RE: [tuning] Re: Variations on the Shrutar tuning: three-stepsize -omintetrachordality; Graham Bre

Dave Keenan wrote,

>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.

I see nothing wrong with the latter -- and there would only be a few cents
difference between the various versions. Besides, the strings are likely to
be tuned to a pure 3:2 on average, given the way I tune at least.

>This sure seems like what you are looking for.

Well, rather than "breaking" the chains, I was thinking of applying an
unequal tempering so that the proximate intervals would be closer to just,
and those around the edges farthest from just (but still usable). Kind of
like a well-temperament, but with an extra dimension.

Hi Dave K.,

>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.

Can you elucidate your logic here? Does it affect the logic of your
derivation that I will by no means restrict myself to those frets? Also, it
looks like s4-s17 is just, but that's not in your lattice . . . I think we
may be closing in on the final Shrutar, though!

Dave Keenan wrote,

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

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.

I'll look at your full proposal more closely when I get a chance . . .

Dave Keenan wrote,

>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).

But weren't some of the major thirds, as between shruit 4 and shruti 11,
better before?

>Here's the fret arrangement expressed as ratios.

>19 64/35

Hmm . . . if you don't have a decent 9/5 on the 1/1 string, you won't have a
decent 6/5 on the 3/2 string, which is a problem, because you're supposed to
have the whole Modern Indian Gamut on the 3/2 string.

I wrote,

>Hmm . . . if you don't have a decent 9/5 on the 1/1 string, you won't have
a
>decent 6/5 on the 3/2 string, which is a problem, because you're supposed
to
>have the whole Modern Indian Gamut on the 3/2 string.

Oops -- I'm sorry, the 8/5 on the 1/1 string is what produces the 6/5 on the
3/2 string -- so it's there -- sorry!

>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.

Here's the full lattice of all the just triads and tetrads:

m16========7========s20
/|\ /|\ / \
/ | \ / | \ / \
/ m5--------s18 \ / \
/,' `.\ /,' `.\ / \
9=========0=========13========4
\ / \`. .'/ \`. ,'/
\ / \ m17-/---\-s8 /
\ / \ | / \ | /
\ / \|/ \|/
m15========6========s19

Fingerings for just triads:

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

note string fret
7 1/1 7
13 1/1 13
s20 3/2 7

note string fret
m16 1/1 16
0 1/1 0
7 3/2 16

note string fret
9 1/1 9
m15 1/1 15
0 3/2 9

note string fret
m15 1/1 15
0 1/1 0 or 22
6 3/2 15

note string fret
6 1/1 6
13 3/2 0
or 1/1 13
s19 3/2 6

Clearly any just 5-limit triad will be playable, with some drone strings
open to boot (often adding nice intervals like major sevenths, etc.)

Fingerings for just tetrads:

note string fret
0 1/1 0
7 1/1 7
13 3/2 0
s18 3/2 5

note string fret
0 1/1 0
6 3/2 15
13 3/2 0
m17 1/1 17

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?

Dave Keenan wrote,

>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

Oops! Thanks for catching that!

>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.

Not really -- it's only a little over 3 12-tET frets!

>6:9:11 is possible too.
> 4 1/1 4
>s17 3/2 4
>m1 1/1 1

So we can play a full 8:9:10:11:12 chord on 3/2 thus:

13 3/2 0
s20 3/2 7
4 1/1 4
s17 3/2 4
m1 1/1 1

Awesome! Might be difficult to finger, but the fretted notes could be moved
up an octave to make it easier. A resolution from m1 in this chord to an
m0(1/1) in the next chord might be particularly effective.

>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.

The whole thing is:

0 1/1 0
13 1/1 13
7 3/2 15
s18 3/2 18
4 3/2 13
m10 1/1 10

So any 5-note subset that includes 0 is playable.

I still have a question about this:

You wrote,

>>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).

I wrote,

>But weren't some of the major thirds, as between shruit 4 and shruti 11,
better before?

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:
>But weren't some of the major thirds, as between shruit 4 and
>shruti 11, better before?

I don't know how I missed this question before!. I hope I didn't miss
others.

Currently 4-s11 is 4.9 c wide, 4-m11 is 9.8 c wide. I think they were
6.5 c and 13 c in the earlier case where the tempered fifths were
1/3-diaschisma (6.5 c) or 1/2 BP comma (7.1 c). But even if they were
0 c and 6.5 c in the earlier case, why would you care? That major
third is as far from the centre as you can get.

What got worse with this latest tuning are several 4:7's and 6:7's
that do not partake in any tetrad, but merely 4:6:7 triads. Their
errors went up from zero to 4.7 c.

I'm interested in your opinion whether m19 and s3 are playable as 11's
(and if so, whether you _want_ to use them as such),
e.g. m19 in subsets of the 9-0-m16-m5-13-m19 hexad. Because if not,
they should be tuned to improve some 4:6:7's and 1/(7:6:4)'s and make
the scale a little more even melodically.

Regards,
-- Dave Keenan

Dave Keenan wrote,

>Currently 4-s11 is 4.9 c wide, 4-m11 is 9.8 c wide. I think they were
>6.5 c and 13 c in the earlier case where the tempered fifths were
>1/3-diaschisma (6.5 c) or 1/2 BP comma (7.1 c).

Oops -- I guess I read something wrong?

>But even if they were
>0 c and 6.5 c in the earlier case, why would you care? That major
>third is as far from the centre as you can get.

Yes, but it's the only reason for tempering out the diaschisma in the first
place -- I definitely want to use the 4-11-17 triad.

>I'm interested in your opinion whether m19 and s3 are playable as 11's
>(and if so, whether you _want_ to use them as such),

Not really, though I like the idea of having melodically usable fifths from
as many notes as possible, including the 11's. So where does that leave us?

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:
> Dave Keenan wrote,
>
> >Currently 4-s11 is 4.9 c wide, 4-m11 is 9.8 c wide. I think they
were
> >6.5 c and 13 c in the earlier case where the tempered fifths were
> >1/3-diaschisma (6.5 c) or 1/2 BP comma (7.1 c).
>
> Oops -- I guess I read something wrong?

You'd better check. It might be me who's reading it wrong.

> >But even if they were
> >0 c and 6.5 c in the earlier case, why would you care? That major
> >third is as far from the centre as you can get.
>
> Yes, but it's the only reason for tempering out the diaschisma in
the first
> place -- I definitely want to use the 4-11-17 triad.

Oh sure. 4-s17 is a just fifth. Isn't 4-s11-s17 playable?

Ah! But if you want to do a chord progression around the "diaschismic
loop" you'd need to use m11-s2. This is a just fifth, but s17-s2 and
m11-m18 are major thirds that have a 9.8 c error and would have to be
used. Surely this is ok. Given that they will be used with just
fifths, and we are used to the worse errors of 12-tET.

> >I'm interested in your opinion whether m19 and s3 are playable as
11's
> >(and if so, whether you _want_ to use them as such),
>
> Not really, though I like the idea of having melodically usable
fifths from
> as many notes as possible, including the 11's. So where does that
leave us?

It leaves us with this tuning:

On 1/1 Cents Step On 3/2 Cents Difference
string string between strings
-------------------------------------------------------
0 (o) 0.00 53.3 0 0.00 0.00
1 53.30 53.5 1 53.30 0.00
m2 106.84 48.3 s2 101.96 4.89
m3 155.14 48.8 s3 150.61 4.53
4 203.91 63.0 4 203.91 0.00
m5 266.87 48.8 s5 262.14 4.73
6 315.64 70.7 6 315.64 0.00
7 386.31 53.5 7 386.31 0.00
m8 439.81 58.2 s8 435.08 4.73
9 498.04 53.3 9 498.04 0.00
m10 551.35 48.7 s10 546.82 4.53
m11 600.00 48.7 s11 595.11 4.89
12 648.65 53.3 12 648.65 0.00
13 701.96 58.2 13 (o) 701.96 0.00
m14 760.19 53.5 s14 755.26 4.93
m15 813.69 70.7 s15 808.80 4.89
m16 884.36 48.8 s16 857.09 27.26
m17 933.13 63.0 s17 905.87 27.26
m18 996.09 48.8 s18 968.83 27.26
m19 1044.86 48.3 s19 1017.60 27.26
m20 1093.16 53.5 s20 1088.27 4.89
m21 1146.70 53.3 s21 1141.77 4.93

Where (o) indicates the open string.

Unless you want me to try to push some of those 9.8 c errors in the
most remote thirds, onto something else. It will have to make some
fifths worse.

If I get time, I'll try to do some lattices showing the error in every
usable interval.

Regards,
-- Dave Keenan

Oh yeah! That was dumb of me. Here's what you want. Gets rid of those
9.8 c third errors by only tempering some remote fifths. Now no errors
anywhere are more than 5 c.

On 1/1 Cents Step On 3/2 Cents Difference
string string between strings
-------------------------------------------------------
0 (o) 0.00 53.3 0 0.00 0.00
1 53.30 53.5 1 53.30 0.00
m2 106.84 48.3 s2 101.96 4.89
m3 155.14 53.7 s3 150.61 4.53
4 208.80 58.1 4 203.91 4.89
m5 266.87 48.8 s5 262.14 4.73
6 315.64 70.7 6 315.64 0.00
7 386.31 53.5 7 386.31 0.00
m8 439.81 58.2 s8 435.08 4.73
9 498.04 53.3 9 493.16 4.89
m10 551.35 48.7 s10 546.82 4.53
m11 600.00 48.7 s11 595.11 4.89
12 648.65 53.3 12 648.65 0.00
13 701.96 58.2 13 (o) 701.96 0.00
m14 760.19 53.5 s14 755.26 4.93
m15 813.69 70.7 s15 808.80 4.89
m16 884.36 48.8 s16 857.09 27.26
m17 933.13 58.1 s17 910.75 22.38
m18 991.20 53.7 s18 968.83 22.38
m19 1044.86 48.3 s19 1017.60 27.26
m20 1093.16 53.5 s20 1088.27 4.89
m21 1146.70 53.3 s21 1141.77 4.93

Dave Keenan wrote,

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

Wait a minute Dave -- that looks wrong -- wouldn't that actually be:

m16========7========s20
/|\ /|\ / \
/ | \ / | \ / \
/ m5--------s18 \ / \
/,'m19`.\ /,' 10`.\ / 1 \ s14
9=========0=========13========4
s8 \ 21 / \ 12.'/ \`.s3 ,'/
\ / \ m17-/---\-s8 /
\ / \ | / \ | /
\ / \|/ \|/
m15========6========s19

so that one could play the 11-limit hexad as
0-13-7-s18-4-s10 instead of 0-13-7-s18-4-m10:

0 1/1 0
13 3/2 0
7 1/1 29
s18 3/2 18
4 1/1 26
s10 3/2 23

which might be possible to play completely!

--- In tuning@egroups.com, "Dave Keenan" <D.KEENAN@U...> wrote:
Oops!

On the previous tuning I failed to show that there are now separate m
and s varieties for notes 4 and 9. This may affect playability of
chords you've previously OKed.

Paul,

Can you tell me whether making all thirds within 5 c is worth this
price, before we look at where the ratios of 11 are?

Regards,
-- Dave Keenan

I'm really gone today. I wrote:

>0 1/1 0
>13 3/2 0
>7 1/1 7
>s18 3/2 5
>4 1/1 4
>s10 3/2 10

>Ahh . . . should be playable, especially with all fretted notes an octave
higher.

Mistake. Should be:

0 1/1 0
13 3/2 0
7 1/1 29
s18 3/2 27
4 1/1 26
s10 3/2 19

I think I can play that!

I wrote,

>0 1/1 0
>13 3/2 0
>7 1/1 29
>s18 3/2 18
>4 1/1 26
>s10 3/2 23

Oops . . . that's all wrong . . . should be:

0 1/1 0
13 3/2 0
7 1/1 7
s18 3/2 5
4 1/1 4
s10 3/2 10

Ahh . . . should be playable, especially with all fretted notes an octave
higher.

And this one was wrong too:

>0 1/1 0
>13 1/1 13
>7 3/2 15
>s18 3/2 18
>4 3/2 13
>m10 1/1 10

Should be:

0 1/1 0
13 3/2 0
7 1/1 7
s18 3/2 5
4 3/2 13
m10 1/1 10

This would involve a serious stretch. So I think s10 is a bit more important
than m10 as a fingering for 11/8.

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:

http://www.egroups.com/message/tuning/16910

> Well, rather than "breaking" the chains, I was thinking of applying
an unequal tempering so that the proximate intervals would be closer
to just, and those around the edges farthest from just (but still
usable). Kind of like a well-temperament, but with an extra dimension.

Isn't this reminiscent of a tuning that Paul Erlich was talking about
several months ago... and it was associated with a little "riddle"
that Paul gave the group and which nobody solved (??)

Am I just totally off base here and dreaming, or was this really
somthing that happened??

_________ _______ _____ ___
Joseph Pehrson

I wrote:
>
> 0 1/1 0
> 13 3/2 0
> 7 1/1 29
> s18 3/2 27
> 4 1/1 26
> s10 3/2 19
>
> I think I can play that!

I tried it on my 22-tET guitar, and couldn't play it.

I wrote,
>
> 0 1/1 0
> 13 3/2 0
> 7 1/1 7
> s18 3/2 5
> 4 3/2 13
> m10 1/1 10
>
> This would involve a serious stretch. So I think s10 is a bit more important
> than m10 as a fingering for 11/8.

On my 22-tET guitar, I _could_ play this fingering. So I now think m10 is more important
than s10 as a fingering for 11/8.

--- In tuning@egroups.com, "Joseph Pehrson" <pehrson@p...> wrote:
> --- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:
>
> http://www.egroups.com/message/tuning/16910
>
> > Well, rather than "breaking" the chains, I was thinking of applying
> an unequal tempering so that the proximate intervals would be closer
> to just, and those around the edges farthest from just (but still
> usable). Kind of like a well-temperament, but with an extra dimension.
>
> Isn't this reminiscent of a tuning that Paul Erlich was talking about
> several months ago... and it was associated with a little "riddle"
> that Paul gave the group and which nobody solved (??)
>
> Am I just totally off base here and dreaming, or was this really
> somthing that happened??
>
Sorry, Joseph, that was something completely different -- they were real 12-tone
well-temperaments -- and the answer to the riddle was that all the well-temperaments I posted
had their widest fifths interrupted by a couple of less-wide fifths at the far end of the circle of fifths,
if I recall correctly.

In any case, I'll have more to say on the actual topic of this message shortly.

Hi Dave K. -- I don't have time to look at your messages from Jan. 1st right now, but one think I
did think of is that if fret 11 is at 600 cents and fret 7 is at 386 cents, then if you try to play a
perfect fifth s20 (on 7) - m11 (on 11), it will be 10 cents sharp -- too sharp. I'll have to look at
your messages later . . .

I wrote,

> Sorry, Joseph, that was something completely different -- they were
real 12-tone
> well-temperaments -- and the answer to the riddle was that all the
well-temperaments I posted
> had their widest fifths interrupted by a couple of less-wide fifths
at the far end of the circle of fifths,
> if I recall correctly.

I didn't recall that correctly. The pattern was that they has their
narrowest fifths interrupted by some slightly less-narrow fifths at
the near end of the circle of fifths. As an idealized example, I'll
propose the following well-temperament:

Ab-Eb: wide 1/8 Pyth. comma
Eb-Bb: wide 1/8 Pyth. comma
Bb-F: narrow 1/4 Pyth. comma
F-C: narrow 1/6 Pyth. comma
C-G: narrow 1/6 Pyth. comma
G-D: narrow 1/6 Pyth. comma
D-A: narrow 1/6 Pyth. comma
A-E: narrow 1/6 Pyth. comma
E-B: narrow 1/6 Pyth. comma
B-F#: narrow 1/4 Pyth. comma
F#-C#: wide 1/8 Pyth. comma
C#-G#: wide 1/8 Pyth. comma

This gives the following major thirds:

Db-F: 410.5083
Ab-C: 404.2356
Eb-G: 397.9630
Bb-D: 391.6903
F-A: 393.4825
C-E: 393.4825
G-B: 393.4825
D-F#: 391.6903
A-C#: 397.9630
E-G#: 404.2356
B-D#: 410.5083
F#-A#: 418.5731

This could work well for a lot of Mozart and early Bach where the key
signature mainly stays within 2 sharps to 2 flats but G# gets used as
a leading tone in A minor while Ab gets some play as the flattened
sixth in C major . . .