Yahoo Tuning Groups Ultimate Backup tuning A circulating good temperament from N equal divisions

This is the result of trying to construct a chromatic scale which can
be used to play in every key without the thirds or fifths being
noticeably out of tune, but where every key sounds different, using
intervals taken from the smallest possible number of equal divisions.
Well, of course it is possible if you take 300 equal divisions of 4
cents each, then you can produce Vallotti and Barnes-Bach and
anything else using only (multiples of) 1/6 Pythagorean comma
temperament. But is it possible to do it without so many tiny
divisions? The divisions into 53 and 55 equal parts are well known,
but produce temperaments with one or more 'wolf' intervals when 12
pitches are selected to form the scale.

The best I could come up with is 150 equal divisions (8 cents) where
there are two sizes of fifth, 87 and 88 divisions. Then the circle of
fifths is made up of six narrow fifths (-) and six wide ones (+).
There are various ways of distributing them, here is one:

C (-) G (+) D (-) A (-) E (-) B (+) F# (+) C# (+) G# (-) Eb (+) Bb
(+) F (-) C

which gives the following numbers:

C C#/Db D D#/Eb E F F# G G#/Ab A Bb B C
0 12 25 37 49 63 74 87 100 112 125 136 150
0 96 200 296 392 504 592 696 800 896 1000 1088 1200

For comparison here are the just intervals in cents:

0 71/112 204 275/316 386 498 590 702 773/814 884 996 1088

The semitones are of size

C 12, 13 D 12, 12 E 14 F 11, 13 G 13, 12 A 13, 11 B 14 C

So this is fairly near equal temperament, but it has a noticeable
difference in the quality of thirds and fifths. The widest major 3rds
are 208 cents (i.e. Pythagorean), the narrowest are 192.

Still, 150 is a lot of equal divisions... I wonder if there is some
clever choice of number which will do the job with fewer while
avoiding the 'wolf' - which I would define as any major 3rd wider
than 208c.

~~~Thomas~~~

🔗Ozan Yarman <ozanyarman@superonline.com>

How about 29tET? Does this accurately reflect current Western 12-tone practice with non-fretted instruments?

0: 0.000 cents 0.000 0 0 commas C
17: 703.448 cents 1.493 46 G
5: 703.448 cents 2.987 92 D
22: 703.448 cents 4.480 137 5/24 synt. commas A
10: 703.448 cents 5.973 183 5/18 synt. commas E
27: 703.448 cents 7.466 229 B
15: 703.448 cents 8.960 275 5/12 synt. commas F#
3: 703.448 cents 10.453 321 C#
20: 703.448 cents 11.946 367 G#
8: 703.448 cents 13.439 412 D#
25: 703.448 cents 14.933 458 A#
13: 703.448 cents 16.426 504 F/
1: 703.448 cents 17.919 550 C/
18: 703.448 cents 19.413 596 G/
6: 703.448 cents 20.906 642 D/
23: 703.448 cents 22.399 687 A/
11: 703.448 cents 23.892 733 F\
28: 703.448 cents 25.386 779 C\
16: 703.448 cents 26.879 825 G\
4: 703.448 cents 28.372 871 D\
21: 703.448 cents 29.865 917 A\
9: 703.448 cents 31.359 962 E\
26: 703.448 cents 32.852 1008 B\
14: 703.448 cents 34.345 1054 Gb
2: 703.448 cents 35.839 1100 Db
19: 703.448 cents 37.332 1146 Ab
7: 703.448 cents 38.825 1192 Eb
24: 703.448 cents 40.318 1237 Bb
12: 703.448 cents 41.812 1283 F
29: 703.448 cents 43.305 1329 C
Average absolute difference: 22.3991 cents
Root mean square difference: 26.1018 cents
Maximum absolute difference: 43.3050 cents
Maximum formal fifth difference: 1.4933 cents

----- Original Message -----
From: Tom Dent
To: tuning@yahoogroups.com
Sent: 21 Nisan 2005 Perşembe 2:51
Subject: [tuning] A circulating good temperament from N equal divisions

C (-) G (+) D (-) A (-) E (-) B (+) F# (+) C# (+) G# (-) Eb (+) Bb
(+) F (-) C

which gives the following numbers:

C C#/Db D D#/Eb E F F# G G#/Ab A Bb B C
0 12 25 37 49 63 74 87 100 112 125 136 150
0 96 200 296 392 504 592 696 800 896 1000 1088 1200

For comparison here are the just intervals in cents:

0 71/112 204 275/316 386 498 590 702 773/814 884 996 1088

The semitones are of size

C 12, 13 D 12, 12 E 14 F 11, 13 G 13, 12 A 13, 11 B 14 C

So this is fairly near equal temperament, but it has a noticeable
difference in the quality of thirds and fifths. The widest major 3rds
are 208 cents (i.e. Pythagorean), the narrowest are 192.

~~~Thomas~~~

🔗Carl Lumma <ekin@lumma.org>

>This is the result of trying to construct a chromatic scale which
>can be used to play in every key without the thirds or fifths being
>noticeably out of tune, but where every key sounds different, using
>intervals taken from the smallest possible number of equal divisions.
>Well, of course it is possible if you take 300 equal divisions of 4
>cents each, then you can produce Vallotti and Barnes-Bach and
>anything else using only (multiples of) 1/6 Pythagorean comma
>temperament. But is it possible to do it without so many tiny
>divisions? The divisions into 53 and 55 equal parts are well known,
>but produce temperaments with one or more 'wolf' intervals when 12
>pitches are selected to form the scale.
>
>The best I could come up with is 150 equal divisions (8 cents) where
>there are two sizes of fifth, 87 and 88 divisions. Then the circle of
>fifths is made up of six narrow fifths (-) and six wide ones (+).
>There are various ways of distributing them, here is one:
>
>C (-) G (+) D (-) A (-) E (-) B (+) F# (+) C# (+) G# (-) Eb (+) Bb
>(+) F (-) C
>
>which gives the following numbers:
>
>C C#/Db D D#/Eb E F F# G G#/Ab A Bb B C
>0 12 25 37 49 63 74 87 100 112 125 136 150
>0 96 200 296 392 504 592 696 800 896 1000 1088 1200
>
>For comparison here are the just intervals in cents:
>
>0 71/112 204 275/316 386 498 590 702 773/814 884 996 1088
>
>The semitones are of size
>
>C 12, 13 D 12, 12 E 14 F 11, 13 G 13, 12 A 13, 11 B 14 C
>
>So this is fairly near equal temperament, but it has a noticeable
>difference in the quality of thirds and fifths. The widest major
>3rds are 208 cents (i.e. Pythagorean), the narrowest are 192.

I think you mean 408 and 392.

>Still, 150 is a lot of equal divisions... I wonder if there is
>some clever choice of number which will do the job with fewer
>while avoiding the 'wolf' - which I would define as any major
>3rd wider than 208c.

I'm not clear why you're thinking in terms of equal divisions
here... you're planning on using this as a 12-tone tuning, right?

To my ear, Pythagorean major thirds are noticeably worse than
equal-tempered major thirds. So I tried to find tunings where
some thirds were noticeably better but none noticeably worse.
I did this by using octaves that are slightly flat. When an
octave stretch is desired, it can still be applied to the
prescribed flat octave.

In practice, the tunings I came up with have worked very well
on synthesizers of various kinds. A series of experiments with
my piano were promising but not overwhelmingly so... but the
stretch algorithm used was thrown together. . .

-Carl

🔗Tom Dent <tdent@auth.gr>

Although I don't quite understand your table (surely it would be
easier to put all the pitches within a single octave? what are the
numbers 1.493 etc.?) - this seems to be a reasonable approximation to
Pythagorean tuning, except that all intervals are slightly sharp. As
such, it is not really what any string players/singers would use. Now
53tET *is* a good approximation to exact Pythagorean tuning, which
would be useful for string instruments, although one would have to
put the syntonic comma somewhere when playing in any given key. 53-
tone also has a very good major 3rd. So if you had a 53-note keyboard
you could probably find the notes used by good string players and
singers.

~~~Thomas~~~

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
> How about 29tET? Does this accurately reflect current Western 12-
tone practice with non-fretted instruments?
>
> 0: 0.000 cents 0.000 0 0
commas C
> 17: 703.448 cents 1.493
46 G
> 5: 703.448 cents 2.987
92 D
> 22: 703.448 cents 4.480 137 5/24 synt. commas A
> 10: 703.448 cents 5.973 183 5/18 synt. commas E
> 27: 703.448 cents 7.466
229 B
> 15: 703.448 cents 8.960 275 5/12 synt. commas F#
> 3: 703.448 cents 10.453
321 C#
> 20: 703.448 cents 11.946
367 G#
> 8: 703.448 cents 13.439
412 D#
> 25: 703.448 cents 14.933
458 A#
> 13: 703.448 cents 16.426
504 F/
> 1: 703.448 cents 17.919
550 C/
> 18: 703.448 cents 19.413
596 G/
> 6: 703.448 cents 20.906
642 D/
> 23: 703.448 cents 22.399
687 A/
> 11: 703.448 cents 23.892
733 F\
> 28: 703.448 cents 25.386
779 C\
> 16: 703.448 cents 26.879
825 G\
> 4: 703.448 cents 28.372
871 D\
> 21: 703.448 cents 29.865
917 A\
> 9: 703.448 cents 31.359
962 E\
> 26: 703.448 cents 32.852
1008 B\
> 14: 703.448 cents 34.345
1054 Gb
> 2: 703.448 cents 35.839
1100 Db
> 19: 703.448 cents 37.332
1146 Ab
> 7: 703.448 cents 38.825
1192 Eb
> 24: 703.448 cents 40.318
1237 Bb
> 12: 703.448 cents 41.812
1283 F
> 29: 703.448 cents 43.305
1329 C
> Average absolute difference: 22.3991 cents
> Root mean square difference: 26.1018 cents
> Maximum absolute difference: 43.3050 cents
> Maximum formal fifth difference: 1.4933 cents
>

🔗Carl Lumma <ekin@lumma.org>

🔗pgreenhaw@nypl.org

🔗Ozan Yarman <ozanyarman@superonline.com>

Thomas, 1.493 cents are the deviation amount of the 29tET fifths from the pure fifth. The table is a copy-paste of Scala's cycle of fifths analysis.

I have been reminded that you want a system made up of two sizes of perfect fifths circulating through 12 tones. In this case, why not 90tET with two fifths with the sizes 693 and 703 cents each? The pure thirds are also quite perfect with only 0.353 cents error. The you may try the following out of 90tET:

Notes
C, C#/Db, D, D#/Eb, E, F, F#/Gb, A, A#/Bb, B, C

Steps
1, 7, 15, 22, 29, 37, 45, 53, 60, 68, 75, 82, 90

Also, 41tET seems as good as 53tET regarding major third approximations.

Cordially,
Ozan
----- Original Message -----
From: Tom Dent
To: tuning@yahoogroups.com
Sent: 21 Nisan 2005 Perşembe 22:12
Subject: [tuning] Re: 29tET

🔗Ozan Yarman <ozanyarman@superonline.com>

I always thought the tendency was due to the fact that the E has to be 81:64 away from C5 in order to be in perfect tune with the 440 Hz A string which produces it. If C4 is taken as 1/1, then the tuning of the violin strings are:

3/4 for G
9/8 for D
27/16 for A
81/32 for E

Obviously, it is impossible for the violinist to sound a 5/4 major third on C5, unless he pushes this tone a syntonic comma further with his finger, which would result in an intonation drift. To avoid this, the ear searches for the next best thing, which is the pythagorean third. I do so myself with the tanbur.

Cordially,
Ozan
----- Original Message -----
From: pgreenhaw@nypl.org
To: tuning@yahoogroups.com
Sent: 21 Nisan 2005 Perşembe 22:51
Subject: Re: [tuning] Re: 29tET

>>which would be useful for string instruments,

>How so? I've heard that strings somehow play in pythagorean
>tuning, but I can't imagine why or how that would be the case.

-Carl

I think the notion of string players playing in a more or less Pythagorean tuning comes out of a tendency to push leading tones (or better yet, the "3rds" of chords) more toward the sharp side. Its the strident pull toward the resolution that gives rise to the sharpening....

Paul

🔗Tom Dent <tdent@auth.gr>

Apologies for the typo in previous post... the cents values for major
3rd should indeed be 200 larger (392/408). Thanks for your comments

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >This is the result of trying to construct a chromatic scale which
> >can be used to play in every key without the thirds or fifths
being
> >noticeably out of tune, but where every key sounds different,
using
> >intervals taken from the smallest possible number of equal
divisions. (...)

>
> I'm not clear why you're thinking in terms of equal divisions
> here... you're planning on using this as a 12-tone tuning, right?

Just for curiosity - to try to find the smallest number of equal
divisions from which one can choose a circulating temperament. It is
probably impractical, but a nice mathematical problem.

> To my ear, Pythagorean major thirds are noticeably worse than
> equal-tempered major thirds.

I do want the difference in the 3rds to be *noticeable*. The question
is whether the Pythagorean 3rds are so bad that you cannot stand
them. This depends really on what piece you are playing. It probably
does not work very well for pieces in F#/Gb major with prominent
major 3rds... e.g. Schubert G flat impromptu. However if a piece goes
into F# major for a few bars it would be reasonable.

> So I tried to find tunings where
> some thirds were noticeably better but none noticeably worse.

Isn't that like a town where all of the children are above average?
Well, it is almost possible within a standard framework, I think
Neidhardt did it in the early 18th century by having most thirds very
slightly worse than ET, a few equal, and a few noticeably better.

> I did this by using octaves that are slightly flat. When an
> octave stretch is desired, it can still be applied to the
> prescribed flat octave.

What is an 'octave stretch' here and how does it feed through into
the 3rds? I've often wondered what happens when you don't have pure
octaves - it would be very difficult to apply this to an actual
stringed instrument or organ. (I know pianos don't have proper
octaves, but they try to make us believe that they do... i.e. the
octave beats are tuned out.)

> In practice, the tunings I came up with have worked very well
> on synthesizers of various kinds.

Any examples?

~~~Thomas~~~

🔗Tom Dent <tdent@auth.gr>

Thanks for the tip! Now 90tET has divisions of 13.333.. cents which is
a little over 1/2 Pythagorean comma. I find possible 5ths of size 52
and 53 which give 693.3 and 706.7 cents resp. - i.e. 8.6 cents below
and 4.7 cents above pure. The former is nearly -2/5 comma, the latter
is just over +1/5 comma, where the comma is 22.222.. cents. I think
2/5 comma tempered 5ths would be pretty rough, but I suppose one can
try and see. With respect to ET the fifths are just +- 6.666.. cents
and we have the sum

6x53 + 6x52 = 630.

The semitones in your proposed setup are of the sizes

C 7, 8 D 7, 7, E 8 F 8, 8 G 7, 8 A 7, 7, 8 C

which leads to a major 3rd E-Ab of size 8+8+8+7 = 31 divisions, i.e.
413.3 cents. So the 'bad' thirds in this setup are really quite bad,
although the basic scale in C major is good.

If instead we take

C 7, 8 D 7, 7, E 8 F 7, 8 G 7, 8 A 8, 7, 8 C

then we still have one 'bad' third G#-C... it seems difficult to avoid
wolf-like phenomena here. Still, it is good to know what the
possibilities are.

~~~Thomas~~~

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:

> I have been reminded that you want a system made up of two sizes of
perfect fifths circulating through 12 tones. In this case, why not
90tET with two fifths with the sizes 693 and 703 cents each? The pure
thirds are also quite perfect with only 0.353 cents error. The you may
try the following out of 90tET:
>
> Notes
> C, C#/Db, D, D#/Eb, E, F, F#, G, G#/Ab, A, A#/Bb, B, C
>
> Steps
> 0, 7, 15, 22, 29, 37, 45, 53, 60, 68, 75, 82, 90
>

🔗Ozan Yarman <ozanyarman@superonline.com>

Glad you found my humble suggestion useful dear Thomas. I see how horrible the 12-tone scale out of 90tET I proposed was, but it was too late once I sent it. I'm sure more knowledgeble theorists here will pluck out the most desirable tones so that the thirds and fifths will be in good tune.

Cordially,
Ozan
----- Original Message -----
From: Tom Dent
To: tuning@yahoogroups.com
Sent: 22 Nisan 2005 Cuma 2:46
Subject: [tuning] Re: 29tET

6x53 + 6x52 = 630.

The semitones in your proposed setup are of the sizes

C 7, 8 D 7, 7, E 8 F 8, 8 G 7, 8 A 7, 7, 8 C

which leads to a major 3rd E-Ab of size 8+8+8+7 = 31 divisions, i.e.
413.3 cents. So the 'bad' thirds in this setup are really quite bad,
although the basic scale in C major is good.

If instead we take

C 7, 8 D 7, 7, E 8 F 7, 8 G 7, 8 A 8, 7, 8 C

then we still have one 'bad' third G#-C... it seems difficult to avoid
wolf-like phenomena here. Still, it is good to know what the
possibilities are.

~~~Thomas~~~

🔗Carl Lumma <ekin@lumma.org>

>> I'm not clear why you're thinking in terms of equal divisions
>> here... you're planning on using this as a 12-tone tuning, right?
>
>Just for curiosity - to try to find the smallest number of equal
>divisions from which one can choose a circulating temperament. It is
>probably impractical, but a nice mathematical problem.

Cool.

>> To my ear, Pythagorean major thirds are noticeably worse than
>> equal-tempered major thirds.
>
>I do want the difference in the 3rds to be *noticeable*. The question
>is whether the Pythagorean 3rds are so bad that you cannot stand
>them.

For me, they basically are. I think they can work in medieval
music, though.

>> So I tried to find tunings where
>> some thirds were noticeably better but none noticeably worse.
>
>Isn't that like a town where all of the children are above average?
>Well, it is almost possible within a standard framework, I think
>Neidhardt did it in the early 18th century by having most thirds very
>slightly worse than ET, a few equal, and a few noticeably better.
>
>> I did this by using octaves that are slightly flat. When an
>> octave stretch is desired, it can still be applied to the
>> prescribed flat octave.
>
>What is an 'octave stretch' here

Piano tuners typically stretch octaves, probably because the
partials of piano strings are stretched.

>and how does it feed through into the 3rds? I've often wondered
>what happens when you don't have pure octaves

In an equal tuning with a flat octave, the major thirds will
benefit from 1/3 of the flatness. I just made circulating
versions of this.

>it would be very difficult to apply this to an actual stringed
>instrument or organ. (I know pianos don't have proper octaves,
>but they try to make us believe that they do... i.e. the
>octave beats are tuned out.)

It's difficult to tune in practice, for sure. List member and
master tuner Paul Bailey did the experiments in question on
my spinet with a Verituner (veritune.com).

>> In practice, the tunings I came up with have worked very well
>> on synthesizers of various kinds.
>
>Any examples?

Yes, I did some. I'll try to dig them up, but it might take
me about two months.

-Carl

🔗Carl Lumma <ekin@lumma.org>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Herman Miller <hmiller@IO.COM>

Tom Dent wrote:

> This is the result of trying to construct a chromatic scale which can > be used to play in every key without the thirds or fifths being > noticeably out of tune, but where every key sounds different, using > intervals taken from the smallest possible number of equal divisions. > Well, of course it is possible if you take 300 equal divisions of 4 > cents each, then you can produce Vallotti and Barnes-Bach and > anything else using only (multiples of) 1/6 Pythagorean comma > temperament. But is it possible to do it without so many tiny > divisions? The divisions into 53 and 55 equal parts are well known, > but produce temperaments with one or more 'wolf' intervals when 12 > pitches are selected to form the scale.

64-ET has some interesting properties, which I discovered by accident when playing around with retuned versions of Ravel's _Pavane_:

http://www.io.com/~hmiller/midi/pavane-64.mid

This retuned version uses only a 12-note subset of 64-ET, plus an extra manually-retuned 13th note in a couple of places (where G# and Ab were distinguished in the score).

Compare with the 19-ET version, which is a meantone tuning:

http://www.io.com/~hmiller/midi/pavane-19.mid

Anyway, the odd thing about 64-ET is that the third you get from the cycle of fifths is one step *smaller* than the best third of 64-ET (375 cents, compared with 393.75 cents). 64-ET also has two fairly reasonable sizes of fifths: 693.75 cents and 712.5 cents. So you might want to try different arrangements of the fifths in 64-ET (one possible arrangement is the "diminished" temperament, which repeats every 1/4 octave).

For other possibilities, try downloading Scala, create a 12-ET scale ("equal 12"), and try quantizing it to various divisions of the octave (e.g., "quantize 64"). Then type "show intervals" to examine the sizes of fifths and thirds.

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, "Tom Dent" <tdent@a...> wrote:

Here's a related and possibly relevant list, the list of meantones of
the form 12a+19b, with a and b positive, a<=19 and b<=12. Some
interesting divisions which support meantone are 103, 198, 202, 270,
311, and 342.

31, 43, 50, 55, 67, 69, 74, 79, 81, 88, 91, 98, 103, 105, 107, 112,
115, 117, 119, 122, 126, 127, 131, 136, 139, 141, 143, 145, 146, 151,
153, 157, 160, 163, 164, 167, 169, 170, 174, 175, 177, 179, 181, 183,
184, 187, 188, 189, 191, 193, 194, 195, 198, 199, 202, 203, 205, 208,
211, 212, 213, 218, 219, 221, 223, 225, 226, 227, 229, 231, 232, 233,
235, 236, 239, 240, 241, 242, 245, 246, 247, 249, 251, 253, 255, 256,
257, 260, 261, 263, 265, 266, 267, 269, 270, 274, 277, 280, 281, 284,
285, 287, 288, 289, 291, 293, 298, 299, 303, 304, 305, 308, 311, 312,
313, 317, 318, 322, 323, 325, 327, 329, 332, 337, 339, 342, 346, 349,
353, 356, 360, 361, 363, 365, 375, 377, 380, 384, 389, 394, 399, 401,
413, 418, 425, 432, 437, 456

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗monz <monz@tonalsoft.com>

hi Ozan and Thomas,

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:

> Also, 41tET seems as good as 53tET regarding major third
> approximations.

the point about 53-edo is that it gives an outstanding
approximation to 5-limit just intonation, and an even (much)
better approximation to pythagorean tuning.

now, as for the matter of string players using it ...
let's see the experimental test results, that's when
i'll be convinced.

my bet is that good orchestral players of all pitched
instruments learn how to play in some form of adaptive-JI
... and AFAIK 53-edo does not do adaptive-JI.

i think 152-edo is the smallest EDO which does that, as
multiple chains (8) of 19-edo tuned about 1/3-syntonic-comma
apart. and 217 contains the subset which closely resembles
Vicentino's proposal of 1555, being composed of 7 chains
of 31-edo tuned about 1/4-comma apart.

-monz

🔗monz <monz@tonalsoft.com>

hi Herman and Tom,

--- In tuning@yahoogroups.com,
Herman Miller <hmiller@I...> wrote:

> 64-ET has some interesting
> properties, which I discovered
> by accident when playing
> around with retuned versions
> of Ravel's _Pavane_:
>
> http://www.io.com/~hmiller/midi/pavane-64.mid
>
> This retuned version uses
> only a 12-note subset of 64-ET,
> plus an extra manually-retuned
> 13th note in a couple of places
> (where G# and Ab were
> distinguished in the score).
>
> Compare with the 19-ET version, which is a meantone tuning:
>
> http://www.io.com/~hmiller/midi/pavane-19.mid
>
> Anyway, the odd thing about 64-ET is that the third you
> get from the cycle of fifths is one step *smaller* than
> the best third of 64-ET (375 cents, compared with 393.75
> cents). 64-ET also has two fairly reasonable sizes of
> fifths: 693.75 cents and 712.5 cents. So you might want to
> try different arrangements of the fifths in 64-ET (one
> possible arrangement is the "diminished" temperament,
> which repeats every 1/4 octave).

yes, in fact my favorite version of all the many retunings
you did of the _Pavane_, was the one in 64.

my next favorites are 40 and then 27.

-monz

🔗Tom Dent <tdent@auth.gr>

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> --- In tuning@yahoogroups.com, "Tom Dent" <tdent@a...> wrote:
>
> > But roughly, the basic key-
> > relationships allowed by string instruments are pure 5ths.
>
> String players generally avoid open strings, so this really isn't
such
> a big deal.

It's true that players can avoid (near-)Pythagorean tuning if they
want to, but it tends to be the default for various reasons.

1. Many composers require open strings to be used in multiple-
stopping.

2. Open strings are played in rapid or virtuosic passages especially
in arpeggios.

3. Although the open strings are not played in melodic passages, they
create resonances which allow the player to tell if the note being
played is in tune with the open string.

4. Players tend to use the same fingerings on each string, which
creates pitches a pure 5th apart.

5. Usual teaching is to play leading-notes somewhat sharp.

Now back in the Classical era of extended 1/6-comma meantone, things
were quite different and I am not sure that Mozart would not have
tuned his violin strings by 1/6 comma tempered 5ths.

Has anyone synthesized Mozart and Haydn in extended 1/6-comma
meantone?

~~~T~~~

🔗Tom Dent <tdent@auth.gr>

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
(...)
> the point about 53-edo is that it gives an outstanding
> approximation to 5-limit just intonation, and an even (much)
> better approximation to pythagorean tuning.
>
> now, as for the matter of string players using it ...
> let's see the experimental test results, that's when
> i'll be convinced.
>
> my bet is that good orchestral players of all pitched
> instruments learn how to play in some form of adaptive-JI
> ... and AFAIK 53-edo does not do adaptive-JI.
>
> i think 152-edo is the smallest EDO which does that, as
> multiple chains (8) of 19-edo tuned about 1/3-syntonic-comma
> apart. and 217 contains the subset which closely resembles
> Vicentino's proposal of 1555, being composed of 7 chains
> of 31-edo tuned about 1/4-comma apart.

Where would 53edo fail if being used for adaptive JI? Since it has
virtually pure 5ths and 3rds, any chord in standard classical
repertoire can be played in tune. (Well, apart from chords containing
a syntonic comma e.g. C-D-A.) Is there more to it than that?

~~~T~~~

🔗Carl Lumma <ekin@lumma.org>

>> the point about 53-edo is that it gives an outstanding
>> approximation to 5-limit just intonation, and an even (much)
>> better approximation to pythagorean tuning.
>>
>> now, as for the matter of string players using it ...
>> let's see the experimental test results, that's when
>> i'll be convinced.
>>
>> my bet is that good orchestral players of all pitched
>> instruments learn how to play in some form of adaptive-JI
>> ... and AFAIK 53-edo does not do adaptive-JI.
>>
>> i think 152-edo is the smallest EDO which does that, as
>> multiple chains (8) of 19-edo tuned about 1/3-syntonic-comma
>> apart. and 217 contains the subset which closely resembles
>> Vicentino's proposal of 1555, being composed of 7 chains
>> of 31-edo tuned about 1/4-comma apart.
>
>Where would 53edo fail if being used for adaptive JI? Since it has
>virtually pure 5ths and 3rds, any chord in standard classical
>repertoire can be played in tune. (Well, apart from chords containing
>a syntonic comma e.g. C-D-A.) Is there more to it than that?

Hi Tom,

"Adaptive JI" can be viewed as tempering out a comma in melodic
space rather than harmonic space like traditional temperament.

The answer depends on what comma you want to temper out. Since
53-tET does not split the syntonic comma, it can't be used for
adaptive JI based on that comma.

-Carl

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗monz <monz@tonalsoft.com>

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning@yahoogroups.com, "Tom Dent" <tdent@a...> wrote:
>
>
> This is the result of trying to construct a chromatic scale which
can
> be used to play in every key without the thirds or fifths being
> noticeably out of tune, but where every key sounds different, using
> intervals taken from the smallest possible number of equal
divisions.
> Well, of course it is possible if you take 300 equal divisions of 4
> cents each, then you can produce Vallotti and Barnes-Bach and
> anything else using only (multiples of) 1/6 Pythagorean comma
> temperament. But is it possible to do it without so many tiny
> divisions? The divisions into 53 and 55 equal parts are well known,
> but produce temperaments with one or more 'wolf' intervals when 12
> pitches are selected to form the scale.
>
> The best I could come up with is 150 equal divisions (8 cents)
where
> there are two sizes of fifth, 87 and 88 divisions. Then the circle
of
> fifths is made up of six narrow fifths (-) and six wide ones (+).
> There are various ways of distributing them, here is one:
>
> C (-) G (+) D (-) A (-) E (-) B (+) F# (+) C# (+) G# (-) Eb (+) Bb
> (+) F (-) C
>
> which gives the following numbers:
>
> C C#/Db D D#/Eb E F F# G G#/Ab A Bb B C
> 0 12 25 37 49 63 74 87 100 112 125 136 150
> 0 96 200 296 392 504 592 696 800 896 1000 1088 1200
>
> For comparison here are the just intervals in cents:
>
> 0 71/112 204 275/316 386 498 590 702 773/814 884 996 1088
>
> The semitones are of size
>
> C 12, 13 D 12, 12 E 14 F 11, 13 G 13, 12 A 13, 11 B 14 C
>
> So this is fairly near equal temperament, but it has a noticeable
> difference in the quality of thirds and fifths. The widest major
3rds
> are 208 cents (i.e. Pythagorean), the narrowest are 192.
>
> Still, 150 is a lot of equal divisions... I wonder if there is some
> clever choice of number which will do the job with fewer while
> avoiding the 'wolf' - which I would define as any major 3rd wider
> than 208c.
>
> ~~~Thomas~~~

You would want the tones to be a subset of an ET with one fifth very
close to just and the next smallest fifth no narrower than 1/4-comma
less. Since the difference in size between the two fifths is 1/4-
comma, the single degree of that ET would have to be around 5 cents,
which amounts to >200 tones/octave. My choice is 224-ET: make the 4
fifths from C to E 130 degrees and the remaining 8 fifths 131 degrees
and you will get a very decent well-temperament.

--George

🔗Tom Dent <tdent@auth.gr>

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> > --- In tuning@yahoogroups.com,
> "Tom Dent" <tdent@a...> wrote:
> >
> > > Has anyone synthesized Mozart and Haydn in extended 1/6-comma
> > > meantone?
> > [Gene:]
> > Monz has. Why not pick a Mozart piece you would like someone to
> > take a crack at? I don't have any 55-et versions of anything
> > yet. Or you could explain why you want Haydn in 1/6-comma
> > particularly.
>
> http://tonalsoft.com/monzo/55edo/55edo.htm
>
> Gene, i'm impressed by the quality of the .ogg files you've
> made ... could you please make one of my Mozart MIDI?
> (and a better .mp3 too? ... the MIDI i have there now
> is a bit longer than the old mp3 i made.)
>
> -monz

So this G minor symphony snippet sounds OK, but as you said, within
the limitations of basic MIDI...

The thing with orchestral music is that (barring very sensitive ears)
I don't think one can be very precise with tuning, with 10 violins
playing together, trumpets and horns playing natural harmonics, and
all different types of wind instruments with their peculiar keying
systems... or at least, Mozart's orchestras were not very precise. So
possibly such music was not written specifically to exploit fine
details of tuning.

Now, what I would like to hear first in 55EDO is the E flat quartet
(K428 I think) which has a very chromatic first subject.

I think Gene has some Haydn and Mozart in a more 'unequal' meantone,
but I am having a bit of trouble with the .ogg files just now. (I do
have WinAmp lite installation but the files are not playing after
auto downloading.) Also I prefer my string quartets played by
strings ;) ... seriously, I think it is easier to appreciate tunings
if all voices have similar timbre.

Haydn in 1/6 comma (rather than any stronger temperament) simply
because Haydn and Mozart probably had similar training and Haydn's
piano music was probably intended for something near 12-ET so he may
not have expected heavy temperament.

A good Haydn piece to try would be the 'slow' movement of Op.54 no.1
in G (second movement Allegretto) which has interesting modulations.
Also the first movement of Op.77 no.2 in F which requires an
enharmonic equivalence.

Anyway, all in good time!

~~~T~~~

🔗monz <monz@tonalsoft.com>

hi Tom,

> The thing with orchestral music is that (barring
> very sensitive ears) I don't think one can be very
> precise with tuning, with 10 violins playing together,
> trumpets and horns playing natural harmonics, and
> all different types of wind instruments with their
> peculiar keying systems... or at least, Mozart's
> orchestras were not very precise. So possibly such
> music was not written specifically to exploit fine
> details of tuning.

sure, a real orchestral can have nowhere near the
precision of tuning that a computer provides.

but Johnny Reinhard's players can produce a reasonably
accurate approximation of 55-edo or 1/6-comma meantone,
and since the violin manual written by Mozart's father
stipulates that the whole-tone consists of 9 commas,
the diatonic semitone is 5 commas, and the chromatic
semitone is 4 commas, we must assume that the players
of that time were able to produce it as well.

and i'd prefer to assume the opposite of your last
sentence: Mozart specifically uses F# all over the
beginning of his G-minor Symphony (as one would expect),
but there's a point later in the movement where he
writes Gb, and since his own teaching specified that
these are two different pitches, i think we should
assume that his music *does* "specifically ... exploit
fine details of tuning".

> Haydn in 1/6 comma (rather than any stronger temperament)
> simply because Haydn and Mozart probably had similar
> training and Haydn's piano music was probably intended
> for something near 12-ET so he may not have expected
> heavy temperament.
>
> A good Haydn piece to try would be the 'slow' movement
> of Op.54 no.1 in G (second movement Allegretto) which
> has interesting modulations. Also the first movement
> of Op.77 no.2 in F which requires an enharmonic equivalence.

keep in mind that we're talking about orchestral playing
here ... during the 1700s and later, meantone would only
be expected for non-keyboard music. for keyboards, a
12-tone well-temperament was standard from c.1700-1850,
and 12-edo after that.

-monz

🔗Tom Dent <tdent@auth.gr>

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> hi Herman and Tom,
>
> --- In tuning@yahoogroups.com,
> Herman Miller <hmiller@I...> wrote:
>
>
> > 64-ET has some interesting
> > properties, which I discovered
> > by accident when playing
> > around with retuned versions
> > of Ravel's _Pavane_:
> >
> > http://www.io.com/~hmiller/midi/pavane-64.mid
> >
> > This retuned version uses
> > only a 12-note subset of 64-ET,
> > plus an extra manually-retuned
> > 13th note in a couple of places
> > (where G# and Ab were
> > distinguished in the score).

This sounded pretty good!

> > Compare with the 19-ET version, which is a meantone tuning:
> >
> > http://www.io.com/~hmiller/midi/pavane-19.mid
> >

This sounded pretty awful!

> > Anyway, the odd thing about 64-ET is that the third you
> > get from the cycle of fifths is one step *smaller* than
> > the best third of 64-ET (375 cents, compared with 393.75
> > cents). 64-ET also has two fairly reasonable sizes of
> > fifths: 693.75 cents and 712.5 cents. So you might want to
> > try different arrangements of the fifths in 64-ET (one
> > possible arrangement is the "diminished" temperament,
> > which repeats every 1/4 octave).

It's not so odd if you reflect that 64-ET has a flat 5th which is
nearly 1/3 comma, and 1/3 comma gives you major 6ths in tune and
flatter major 3rds.

I'm quite surprised that it sounded so good if the 5ths are
only 'fairly reasonable'. Perhaps the quality of 3rds is more
important in the Ravel. (Of course the piece does not have extensive
modulation.)

What is meant by 'inconsistent' for the 64-ET tuning?

~~~Thomas~~~

🔗monz <monz@tonalsoft.com>

hi Tom,

--- In tuning@yahoogroups.com,
"Tom Dent" <tdent@a...> wrote:

> What is meant by 'inconsistent'
> for the 64-ET tuning?

from my webpage:
http://tonalsoft.com/enc/consistent.htm

"An equal temperament with an
integer number of notes per octave
is consistent with JI through some
odd limit if a complete chord of
that limit is constructed in that
equal temperament in the same way
no matter which intervals are
approximated.

If for all odd integers a, b, c such that 1 <= a < b < c <= n, the
ET's best approximation of b /a plus the ET's best approximation of c
/b equals the ET's best approximation of c /a, then the ET is
consistent in the n-limit."

the webpage has a link to Patrick Ozzard-Low's book,
which describe consistency in depth.

apparently you're not very familiar with my Encyclopedia.
browsing around in it a bit should be helpful to you.

-monz
http://tonalsoft.com
microtonal music software

🔗monz <monz@tonalsoft.com>

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> hi Tom,
>
>
> --- In tuning@yahoogroups.com,
> "Tom Dent" <tdent@a...> wrote:
>
> > What is meant by 'inconsistent'
> > for the 64-ET tuning?
>
> <snip>
>
> the webpage has a link to Patrick Ozzard-Low's book,
> which describe consistency in depth.

to help explain it a bit more ... from page 62 of
Ozzard-Low's book:

- begin quote -

In the simple goodness-of-fit-to-just-ratios-analysis of
equal-temperaments the relationship of each individual
scale degree is considered relative to 1/1. But if a
scale is "inconsistent" at a certain limit, then it is
not possible (in fixed intonation) to take advantage of
the implied "nearest" scale degree when forming consonant
chords (or particular melodic figures) - that is, within
the specified m-limit. For example, "goodness-of-fit"
shows that 24-ET has a better approximation of the harmonic
seventh than 12-ET. However, suppose that we want to form
a root position "just intonation dominant seventh" chord,
approximating 7-Limit just intonation, in 24-ET. The
just chord is described by the ratios "4:5:6:7" - that is,
the notes 1/1, 5/4, 6/5 and 7/4. There are thus six ratios
implicit in this chord: 5/4, 6/4 (3/2), 7/4, 6/5, 7/5 and 7/6.
In 24-ET the closest approximation to 5/4 (major third,
386 cents) is 400 cents (8 quarter-tones), and the closest
approximation to 7/5 (septimal augmented fourth, 583 cents)
is 600 cents (12 quarter-tones). If the two just intervals
are added the result is equal to 7/4 (harmonic seventh,
969 cents). But the closest approximation to 7/4 in 24-ET
is 950 cents (19 quarter-tones), and 8 + 12 ¹ 19. And
although the same chord (C E G Bb) can be formed by exactly
the same intervals in 24-ET as in 12-ET, there is no single
optimal approximation of the just dominant seventh chord in
24-ET, since, whichever way the chord is tuned, 7/4 and 7/5
cannot both be "optimally approximated" at the same time
within the temperament. Therefore, 24-ET is "inconsistent"
at the (primary) 7-Limit.

In 12-ET the nearest realisation of this chord is of course
C E G Bb (when 1/1 is C). And it can be shown that all of
the six ratios implicit in this chord are given the best
possible approximation in 12-ET by C E G Bb (or 4, 3 and 3
semitones). So although the chord can in fact be tuned better
in 24-ET than in 12-ET, 12-ET has the virtue of being
consistent, whereas 24-ET does not. In fact, it can be
shown that chord formation in 12-ET is actually consistent
to the primary 9-Limit, whereas 24-ET is consistent only
to the 5-limit.

- end quote -

Ozzard-Low's "Appendix IV" contains the most detailed
explanation of "consistency" that i've seen anywhere.

-monz

🔗Tom Dent <tdent@auth.gr>

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> hi Tom,
>
> (...) i'd prefer to assume the opposite of your last
> sentence: Mozart specifically uses F# all over the
> beginning of his G-minor Symphony (as one would expect),
> but there's a point later in the movement where he
> writes Gb, and since his own teaching specified that
> these are two different pitches, i think we should
> assume that his music *does* "specifically ... exploit
> fine details of tuning".

The Gb is only heard for tiny fractions of a second in passing -
but I would certainly agree now that you have pointed out the
specific issue of enharmonics. There are important sustained
chords E-G-Bb-C# and Eb-G-Bb-Db in this movement which resolve to
D and A flat respectively. So you would expect the C# and Db to
be different pitches. But they are separated by a long stretch of
music so that the clash of C# and Db is not at all evident - all
you hear is that each note sounds in tune within its own context.

Incidentally rather few professional flute players get the C# in
tune (bar 15) - the Berlin Phil with Böhm doesn't.

I suppose what I meant is two things. First, major chords would
probably have their 5ths adjusted upwards to be pure rather than
staying with 55edo, simply because it sounds better - so simply
by trusting their ears the orchestra would deviate from the model.

Second, there are no enharmonic modulations in full harmony where
the difference between 12-ET and 55edo becomes really obvious.
(You could try to see bar 101 (etc.) as an enharmonic modulation
since the notated G# is in the context where an Ab would be
expected, and the F resolves as an E# after 3 bars rest, to F#.
This rather disjointed passage might be Mozart recognizing the
fact that a modulation from G minor to F# minor requires you to
hide the wolf somewhere.)

So I might revise my statement to say that, Mozart is very careful
to use accidentals so that they sound 'normal' within each key -
the exotic enharmonic possibilities of 55edo are not used, and
one hears an ordinary mean-tone tuning.

(...)
> > A good Haydn piece to try would be the 'slow' movement
> > of Op.54 no.1 in G (second movement Allegretto) which
> > has interesting modulations. Also the first movement
> > of Op.77 no.2 in F which requires an enharmonic equivalence.
>
> keep in mind that we're talking about orchestral playing
> here ... during the 1700s and later, meantone would only
> be expected for non-keyboard music. for keyboards, a
> 12-tone well-temperament was standard from c.1700-1850,
> and 12-edo after that.
> -monz

Those pieces are *string quartets*, which I chose because you
*can* hear intonation precisely - no question of 8 violins
playing at slightly different pitches - and the meantone teaching
can be applied. (Or not, in the case of enharmonic equivalence.)

Of course, the use of vibrato (particularly synthesized vibrato)
will blur the fine details of tuning...

Best wishes & goodnight,
~~~Thomas~~~

🔗monz <monz@tonalsoft.com>

--- In tuning@yahoogroups.com, "Tom Dent" <tdent@a...> wrote:
>
>
> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
> >
> > hi Tom,
> >
> > (...) i'd prefer to assume the opposite of your last
> > sentence: Mozart specifically uses F# all over the
> > beginning of his G-minor Symphony (as one would expect),
> > but there's a point later in the movement where he
> > writes Gb, and since his own teaching specified that
> > these are two different pitches, i think we should
> > assume that his music *does* "specifically ... exploit
> > fine details of tuning".
>
> The Gb is only heard for tiny fractions of a second in passing -
> but I would certainly agree now that you have pointed out the
> specific issue of enharmonics. There are important sustained
> chords E-G-Bb-C# and Eb-G-Bb-Db in this movement which resolve to
> D and A flat respectively. So you would expect the C# and Db to
> be different pitches. But they are separated by a long stretch of
> music so that the clash of C# and Db is not at all evident - all
> you hear is that each note sounds in tune within its own context.
>
> Incidentally rather few professional flute players get the C# in
> tune (bar 15) - the Berlin Phil with Böhm doesn't.
>
> I suppose what I meant is two things. First, major chords would
> probably have their 5ths adjusted upwards to be pure rather than
> staying with 55edo, simply because it sounds better - so simply
> by trusting their ears the orchestra would deviate from the model.

that's exactly what we mean around here by "adaptive-JI".

it must be recognized that for Mozart, and indeed nearly
the entire "common practice" repertoire, some form of
meantone is intended, since for composers of that
repertoire the nominals (A, B, C, D, E, F, G) are never
supposed to represent more than one specific pitch.
thus, the syntonic comma must be tempered out, and that
indicates meantone.

if orchestral players adjust the 5ths to be closer to "pure"
(ratio 3:2), then they are emulating an adaptive-JI scheme
such as that of Vicentino:

http://tonalsoft.com/monzo/vicentino/vicentino.htm

in Vicentino's scheme, there is one regular 19-tone chain
of 1/4-comma meantone, and then there is another 17-tone
chain of 1/4-comma meantone tuned 1/4-comma higher than
the 19-tone chain.

as examples of how this works:

1)
by picking the root and major-3rd from the first chain and
the 5th from the second chain, you get a perfectly tuned
JI major triad;

2)
by picking the root from the first chain and the minor-3rd
and 5th from the second chain, you get a perfectly tuned
minor triad.

by following these templates, you get music where vertical
harmonic sonorities are always exact JI ratios, and
horizontonal melodic movement is never shifted by more
than 1/4-comma (~5.376572399 cents) from true JI. and it's
generally assumed that normal human tuning error is ~5 cents.
so essentially, you get JI without the problem of comma drift.

> So I might revise my statement to say that, Mozart is very careful
> to use accidentals so that they sound 'normal' within each key -
> the exotic enharmonic possibilities of 55edo are not used, and
> one hears an ordinary mean-tone tuning.

right, i would not argue that point at all.

-monz

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗monz <monz@tonalsoft.com>

hi Gene,

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> > Gene, i'm impressed by the quality of the .ogg files you've
> > made ... could you please make one of my Mozart MIDI?
> > (and a better .mp3 too? ... the MIDI i have there now
> > is a bit longer than the old mp3 i made.)
>
> Here are four versions, two mp3 and two ogg. The "elite" are
rendered
> using Merlin Elite, the "sgm" using SGM180. I'd be interested in
> hearing opinions on the relative merits of these fonts.
>
> http://66.98.148.43/~xenharmo/elite.mp3
> http://66.98.148.43/~xenharmo/elite.ogg
> http://66.98.148.43/~xenharmo/sgm.mp3
> http://66.98.148.43/~xenharmo/sgm.ogg

thanks!

mostly i prefer the SGM over the Elite ... but the Elite
has a brighter sound which is appropriate sometimes too.
the Elite version sounds especially harsh near the beginning,
but mellows out after the winds come in.

-monz

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

> mostly i prefer the SGM over the Elite ... but the Elite
> has a brighter sound which is appropriate sometimes too.
> the Elite version sounds especially harsh near the beginning,
> but mellows out after the winds come in.

Ha! I pushed a soundfont person into getting the Elite created after I
claimed the other Merlin fonts had too harsh a string sound. Elite
improves on that, and has some advantages over SGM, so this seems like
progress to me even though the guy who made it claims Elite is too
much like SGM. This fellow got mad when I told him SGM had a more
realistic string sound than his fonts did. He told me I was an idiot
and didn't know what I was talking about, so it is interesting to hear
some other opinions.

People who design soundfonts often seem to think only they know how to
do it right and everyone else's font is all wrong, I notice.
Apparently tastes and perceptions differ quite a bit.

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

Paul Greenhaw wrote, in answer to Carl:
________________________________________________________________________
Date: Thu, 21 Apr 2005 15:51:34 -0400
From: pgreenhaw@...
Subject: Re: Re: 29tET

>... I've heard that strings somehow play in pythagorean
>tuning, but I can't imagine why or how that would be the case.

-Carl

I think the notion of string players playing in a more or less Pythagorean
tuning comes out of a tendency to push leading tones (or better yet, the
"3rds" of chords) more toward the sharp side. Its the strident pull
toward the resolution that gives rise to the sharpening....

Paul
________________________________________________________________________

Hi Paul,

I don't buy this.

Supposing your composer only knows the chord
progression I IV V I. Then in C, that would be
C F G C. The thirds of the C and G chords, namely
E and B, certainly do lead by a semitone to a note
in the next chord, namely F and C. But the third
of the F chord, A, does not lead by a semitone to
either of its nearest neighbours in the C chord,
namely G and C.

So, if your notion of pushing the thirds holds, we
should expect both the E and the B to be sharper,
but not the A.

Now suppose your composer is just a tad more
sophisticated; he knows he can use the progression
I vi ii V7 I. In C, that's C Am Dm G7 C.
In passing from the C chord to the Am chord, there
is NO need for the E to move, hence no tendency
for it to sharpen. Should the player now play a
_different_ E than he played in passing from I to IV?
(We're not talking adaptive tunings here.) And
should the Fs of the Dm and G7 chords be different
too, because of semitone leading back to the E in the
final C chord?

Regards,
Yahya

--
Internal Virus Database is out-of-date.
Checked by AVG Anti-Virus.
Version: 7.0.308 / Virus Database: 266.9.15 - Release Date: 16/4/05

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

Ozan,

You wrote:
________________________________________________________________________
Date: Thu, 21 Apr 2005 23:05:31 +0300
From: "Ozan Yarman" <ozanyarman@...>
Subject: Re: Re: 29tET

3/4 for G
9/8 for D
27/16 for A
81/32 for E

There is a strong difference between the Rast and Suz-i Dilara maqams (just versus phythagorean major scales) as such, the former being more suitable for the major chord 4:5:6. That is probably why the E string of the violings hereabouts is tuned a little lower.
...
________________________________________________________________________
[YA] It's this sort of _practical_ information
on tuning that I find of most musical interest on
this list. Thank you!

To clarify, do you mean that violinists will tune their
instruments with E at 5/4 for Rast, but at 81/64
for Suz-i-Dilara?

More generally, will they retune their instruments
for different maqama?

Regards,
Yahya

--
Internal Virus Database is out-of-date.
Checked by AVG Anti-Virus.
Version: 7.0.308 / Virus Database: 266.9.15 - Release Date: 16/4/05

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

Carl replied to Tom:
_______________________________________________________________________
Date: Thu, 21 Apr 2005 17:37:53 -0700
From: Carl Lumma <ekin@...>
Subject: Re: Re: 29tET

...

>As Ozan says, usual string instruments (when not playing with
>keyboards) are tuned in pure 5ths from C through to E, which is
>then a Pythagorean third (plus some octaves) higher.

It's my understanding that many acoustic guitar players tune
the open strings pure.

>This does not mean that *all* intervals are Pythagorean
>- thankfully!

Yes, exactly. I think it's a bit of a red herring.
________________________________________________________________________
[YA] I agree with both points. When playing the guitar solo -
acoustic OR electric - I invariably tune the open strings as
pure fourths and octaves. However, when accompanied by
keyboard, I tune to the keyboard. Saves nastinesses!

Regards,
Yahya

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🔗Ozan Yarman <ozanyarman@superonline.com>

Monz, the fact that Leopold Mozart assumed that the whole tone consists of 9 commas, the diatonic semitone, 5, and the chromatic semitone, 4, is exactly the kind of numerology one finds in the Yekta-Arel-Ezgi school, where a cycle of 24 fifths are taken as the perfect Turkish Music system of tuning and where the comma steps defining the two distinct semitones above are reversed. I would be much happier if it was said that the major whole tone is the 9/8th of an open string, but could be tempered depending on the tuning.

Cordially,
Ozan
----- Original Message -----
From: monz
To: tuning@yahoogroups.com
Sent: 23 Nisan 2005 Cumartesi 2:26
Subject: [tuning] Re: Mozart -Haydn in 55-edo

🔗Ozan Yarman <ozanyarman@superonline.com>

🔗monz <monz@tonalsoft.com>

hi Ozan,

--- In tuning@yahoogroups.com,
"Ozan Yarman" <ozanyarman@s...>
wrote:

> Monz, the fact that Leopold
> Mozart assumed that the
> whole tone consists of 9
> commas, the diatonic semitone,
> 5, and the chromatic semitone,
> 4, is exactly the kind of
> numerology one finds in the
> Yekta-Arel-Ezgi school, where
> a cycle of 24 fifths are taken
> as the perfect Turkish Music
> system of tuning and where
> the comma steps defining the
> two distinct semitones above
> are reversed.

yes, in many of my posts on this subject over the last
few days, i've pointed out that when the comma steps
for the semitones are reversed, the result is 53-edo,
which has been a theoretical standard for Turkish music
for a long time.

i'll repeat in case you missed it:

where the octave = 5 tones + 2 diatonic semitones,

me, monz,
/tuning/topicId_58099.html#58141

> assuming a closed tuning, this division into 9 commas
> produces either 53-edo or 55-edo, depending on which
> of the two different semitones (chromatic and diatonic)
> is the larger one. making the chromatic semitone larger
> (thus, a pythagorean system) gives 53-edo, and making
> the diatonic larger (thus, a meantone) gives 55:
>
> .......................... 53-edo ........... 55-edo
>
> t = tone .................... 9 ................ 9
> s = diatonic semitone ....... 4 ................ 5
> octave = 5t + 2s ..... (5*9)+(2*4)=53 ... (5*9)+(2*5)=55

> I would be much happier if it was said that the major
> whole tone is the 9/8th of an open string, but could
> be tempered depending on the tuning.

i'm sure that would be fine for Turkish music ... but
not for either of the Mozarts. European common-practice
music always tempered out the syntonic-comma, which means
that there has to be a mean-tone, not a major and minor
whole-tone. it's important to this repertoire that there
is no distinction between different whole-tones -- there
is only one size of whole-tone, and thus it is a meantone.

-monz

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

🔗monz <monz@tonalsoft.com>

hi Yayha,

--- In tuning@yahoogroups.com,
"Yahya Abdal-Aziz" <yahya@m...>
wrote:
>
> Carl,
>
> Could you please explain what you meant when you wrote:
>
______________________________________________________________________
__
> Date: Fri, 22 Apr 2005 10:04:25 -0700
> From: Carl Lumma <ekin@>
> ...
> "Adaptive JI" can be viewed as tempering out a comma in melodic
> space rather than harmonic space like traditional temperament.
>
______________________________________________________________________
__
>
> ???
>
> Any note in a melody is part of the harmonic fabric;
> conversely, moving harmonies inevitably create melodies.
>
> So, how can you divorce the two "spaces"?

the concept behind adaptive-JI is that vertical harmonic
sonorities are produced as small-integer JI ratios, exactly
as in true JI. but true JI might produce commatic drift
with a large enough set of pitches, or conversely, if the
pitch set is kept small enough to eliminate the drift, then
there will be commatic shifts melodically. adaptive-JI
simultaneously eliminates both the drift and the shifts,
circulating the commatic shifts strategically by tempering
melodic intervals by fractions of the comma.

a long time ago, i wrote a webpage about various tunings of
the dominant-7th chord, and some of the examples use
adaptive-JI or adaptive-tuning ... the difference between
those two terms is that in adaptive-tuning the vertical
harmonic sonorities approximate JI but are not exact ratios.
here's the page:

http://tonalsoft.com/td/monzo/i-iv-v7-i/i-iv-v-i.htm

and if you haven't already looked at my page on Vicentino,
you should:

http://tonalsoft.com/monzo/vicentino/vicentino.htm

-monz
http://tonalsoft.com
microtonal music software

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

monz,

You answered my question to Carl:
________________________________________________________________________
> Could you please explain what you meant when you wrote:
______________________________________________________________________
> Date: Fri, 22 Apr 2005 10:04:25 -0700
> From: Carl Lumma <ekin@>
> ...
> "Adaptive JI" can be viewed as tempering out a comma in melodic
> space rather than harmonic space like traditional temperament.
>
______________________________________________________________________
>
> ???
>
> Any note in a melody is part of the harmonic fabric;
> conversely, moving harmonies inevitably create melodies.
>
> So, how can you divorce the two "spaces"?
______________________________________________________________________

thus:

[monz]
the concept behind adaptive-JI is that vertical harmonic
sonorities are produced as small-integer JI ratios, exactly
as in true JI. but true JI might produce commatic drift
with a large enough set of pitches, or conversely, if the
pitch set is kept small enough to eliminate the drift, then
there will be commatic shifts melodically. adaptive-JI
simultaneously eliminates both the drift and the shifts,
circulating the commatic shifts strategically by tempering
melodic intervals by fractions of the comma.

[Yahya] I do not understand the last sentence.

You say "there will be commatic shifts melodically" with
true JI when we keep the pitch set "small enough to
eliminate the [commatic] drift".

You use the term shift for melodic - horizontal - pitch
determination. OK

You use the term drift for harmonic - vertical - pitch
determination. OK

Adaptive-JI "circulat[es] the commatic shifts
strategically by tempering melodic intervals by fractions
of the comma." That means the melodies must move up and
down a little from the true JI notes.

In the scale of C major in JI, let's take the harmonic
progression I - V - ii - IV - I. We have a good fifth G-D
in the G major triad, a good fifth F-C in the F major triad,
and a good major third F-A and a good minor third A-C in
the same chord; but the resulting D gives a bad fifth with
that A, and a bad minor third with that F, in the D minor
triad.

Let's instance this progression in three parts as follows,
with a descending bass line:

G - G - F - F- E
E - D - D - C - C
C - B - A - A - G

I - V - ii - IV - I

You say we're going to temper melodic intervals by
fractions of that comma. Assuming we hold the C - E - G,
G - B - D, and F- A - C notes to JI, *and use the same F and
A in the D minor triad*, we're going to have to change that
D (a 9/8) by a comma if we want the D minor to sound good.
Right? (Not sure about the bit marked *...*; the alternative
seems to be to hold to the same D, and move _both_ the F
and the A.)

These are the things that I still don't get:
Where is the fraction of a comma? How do we circulate
the shift? What is the strategy? And how, if the melodies
still shift - albeit by smaller amounts than a comma - can we
prevent a drift in intonation?

[monz, continued]
a long time ago, i wrote a webpage about various tunings of
the dominant-7th chord, and some of the examples use
adaptive-JI or adaptive-tuning ... the difference between
those two terms is that in adaptive-tuning the vertical
harmonic sonorities approximate JI but are not exact ratios.
here's the page:

http://tonalsoft.com/td/monzo/i-iv-v7-i/i-iv-v-i.htm

and if you haven't already looked at my page on Vicentino,
you should:

http://tonalsoft.com/monzo/vicentino/vicentino.htm

________________________________________________________________________

[Yahya]
Yes, thanks, monz, I've had that pleasure! I've read
and reread both these pages, and still have difficulty
in understanding just HOW to apply adaptive-JI in
practice. Let me ask this: is there just ONE adaptive-
JI rendition of any triadic progression, or are there
choice points, where one could equally correctly choose
a different realization of the next chord?

(The links to cawapu, bike-chain and even comma-pump
on those pages seem to be broken every time I try them.)

Regards,
Yahya

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🔗monz <monz@tonalsoft.com>

hi Yahya,

--- In tuning@yahoogroups.com,
"Yahya Abdal-Aziz" <yahya@m...>
wrote:

> In the scale of C major in JI,
> let's take the harmonic progression
> I - V - ii - IV - I. We have
> a good fifth G-D in the G major
> triad, a good fifth F-C in the
> F major triad, and a good major
> third F-A and a good minor third
> A-C in the same chord; but the
> resulting D gives a bad fifth with
> that A, and a bad minor third with
> that F, in the D minor triad.
>
> Let's instance this progression
> in three parts as follows,
> with a descending bass line:
>
> G - G - F - F- E
> E - D - D - C - C
> C - B - A - A - G
>
> I - V - ii - IV - I
>
> You say we're going to temper melodic intervals by
> fractions of that comma. Assuming we hold the C - E - G,
> G - B - D, and F- A - C notes to JI, *and use the same F and
> A in the D minor triad*, we're going to have to change that
> D (a 9/8) by a comma if we want the D minor to sound good.
> Right? (Not sure about the bit marked *...*; the alternative
> seems to be to hold to the same D, and move _both_ the F
> and the A.)
>
> These are the things that I still don't get:
> Where is the fraction of a comma? How do we circulate
> the shift? What is the strategy? And how, if the melodies
> still shift - albeit by smaller amounts than a comma - can we
> prevent a drift in intonation?

you're thinking too much in true JI, that's why you
don't get it.

let's use Vicentino's adaptive-JI tuning for your example.
here's your chord progression again, with cents values
for the notes taken from Vicentino, but shifted so that
C is 0 (zero) cents, since it's obvious that we're in
the key of C here:

G 0702 - G 0697 - F 0509 - F 0503 - E 0386
E 0386 - D 0199 - D 0193 - C 0005 - C 0000
C 0000 - B 1083 - A 0895 - A 0890 - G 0702

I ------ V ------ ii ----- IV ----- I

first, look at the cents values vertically:
every triad here is a perfectly in-tune JI
major or minor triad.

now, look at the cents values horizontally:

on the top line:

. the first G (~702 cents) is a 3:2 ratio above C (0 cents),
but the second G (~697 cents) is tempered flat by 1/4-comma
(~5 cents);

. the following F (~509 cents) is tempered sharp by 1/2-comma,
but it is a perfect 3:2 above the root of the chord, which is
a D (~503 cents) tuned 1/4-comma sharp; the next F (~503 cents)
is the root of that chord.

the maximum melodic shift on this line is ~5 cents.

similarly for the other lines.

now let's look at the cents values for the roots of the chords:

C 0000 - G 0697 - D 0193 - F 0503 - C 0000

I ------ V ------ ii ----- IV ----- I

again, the maximum deviation from true JI is 1/4-comma, ~5 cents.

do you see how it works now? the theory is that most ears
will accept this because all of the chords sound like real JI
-- BECAUSE THEY ARE! -- but the melodic shifting necessary
to acheive this is nowhere great enough to really be noticeable.

> Let me ask this: is there just ONE adaptive-
> JI rendition of any triadic progression, or are there
> choice points, where one could equally correctly choose
> a different realization of the next chord?

there are an infinite number of adaptive-JI possibilities.
the basic concept is that you create a linear temperament
chain, then create another one at a distance of the amount
of tempering. by doing that, your tuning has the JI ratios
embedded in it, and you just choose the notes from either
chain which you need to get JI chords.

> (The links to cawapu, bike-chain and even comma-pump
> on those pages seem to be broken every time I try them.)

sorry about that. i can't fix broken links now, because
the Encyclopedia is in a state of transition while i work
on converting all of the pages to the new format. if you
all can just bear with us, in a few weeks the whole job
should be done, and the new Encyclopedia uploaded.

-monz

🔗monz <monz@tonalsoft.com>

--- In tuning@yahoogroups.com,
"monz" wrote:

> > [Yayha:]
> > (The links to cawapu,
> > bike-chain and even
> > comma-pump on those pages
> > seem to be broken every
> > time I try them.)
>
> sorry about that. i can't
> fix broken links now, because
> the Encyclopedia is in a
> state of transition while i
> work on converting all of the
> pages to the new format. if
> you all can just bear with us,
> in a few weeks the whole job
> should be done, and the new
> Encyclopedia uploaded.

generally, the current problems with the Encyclopedia links
result from the fact that there is no "monzo" directory
on the Tonalsoft site right now, and so anything that links
into that directory is being redirected to sonic-arts.org/monzo.
if you click on a link on one of *those* pages, it tries
to refer to the non-existent "enc" directory at sonic-arts.

the pages you're looking for (cawapu, bike-chain, comma-pump)
do exist, but on the tonalsoft.com site -- you can get to
them from the Encyclopedia index. sorry about the rigamarole,
it's temporary.

-monz
http://tonalsoft.com
microtonal music software