29-tone for maqams (was: Danny's maqams in pi ratio scales)

Ozan Yarman wrote:

> Can we work on 29tET and make it an adaptive tuning so as to preserve the
> thirds? This way, we may reach an optimal system where all maqams can be
> expressed in any key with as little error as possible. Perhaps your
> trancendental numbers may help us here.

This is 29-tone symmetrical Pythagorean with equivalent JI ratios (7-limit):

0: 1/1
1: 81/80 or 64/63
2: 256/243 or 21/20
3: 16/15 or 15/14
4: 10/9
5: 9/8
6: 8/7
7: 32/27
8: 6/5
9: 5/4
10: 81/64 or 80/63
11: 21/16
12: 4/3
13: 27/20
14: 45/32 or 7/5
15: 64/45 or 10/7
16: 40/27
17: 3/2
18: 32/21
19: 128/81 or 63/40
20: 8/5
21: 5/3
22: 27/16
23: 7/4
24: 16/9
25: 9/5
26: 15/8 or 28/15
27: 243/128 or 40/21
28: 160/81 or 63/32
29: 2/1

So what we want is a temperament that fairly approximates all these intervals. I'm thinking some sort of well-temperament. I need to familiarize myself with the Pythagorean By Sliders function in Scala.

Now if you want 11-limit, you'll need 41-tone; for 13-limit, 53-tone.

Danny, let us not exceed 29 tones per octave and implement such an adaptive
tuning that will allow up to 13-limit intervals, especially 13/12 and 14/13.

Cordially,
Ozan

----- Original Message -----
From: "Danny Wier" <dawiertx@sbcglobal.net>
To: <tuning-math@yahoogroups.com>
Sent: 27 Nisan 2005 �ar�amba 12:55
Subject: [tuning-math] 29-tone for maqams (was: Danny's maqams in pi ratio
scales)

>
> Ozan Yarman wrote:
>
> > Can we work on 29tET and make it an adaptive tuning so as to preserve
the
> > thirds? This way, we may reach an optimal system where all maqams can be
> > expressed in any key with as little error as possible. Perhaps your
> > trancendental numbers may help us here.
>
> This is 29-tone symmetrical Pythagorean with equivalent JI ratios
(7-limit):
>
> 0: 1/1
> 1: 81/80 or 64/63
> 2: 256/243 or 21/20
> 3: 16/15 or 15/14
> 4: 10/9
> 5: 9/8
> 6: 8/7
> 7: 32/27
> 8: 6/5
> 9: 5/4
> 10: 81/64 or 80/63
> 11: 21/16
> 12: 4/3
> 13: 27/20
> 14: 45/32 or 7/5
> 15: 64/45 or 10/7
> 16: 40/27
> 17: 3/2
> 18: 32/21
> 19: 128/81 or 63/40
> 20: 8/5
> 21: 5/3
> 22: 27/16
> 23: 7/4
> 24: 16/9
> 25: 9/5
> 26: 15/8 or 28/15
> 27: 243/128 or 40/21
> 28: 160/81 or 63/32
> 29: 2/1
>
> So what we want is a temperament that fairly approximates all these
> intervals. I'm thinking some sort of well-temperament. I need to
familiarize
> myself with the Pythagorean By Sliders function in Scala.
>
> Now if you want 11-limit, you'll need 41-tone; for 13-limit, 53-tone.
>
>

On 4/27/05, Danny Wier <dawiertx@sbcglobal.net> wrote:

> So what we want is a temperament that fairly approximates all these
> intervals. I'm thinking some sort of well-temperament. I need to familiarize
> myself with the Pythagorean By Sliders function in Scala.
>
> Now if you want 11-limit, you'll need 41-tone; for 13-limit, 53-tone.

Have you looked at 58? I don't know how 29 came up, because when I
was at school maqams always had neutral thirds. If you add neutral
thirds to 29, you end up with 58. It's also a nifty 15-limit equal
temperament.

The next stage is mystery. You start with 29-equal, and divide each
step unequally. That gives a "mystery comma" which is a little less
then one step of 58-equal. The rules for converting from 29-equal to
mystery are simple:

1) For each factor of a prime above 3 in the numerator, sharpen by a
mystery comma

2) For each factor of a prime above 3 in the denominator, flatten by a
mystery comma

Or maybe vice-versa, but I think that's it. You can either set the
mystery comma to give pure 5:4s, or compromise it for the 15-limit.

I haven't used it in anger, but it looks like it could work like
Vicentino's second archicembalo tuning for adaptive temperament. A
complete 15-limit chord can be played over 58 notes. That is, you can
set up a 58 note keyboard and every chord in your 15-limit piece can
be played on it, with some comma shifts but no overall pitch drift.
It's also unique in the 15-limit, and so works as a first step towards
adaptive JI.

Graham

Dear Graham,

I apologize for the late reply. I just had a moment with 58tET, and already feel that it's most reliable for expressing maqams. The fact that it is double the size of 29 makes me more comfortable, since by choosing 58, I can somewhat express every tone through half as much notes.

Cordially,
Ozan

----- Original Message -----
From: Graham Breed
To: tuning-math@yahoogroups.com
Sent: 01 Mayıs 2005 Pazar 12:49
Subject: Re: [tuning-math] 29-tone for maqams (was: Danny's maqams in pi ratio scales)

On 4/27/05, Danny Wier <dawiertx@sbcglobal.net> wrote:

> So what we want is a temperament that fairly approximates all these
> intervals. I'm thinking some sort of well-temperament. I need to familiarize
> myself with the Pythagorean By Sliders function in Scala.
>
> Now if you want 11-limit, you'll need 41-tone; for 13-limit, 53-tone.

Have you looked at 58? I don't know how 29 came up, because when I
was at school maqams always had neutral thirds. If you add neutral
thirds to 29, you end up with 58. It's also a nifty 15-limit equal
temperament.

The next stage is mystery. You start with 29-equal, and divide each
step unequally. That gives a "mystery comma" which is a little less
then one step of 58-equal. The rules for converting from 29-equal to
mystery are simple:

1) For each factor of a prime above 3 in the numerator, sharpen by a
mystery comma

2) For each factor of a prime above 3 in the denominator, flatten by a
mystery comma

Or maybe vice-versa, but I think that's it. You can either set the
mystery comma to give pure 5:4s, or compromise it for the 15-limit.

I haven't used it in anger, but it looks like it could work like
Vicentino's second archicembalo tuning for adaptive temperament. A
complete 15-limit chord can be played over 58 notes. That is, you can
set up a 58 note keyboard and every chord in your 15-limit piece can
be played on it, with some comma shifts but no overall pitch drift.
It's also unique in the 15-limit, and so works as a first step towards
adaptive JI.

Graham