.

Graham

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

Graham Breed wrote,

>It would have been more relevant if I'd mentioned the double positive
>mapping that fits 22 and 46-equal:

> C D E F G A B C
> s s s s r s s s s s s r

>This isn't so useful as a keyboard tuning, but makes some sense as a basis
>for notation.

This is a tremendously useful keyboard tuning for 22-tET, since you have
three decatonic scales: two pentachordal and one symmetrical.

>68-equal is a good 7-limit approximation, and
>you might get it to work with a clever tuning of a guitar fretted for
>34-equal.

That's very clever, but you'd be skipping around the fingerboard just to
play a simple melody. Plus, I find the melodic difference between a major
whole tone and minor whole tone just a little too large in 34-equal.

🔗graham@microtonal.co.uk

In-Reply-To: <CE80F17667E4D211AE53009027466272012A33D3@acadian-asset.com>
Paul Erlich wrote:

> >It would have been more relevant if I'd mentioned the double positive
> >mapping that fits 22 and 46-equal:
>
> > C D E F G A B C
> > s s s s r s s s s s s r
>
> >This isn't so useful as a keyboard tuning, but makes some sense as a
> basis >for notation.
>
> This is a tremendously useful keyboard tuning for 22-tET, since you have
> three decatonic scales: two pentachordal and one symmetrical.

And also some diatonic scales. There are kinds of E and D major in there.
But you can do better with meantones.

The alternative with 12 notes is:

C C# D Eb E F F# G G# A Bb B C
s s r s s s s s r s s s

Where you get a kind of C major scale straight off, and sharps and flats
are consistent relative to it. All white-note thirds except D-F are
5-limit. There are also four decatonic scales in 22-equal, two of each
kind. And ... it looks like the pattern of decatonics is preserved
(transposed) for the other mapping. So it looks better all round as a
decatonic keyboard tuning.

It has some advantages as a basis for notation. It would be more
compatible with a schismic or JI notation that assumed the basis to be a
just C major scale. However, as a Pythgorean basis seems to be in favour,
that would count against it. Another wrinkle is that Ab won't naturally
equal G#. So overall, I don't favour it for notation, although it seems
to be preferable as a 12-from-22 tuning.

Note that the 7-limit intervals in the decatonics don't translate to other
tunings like 46-equal. So, in the general case, these mappings are only
useful in the 5-limit. Whereas the standard 12-note meantone tuning is
generally useful in the 7-limit, as is DK's 22-note 2-interval scale.

> >68-equal is a good 7-limit approximation, and
> >you might get it to work with a clever tuning of a guitar fretted for
> >34-equal.
>
> That's very clever, but you'd be skipping around the fingerboard just to
> play a simple melody. Plus, I find the melodic difference between a
> major
> whole tone and minor whole tone just a little too large in 34-equal.

I've done the same kind of thing to get neutral thirds from a
spiral-of-fifths fretting. You don't need to "skip around the
fingerboard" but only "skip between strings". It's cumbersome, but easier
than refretting the guitar for simple experiments ... in this case only if
you happen to have a 34-equally fretted guitar lying around. I thought it
was worth mentioning, as 68= is very close to the optimum.

Clever tuning with 17= really would mean skipping round the fingerboard.

Graham

🔗Paul Erlich <PERLICH@ACADIAN-ASSET.COM>

--- In tuning@egroups.com, graham@m... wrote:
> In-Reply-To: <CE80F17667E4D211AE53009027466272012A33D3@a...>
> Paul Erlich wrote:
>
> > >It would have been more relevant if I'd mentioned the double positive
> > >mapping that fits 22 and 46-equal:
> >
> > > C D E F G A B C
> > > s s s s r s s s s s s r
> >
> > >This isn't so useful as a keyboard tuning, but makes some sense as a
> > basis >for notation.
> >
> > This is a tremendously useful keyboard tuning for 22-tET, since you have
> > three decatonic scales: two pentachordal and one symmetrical.
>
> And also some diatonic scales. There are kinds of E and D major in there.

Ugh -- these are melodically unpleasant (IMO); in E major you have a putrecent V chord,
and in D major a yucky IV chord. The closest you can get to the usual "JI major" scale,
with an out-of-tune ii chord and good triads on I, iii, IV, V, and vi, are A major and Ab
major, though the melodic unpleasantness is still there.

> But you can do better with meantones.

Infinitely better.
>
> The alternative with 12 notes is:
>
> C C# D Eb E F F# G G# A Bb B C
> s s r s s s s s r s s s
>
> Where you get a kind of C major scale straight off, and sharps and flats
> are consistent relative to it. All white-note thirds except D-F are
> 5-limit. There are also four decatonic scales in 22-equal, two of each
> kind. And ... it looks like the pattern of decatonics is preserved
> (transposed) for the other mapping.

Not following.

> So it looks better all round as a
> decatonic keyboard tuning.

The other mapping has the advantage that the color relationships immediately tell you
the sound relationships, and the colors are reversible, so to speak. True, yours has more
decatonics, but the whole chromatic scale on mine can be seen as a hexachordal
dodecatonic scale, which I like.
>
> > >68-equal is a good 7-limit approximation, and
> > >you might get it to work with a clever tuning of a guitar fretted for
> > >34-equal.
> >
> > That's very clever, but you'd be skipping around the fingerboard just to
> > play a simple melody. Plus, I find the melodic difference between a
> > major
> > whole tone and minor whole tone just a little too large in 34-equal.
>
> I've done the same kind of thing to get neutral thirds from a
> spiral-of-fifths fretting. You don't need to "skip around the
> fingerboard" but only "skip between strings".

You have to skip around the fingerboard if you're always using all available open strings
as a drone, which is what I do in this style. Plus the melodic nuances I use really require
the melody to be played on one string at a time.