Re: A 17-ish tuning

Hello, there, everyone, and in view of some recent discussions, here
is a somewhat "17-ish" scale for people to consider, with special
salutations to Robert Walker.

There are some ratios of 17, although the definition of "17-ness" may
rest largely in the eye of the beholder, or the ears of the listener.

This 24-note tuning is designed for two 12-note manuals, with
alternating notes placed on the two manuals. I give a Scala file, then
Scala's output showing the value of each ratio in cents.

There is also a tempered version of this scale which I have devised,
and will post after giving people a chance to consider this rational
version.

Here I might comment that one feature of a tuning proposed by
Kirnberger in 1766 plays a role by analogy in this scale, and also
that on looking through the Scala archives, I found that Manuel Op de
Coul had designed a scale with a certain conceptual affinity on some
level which actually shares one ratio in common placed in a
corresponding position.

My special thanks go also to Jacky Ligon and Dave Keenan, who in
sharing their often very different philosophies concerning rational
and/or just intonation systems, and providing most illustrative
examples, have both played a vital role in this creative process.

Here I present first a Scala file of the tuning showing only the
integer ratios, and suitable for import into the free Scala program
created by Manuel Op de Coul; and then output from Scala showing the
value of these ratios in cents.

--------------- Scala file starts on next line of text -------------

! ri17isha.scl
!
Rational intonation (RI) scale with some "17-ish" features (24 notes)
24
!
32/31
243/224
243/217
26/23
7/6
20/17
17/14
23/18
368/279
11160261/8388608
11160261/8126464
13/9
416/279
16777216/11160261
26040609/16777216
31/19
32/19
17/10
561/320
23/13
736/403
48/25
119/60
2/1

------------- Scala output showing value of ratios in cents ----------

|
Rational intonation (RI) scale with some "17-ish" features (24 notes)
0: 1/1 0.000000 unison, perfect prime
1: 32/31 54.96445 Greek enharmonic 1/4-tone
2: 243/224 140.9491
3: 243/217 195.9136
4: 26/23 212.2534
5: 7/6 266.8710 septimal minor third
6: 20/17 281.3584
7: 17/14 336.1296 supraminor third
8: 23/18 424.3645
9: 368/279 479.3289
10: 11160261/8388608 494.2411
11: 11160261/8126464 549.2055
12: 13/9 636.6179
13: 416/279 691.5823
14: 16777216/11160261 705.7594
15: 26040609/16777216 761.1121
16: 31/19 847.5230
17: 32/19 902.4874 19th subharmonic
18: 17/10 918.6421
19: 561/320 971.9151
20: 23/13 987.7471
21: 736/403 1042.711
22: 48/25 1129.328 classic diminished octave
23: 119/60 1185.513
24: 2/1 1200.000 octave

Most respectfully,

Margo Schulter
mschulter@value.net

Margo
can you explain the "manuals"
i am not familiar so much with the Kirnberger scale as i have been hard @
work with the Pagano/Beardsley 17 limit scale.
can you post a brief history of the scale?

Pat Pagano, Director
South East Just Intonation Society
http://indians.australians.com/meherbaba/
http://www.screwmusicforever.com/SHREESWIFT/
----- Original Message -----
From: M. Schulter <MSCHULTER@VALUE.NET>
To: <tuning@yahoogroups.com>
Sent: Thursday, March 15, 2001 11:49 PM
Subject: [tuning] Re: A 17-ish tuning

> Hello, there, everyone, and in view of some recent discussions, here
> is a somewhat "17-ish" scale for people to consider, with special
> salutations to Robert Walker.
>
> There are some ratios of 17, although the definition of "17-ness" may
> rest largely in the eye of the beholder, or the ears of the listener.
>
> This 24-note tuning is designed for two 12-note manuals, with
> alternating notes placed on the two manuals. I give a Scala file, then
> Scala's output showing the value of each ratio in cents.
>
> There is also a tempered version of this scale which I have devised,
> and will post after giving people a chance to consider this rational
> version.
>
> Here I might comment that one feature of a tuning proposed by
> Kirnberger in 1766 plays a role by analogy in this scale, and also
> that on looking through the Scala archives, I found that Manuel Op de
> Coul had designed a scale with a certain conceptual affinity on some
> level which actually shares one ratio in common placed in a
> corresponding position.
>
> My special thanks go also to Jacky Ligon and Dave Keenan, who in
> sharing their often very different philosophies concerning rational
> and/or just intonation systems, and providing most illustrative
> examples, have both played a vital role in this creative process.
>
> Here I present first a Scala file of the tuning showing only the
> integer ratios, and suitable for import into the free Scala program
> created by Manuel Op de Coul; and then output from Scala showing the
> value of these ratios in cents.
>
>
> --------------- Scala file starts on next line of text -------------
>
> ! ri17isha.scl
> !
> Rational intonation (RI) scale with some "17-ish" features (24 notes)
> 24
> !
> 32/31
> 243/224
> 243/217
> 26/23
> 7/6
> 20/17
> 17/14
> 23/18
> 368/279
> 11160261/8388608
> 11160261/8126464
> 13/9
> 416/279
> 16777216/11160261
> 26040609/16777216
> 31/19
> 32/19
> 17/10
> 561/320
> 23/13
> 736/403
> 48/25
> 119/60
> 2/1
>
>
> ------------- Scala output showing value of ratios in cents ----------
>
> |
> Rational intonation (RI) scale with some "17-ish" features (24 notes)
> 0: 1/1 0.000000 unison, perfect prime
> 1: 32/31 54.96445 Greek enharmonic 1/4-tone
> 2: 243/224 140.9491
> 3: 243/217 195.9136
> 4: 26/23 212.2534
> 5: 7/6 266.8710 septimal minor third
> 6: 20/17 281.3584
> 7: 17/14 336.1296 supraminor third
> 8: 23/18 424.3645
> 9: 368/279 479.3289
> 10: 11160261/8388608 494.2411
> 11: 11160261/8126464 549.2055
> 12: 13/9 636.6179
> 13: 416/279 691.5823
> 14: 16777216/11160261 705.7594
> 15: 26040609/16777216 761.1121
> 16: 31/19 847.5230
> 17: 32/19 902.4874 19th subharmonic
> 18: 17/10 918.6421
> 19: 561/320 971.9151
> 20: 23/13 987.7471
> 21: 736/403 1042.711
> 22: 48/25 1129.328 classic diminished octave
> 23: 119/60 1185.513
> 24: 2/1 1200.000 octave
>
>
> Most respectfully,
>
> Margo Schulter
> mschulter@value.net
>
>
>
>
>
>
>
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--- In tuning@y..., "M. Schulter" <MSCHULTER@V...> wrote:
> Hello, there, everyone, and in view of some recent discussions, here
> is a somewhat "17-ish" scale for people to consider, with special
> salutations to Robert Walker.
>
> There are some ratios of 17, although the definition of "17-ness"
may
> rest largely in the eye of the beholder, or the ears of the
listener.
>
> This 24-note tuning is designed for two 12-note manuals, with
> alternating notes placed on the two manuals. I give a Scala file,
then
> Scala's output showing the value of each ratio in cents.
>
> There is also a tempered version of this scale which I have devised,
> and will post after giving people a chance to consider this rational
> version.
>
> Here I might comment that one feature of a tuning proposed by
> Kirnberger in 1766 plays a role by analogy in this scale, and also
> that on looking through the Scala archives, I found that Manuel Op
de
> Coul had designed a scale with a certain conceptual affinity on some
> level which actually shares one ratio in common placed in a
> corresponding position.
>
> My special thanks go also to Jacky Ligon and Dave Keenan, who in
> sharing their often very different philosophies concerning rational
> and/or just intonation systems, and providing most illustrative
> examples, have both played a vital role in this creative process.
>
> Here I present first a Scala file of the tuning showing only the
> integer ratios, and suitable for import into the free Scala program
> created by Manuel Op de Coul; and then output from Scala showing the
> value of these ratios in cents.
>

Margo,

Hello!

I've did a little deconstruction of your scale to view the inner
symmetries, and it is very compelling to look inside!

Let's look at the rounded cents values for the consecutive intervals:

Degree Ratio Rounded Consecutive
0: 1/1 0
1: 32/31 55
2: 243/224 86
3: 243/217 55
4: 26/23 16
5: 7/6 55
6: 20/17 14
7: 17/14 55
8: 23/18 88
9: 368/279 55
10: 11160261/8388608 15
11: 11160261/8126464 55
12: 13/9 87
13: 416/279 55
14: 16777216/11160261 14
15: 26040609/16777216 55
16: 31/19 86
17: 32/19 55
18: 17/10 16
19: 561/320 53
20: 23/13 16
21: 736/403 55
22: 48/25 87
23: 119/60 56
24: 2/1 14

And by considering the consecutive intervals (a somewhat melodic
consideration for me) we find:

1. Seven @ and average of 15 cents.
2. Twelve @ an average of 55 cents.
3. Five @ an average of 87 cents.

Now, I'm supposing that the odd scale degrees are for the lower
manual, which has the following structure:

Degree Ratio Cents Rounded Consecutive
0: 1/1 0
1: 32/31 54.96445 55
3: 243/217 195.9136 141
5: 7/6 266.871 71
7: 17/14 336.1296 69
9: 368/279 479.3289 143
11: 11160261/8126464 549.2055 70
13: 416/279 691.5823 142
15: 26040609/16777216 761.1121 70
17: 32/19 902.4874 141
19: 561/320 971.9151 69
21: 736/403 1042.711 71
23: 119/60 1185.513 143
24: 2/1 1200 14

And the even degrees giving:

Degree Ratio Cents Rounded Consecutive
1/1 0 0
2: 243/224 140.9491 141
4: 26/23 212.2534 71
6: 20/17 281.3584 69
8: 23/18 424.3645 143
10: 11160261/8388608 494.2411 70
12: 13/9 636.6179 142
14: 16777216/11160261 705.7594 69
16: 31/19 847.523 142
18: 17/10 918.6421 71
20: 23/13 987.7471 69
22: 48/25 1129.328 142
24: 2/1 1200 71

A "near MOS" for this manual.

Looking at the scale broken this way onto the two manuals, reveals
other interesting properties as we can see above:

1. One @ 14 cents.
2. Thirteen @ an average of 70 cents
3. Ten @ an average of 142 cents.

With this exploded view of the consecutive intervals, we can see how
the 15 cents commas, play into the real-time "adaptive" ability of
this scale for two manuals.

32/31 is it's most common interval, and has the remarkable property
of having 29 fifths @ an average of 702.293 - and is a constant
structure (I favor constant structures too).

A lovely structure Margo, and thanks for sharing. Please let us know
when the tempered version is available - I'll be eargerly awaiting
this.

In gratitude,

Jacky Ligon

P.S. Please let me know if I've missed anything special here.

> ------------- Scala output showing value of ratios in cents --------
--
>
> |
> Rational intonation (RI) scale with some "17-ish" features (24
notes)
> 0: 1/1 0.000000 unison, perfect prime
> 1: 32/31 54.96445 Greek enharmonic 1/4-tone
> 2: 243/224 140.9491
> 3: 243/217 195.9136
> 4: 26/23 212.2534
> 5: 7/6 266.8710 septimal minor third
> 6: 20/17 281.3584
> 7: 17/14 336.1296 supraminor third
> 8: 23/18 424.3645
> 9: 368/279 479.3289
> 10: 11160261/8388608 494.2411
> 11: 11160261/8126464 549.2055
> 12: 13/9 636.6179
> 13: 416/279 691.5823
> 14: 16777216/11160261 705.7594
> 15: 26040609/16777216 761.1121
> 16: 31/19 847.5230
> 17: 32/19 902.4874 19th subharmonic
> 18: 17/10 918.6421
> 19: 561/320 971.9151
> 20: 23/13 987.7471
> 21: 736/403 1042.711
> 22: 48/25 1129.328 classic diminished octave
> 23: 119/60 1185.513
> 24: 2/1 1200.000 octave
>
>
> Most respectfully,
>
> Margo Schulter
> mschulter@v...