piano stretching

While looking at Backus' book "The Acoustical Foundations of Music" I
was a little surprised by the amount and breadth of stretching going on
in piano tuning. The implications for microtonal music seem
problematic. It seems that according to his chart of an average of 16
pianos, only the areas around middle C and A440 are dead on. From
middle C down all the notes go flat. At 2 octaves below middle C they
take a distinct flattening past 10� all the way to past 30� flat. On
the up side everything is in almost exact reverse so that the top 2
notes of the piano are just past 30� sharp.

I guess what I'm thinking is that if you built this stretching into your
music, it might capture some of the brilliance characteristic of the
piano. And the other thought is that an octave in western music may not
be an octave. I mean, all those ear training classes in which the piano
reigns as arbiter . . .

--- In tuning@y..., Rick Tagawa <ricktagawa@e...> wrote:
> While looking at Backus' book "The Acoustical Foundations of Music"
I
> was a little surprised by the amount and breadth of stretching
going on
> in piano tuning. The implications for microtonal music seem
> problematic. It seems that according to his chart of an average of
16
> pianos, only the areas around middle C and A440 are dead on. From
> middle C down all the notes go flat. At 2 octaves below middle C
they
> take a distinct flattening past 10¢ all the way to past 30¢ flat.
On
> the up side everything is in almost exact reverse so that the top 2
> notes of the piano are just past 30¢ sharp.

Yes, but any _octave_ on the piano is just a few cents sharp.
>
> I guess what I'm thinking is that if you built this stretching into
your
> music, it might capture some of the brilliance characteristic of the
> piano.

The reason for the stretching is largely the fact that the partials
of the piano spectrum are themselves stretched. And that's the reason
for the brilliance, to a large extent. A resynthesized piano with
exactly harmonic partials sounds quite "fake", compared with a
resynthesized piano with correctly stretched partials.

> And the other thought is that an octave in western music may not
> be an octave. I mean, all those ear training classes in which the
piano
> reigns as arbiter . . .

When sine waves are used, subjects typically picked octaves of about
1209¢ as melodically "correct". However, with most timbres, such as
voice, bowed strings, brass, and reeds, the partials are exactly
harmonic, and thus the melodically "correct" octave is generally
close to 1200¢. On the piano, the partials are most stretched in the
extreme upper and lower registers, and that is where the octaves are
stretched the most (by a few cents) as well.

--- In tuning@y..., Rick Tagawa <ricktagawa@e...> wrote:

/tuning/topicId_23646.html#23646

> While looking at Backus' book "The Acoustical Foundations of
> Music" I was a little surprised by the amount and breadth of
> stretching going on in piano tuning. The implications for
> microtonal music seem problematic. It seems that according
> to his chart of an average of 16 pianos, only the areas around
> middle C and A440 are dead on. From middle C down all the
> notes go flat. At 2 octaves below middle C they take a
> distinct flattening past 10¢ all the way to past 30¢ flat.
> On the up side everything is in almost exact reverse so that
> the top 2 notes of the piano are just past 30¢ sharp.
>
> I guess what I'm thinking is that if you built this stretching
> into your music, it might capture some of the brilliance
> characteristic of the piano. And the other thought is that
> an octave in western music may not be an octave. I mean, all
> those ear training classes in which the piano reigns as
> arbiter . . .

Hi Rick,

One of my main interests right now is along these lines.
I'm working on creating some stretch-JIs for piano timbres
for some of my compositions, which follow the type of register
curve you talk about here.

We discussed a bit of this last year. John deLabenfels
recently inquired, and I'll repost my answer for your benefit:

/tuning/topicId_22604.html#22688

> Do you remember where you and another list member

It was Allan Myhara.

> discussed a function, rather uniform in the center and
> more extreme at each end of the keyboard, for ideal stretch?

The original message, with links to all the follow-up posts, is:
/tuning/topicId_13830.html#13830

The message with the Excel math formula is:
/tuning/topicId_13830.html#13895

And the graph is:
/tuning/files/monz/rhodes.jpg

-monz
http://www.monz.org
"All roads lead to n^0"

[Rick Tagawa wrote:]
>While looking at Backus' book "The Acoustical Foundations of Music" I
>was a little surprised by the amount and breadth of stretching going on
>in piano tuning. The implications for microtonal music seem
>problematic. It seems that according to his chart of an average of 16
>pianos, only the areas around middle C and A440 are dead on. From
>middle C down all the notes go flat. At 2 octaves below middle C they
>take a distinct flattening past 10› all the way to past 30› flat. On
>the up side everything is in almost exact reverse so that the top 2
>notes of the piano are just past 30› sharp.

>I guess what I'm thinking is that if you built this stretching into
>your music, it might capture some of the brilliance characteristic of
>the piano. And the other thought is that an octave in western music
>may not be an octave. I mean, all those ear training classes in which
>the piano reigns as arbiter . . .

Funny you should mention that. Monz recently challenged me to add that
capability to my tuning program (well, Monz expressed a wish to have
that capability, and I took it as a challenge).

It'd be easy enough to add this to my program. What sequence would you
(or anybody) like to hear stretched? For now, let's keep it to solo
piano. General MIDI.

JdL

Rick Tagawa wrote:

> While looking at Backus' book "The Acoustical Foundations of Music" I
> was a little surprised by the amount and breadth of stretching going on
> in piano tuning. The implications for microtonal music seem
> problematic. It seems that according to his chart of an average of 16
> pianos, only the areas around middle C and A440 are dead on. From
> middle C down all the notes go flat. At 2 octaves below middle C they
> take a distinct flattening past 10� all the way to past 30� flat. On
> the up side everything is in almost exact reverse so that the top 2
> notes of the piano are just past 30� sharp.
>
> I guess what I'm thinking is that if you built this stretching into your
> music, it might capture some of the brilliance characteristic of the
> piano. And the other thought is that an octave in western music may not
> be an octave. I mean, all those ear training classes in which the piano
> reigns as arbiter . . .

Most pianos I tinker with sound totally out of tune on the first play of a scale. Then things come
togetherwith a few chords. Our human auditory systems really are too generous.

Regards.

>

Hello
Does anybody know if an analog oscillator produces exactly harmonic
partials?
I understand that this must depend on the waveform. Sinusoids have only the
first component , what about saws and others ?

--- In tuning@y..., Alexandros Papadopoulos <Alexmoog@o...> wrote:
> Hello
> Does anybody know if an analog oscillator produces exactly harmonic
> partials?
> I understand that this must depend on the waveform. Sinusoids have
only the
> first component , what about saws and others ?

All waveforms that repeat themselves exactly consist of exactly
harmonic partials.

Sine waves produced by oscillators, which are sold as test equipment rather than as musical instruments, consist of fundamental tones only, in theory. They might produce a small amount of harmonic or even inharmonic distortion, but it should be insignificant if the unit is operating properly. This could be tested using a harmonic distortion analyzer. For test purposes, this might be necessary, but for music it shouldn't be. The musical instrument which is related is the theremin (and its variants).
Most of these oscillators also can be switched to produce saw-tooth waves and/or square waves. These have loads of harmonics. The saw tooth
waves look like triangles and have essentially only odd harmonics. Square waves contain both odd and even harmonics. A 20 Hz square wave will contain harmonics at least ten octaves higher (20,480 Hz). These are used to test phono cartridges. This is quite a severe test, and a variety of flaws are usually apparent, when viewed on an oscilloscope. I believe they are seldom used to test loudspeakers because these are so much worse as electro mechanical transducers that the results would be difficult to interpret meaningfully.
Synthesizers sometimes start with square or sawtooth waves and filter them to more closely approximate musical instruments. But combinations of sine waves with selected frequencies (octaves, fifths, thirds, etc.) can be used to build up more complex forms as an alternative method. This is more like what a Hammond organ does, but with rotating tone wheels and octave doublers..
>>> Alexmoog@otenet.gr 05/24/01 06:40PM >>>
Hello
Does anybody know if an analog oscillator produces exactly harmonic
partials?
I understand that this must depend on the waveform. Sinusoids have only the
first component , what about saws and others ?

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--- In tuning@y..., "John F. Sprague" <jsprague@d...> wrote:
> Most of these oscillators also can be switched to produce saw-tooth
waves and/or square waves. These have loads of harmonics. The saw
tooth
> waves look like triangles and have essentially only odd harmonics.

Correction: Sawtooth waves have both odd and even harmonics and look
like this:

|\|\|\|\|\|\|\|\|\|\|\|\|\|\|\|\|\|\|\|\|\|\|\|\

Triangle waves contain only odd harmonics and look like this:

/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\

>Square waves contain both odd and even harmonics.

I'm afraid that's also incorrect. Square waves have only odd
harmonics.

To get more specific:

Square waves have only odd harmonics, and the amplitude of harmonic n
is proportional to 1/n.

Triangle waves have only odd harmonics, and the amplitude of harmonic
n is proportional to 1/n^2.

Sawtooth waves have all harmonics, and the amplitude of harmonic n is
proportional to 1/n.

Parabolic waves have all harmonics, and the amplitude of harmonic n is
proportional to 1/n^2.

[Paul said:
> Correction: Sawtooth waves have both odd and even harmonics and
> look like this:
>
> |\|\|\|\|\|\|\|\|\|\|\|\|\|\|\|\|\|\|\|\|\|\|\|\

Hi Paul, actually that's a Japanese saw.

The Western ones look like this:

/|/|/|/|/|/|/|/|/|/|/|/|/|/|/|/|/|/|/|/|/|/|/|/|/|

The japanese ones are more *efficient* because they use
the -pull- stroke, but the western ones are *safer* for
children and old folks because they cut on the -push-
stroke; when the saw is moving away from the body.

Anyway, hope this is helpful.

- Jeff

Sorry, I did this off the top of my head, without checking references I'd not looked at for a long time. It may be the triangular waves the oscillators switch to. It's a switch I seldom use. However, some composers of electronic and computer music have used these wave forms as alternatives to sine waves.

>>> paul@stretch-music.com 05/25/01 12:49PM >>>
--- In tuning@y..., "John F. Sprague" <jsprague@d...> wrote:
> Most of these oscillators also can be switched to produce saw-tooth
waves and/or square waves. These have loads of harmonics. The saw
tooth
> waves look like triangles and have essentially only odd harmonics.

Correction: Sawtooth waves have both odd and even harmonics and look
like this:

|\|\|\|\|\|\|\|\|\|\|\|\|\|\|\|\|\|\|\|\|\|\|\|\

Triangle waves contain only odd harmonics and look like this:

/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\

>Square waves contain both odd and even harmonics.

I'm afraid that's also incorrect. Square waves have only odd
harmonics.

To get more specific:

Square waves have only odd harmonics, and the amplitude of harmonic n
is proportional to 1/n.

Triangle waves have only odd harmonics, and the amplitude of harmonic
n is proportional to 1/n^2.

Sawtooth waves have all harmonics, and the amplitude of harmonic n is
proportional to 1/n.

Parabolic waves have all harmonics, and the amplitude of harmonic n is
proportional to 1/n^2.

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Following the piano stretching thread it dawns on me that the 13th limit can admitted into the 72-tET
by the umbrella of "stretched" tuning. By substituting Note 51 which is 9.4� too sharp for note 50
which is 7.2� too flat . . . then we have a 13th which may not be "ambiguous" at all?????

Or perhaps you could use note 50 for the 13th whenever it occurs below middle C and 51 whenever you
want to use it above middle C?

Rick Tagawa wrote:

> Dear Tuning List,
> I received Joe's book "Preliminary Studies in the Virtual Pitch Continuum" <snip>
>
> Regarding the other thread about piano stretching, <snip> the plus or minus 5
> cents from pure ET that make up the bulk of piano tuning might be the fudge factor that would
> allow the 13 limit into the 72 or any limit for that matter, since the maximum error of 72 is
> 8.33� or half the 1/12 of a tone. <snip>

P.S. The 8.33� maximum error wouldn't be accurate anymore since instead of looking for the closest
approximation we are factoring in the "stretched" tuning.

I would normally have finished posting Dave Canright's calculations of just intervals on my webpage
but the retyping and html is just a little more work than I want to expend on it for now, so I'll
just post it here.

Subject:
table ordered by harmonic liimit
Date:
Wed, 2 May 2001 11:56:53 -0700
From:
"Canright, David" <dcanright@nps.navy.mil>
To:
'Rick Tagawa' <ricktagawa@earthlink.net>

72-tET Approximation of Just Intervals

Below is a table of all the intervals found in the harmonic series up to
harmonic 64, and their closest approximations by 72-tET. These are ordered
by first appearance in the harmonic series (by numerator, then by
denominator; for example, if you are only interested up to harmonic 20, stop
when you first hit 21). Not all of the "octave complements" appear (if they
involve a numerator greater than 64); of course, their tempered
approximations have equal but opposite errors. Each entry includes the
ratio, cents error relative to nearest tempered (so, for example, 3/2 is
+2.0c relative to note #42), tempered note # (0-71), keyboard # (0-5), key #
(0-11), where keyboards 1-5 are assumed progressively sharper relative to
keyboard 0.

ratio error note kb:key#

1/1 +0.0c 0 0:0
3/2 +2.0c 42 0:7
4/3 -2.0c 30 0:5
5/3 +1.0c 53 5:8
5/4 +3.0c 23 5:3
6/5 -1.0c 19 1:3
7/4 +2.2c 58 4:9
7/5 -0.8c 35 5:5
7/6 +0.2c 16 4:2
8/5 -3.0c 49 1:8
8/7 -2.2c 14 2:2
9/5 +0.9c 61 1:10
9/7 +1.8c 26 2:4
9/8 +3.9c 12 0:2
10/7 +0.8c 37 1:6
10/9 -0.9c 11 5:1
11/6 -0.6c 63 3:10
11/7 -0.8c 47 5:7
11/8 +1.3c 33 3:5
11/9 -2.6c 21 3:3
11/10 -1.7c 10 4:1
12/7 -0.2c 56 2:9
12/11 +0.6c 9 3:1
13/7 +5.0c 64 4:10
13/8 +7.2c 50 2:8
13/9 +3.3c 38 2:6
13/10 +4.2c 27 3:4
13/11 +5.9c 17 5:2
13/12 +5.2c 8 2:1
14/9 -1.8c 46 4:7
14/11 +0.8c 25 1:4
14/13 -5.0c 8 2:1
15/8 +4.9c 65 5:10
15/11 +3.6c 32 2:5
15/13 -2.3c 15 3:2
15/14 +2.8c 7 1:1
16/9 -3.9c 60 0:10
16/11 -1.3c 39 3:6
16/13 -7.2c 22 4:3
16/15 -4.9c 7 1:1
17/9 +1.0c 66 0:11
17/10 +2.0c 55 1:9
17/11 +3.6c 45 3:7
17/12 +3.0c 36 0:6
17/13 -2.2c 28 4:4
17/14 +2.8c 20 2:3
17/15 +0.0c 13 1:2
17/16 +5.0c 6 0:1
18/11 +2.6c 51 3:8
18/13 -3.3c 34 4:5
18/17 -1.0c 6 0:1
19/10 -5.5c 67 1:11
19/11 -3.8c 57 3:9
19/12 -4.4c 48 0:8
19/13 +7.0c 39 3:6
19/14 -4.6c 32 2:5
19/15 -7.4c 25 1:4
19/16 -2.5c 18 0:3
19/17 -7.4c 12 0:2
19/18 -6.4c 6 0:1
20/11 +1.7c 62 2:10
20/13 -4.2c 45 3:7
20/17 -2.0c 17 5:2
20/19 +5.5c 5 5:0
21/11 +2.8c 67 1:11
21/13 -3.1c 50 2:8
21/16 +4.1c 28 4:4
21/17 -0.8c 22 4:3
21/19 +6.6c 10 4:1
21/20 +1.1c 5 5:0
22/13 -5.9c 55 1:9
22/15 -3.6c 40 4:6
22/17 -3.6c 27 3:4
22/19 +3.8c 15 3:2
22/21 -2.8c 5 5:0
23/12 -7.0c 68 2:11
23/13 +4.4c 59 5:9
23/14 -7.2c 52 4:8
23/15 +6.7c 44 2:7
23/16 -5.1c 38 2:6
23/17 +6.7c 31 1:5
23/18 +7.7c 25 1:4
23/19 -2.6c 20 2:3
23/20 -8.0c 15 3:2
23/21 +7.5c 9 3:1
23/22 -6.4c 5 5:0
24/13 -5.2c 64 4:10
24/17 -3.0c 36 0:6
24/19 +4.4c 24 0:4
24/23 +7.0c 4 4:0
25/13 -1.2c 68 2:11
25/14 +3.8c 60 0:10
25/16 +6.0c 46 4:7
25/17 +1.0c 40 4:6
25/18 +2.1c 34 4:5
25/19 -8.2c 29 5:4
25/21 +1.8c 18 0:3
25/22 +4.6c 13 1:2
25/23 -5.6c 9 3:1
25/24 +4.0c 4 4:0
26/15 +2.3c 57 3:9
26/17 +2.2c 44 2:7
26/19 -7.0c 33 3:5
26/21 +3.1c 22 4:3
26/23 -4.4c 13 1:2
26/25 +1.2c 4 4:0
27/14 +3.7c 68 2:11
27/16 +5.9c 54 0:9
27/17 +0.9c 48 0:8
27/19 -8.3c 37 1:6
27/20 +2.9c 31 1:5
27/22 +4.5c 21 3:3
27/23 -5.7c 17 5:2
27/25 -0.1c 8 2:1
27/26 -1.3c 4 4:0
28/15 -2.8c 65 5:10
28/17 -2.8c 52 4:8
28/19 +4.6c 40 4:6
28/23 +7.2c 20 2:3
28/25 -3.8c 12 0:2
28/27 -3.7c 4 4:0
29/15 +8.0c 68 2:11
29/16 -3.8c 62 2:10
29/17 +8.0c 55 1:9
29/18 -7.7c 50 2:8
29/19 -1.3c 44 2:7
29/20 -6.7c 39 3:6
29/21 -7.9c 34 4:5
29/22 -5.1c 29 5:4
29/23 +1.3c 24 0:4
29/24 -5.7c 20 2:3
29/25 +6.9c 15 3:2
29/26 +5.7c 11 5:1
29/27 +7.0c 7 1:1
29/28 -5.9c 4 4:0
30/17 +0.0c 59 5:9
30/19 +7.4c 47 5:7
30/23 -6.7c 28 4:4
30/29 -8.0c 4 4:0
31/16 -5.0c 69 3:11
31/17 +6.7c 62 2:10
31/18 +7.8c 56 2:9
31/19 -2.5c 51 3:8
31/20 -7.9c 46 4:7
31/21 +7.6c 40 4:6
31/22 -6.3c 36 0:6
31/23 +0.1c 31 1:5
31/24 -6.9c 27 3:4
31/25 +5.7c 22 4:3
31/26 +4.5c 18 0:3
31/27 +5.8c 14 2:2
31/28 -7.1c 11 5:1
31/29 -1.2c 7 1:1
31/30 +6.8c 3 3:0
32/17 -5.0c 66 0:11
32/19 +2.5c 54 0:9
32/21 -4.1c 44 2:7
32/23 +5.1c 34 4:5
32/25 -6.0c 26 2:4
32/27 -5.9c 18 0:3
32/29 +3.8c 10 4:1
32/31 +5.0c 3 3:0
33/17 -1.7c 69 3:11
33/19 +5.8c 57 3:9
33/20 +0.3c 52 4:8
33/23 +8.3c 37 1:6
33/25 -2.7c 29 5:4
33/26 -3.9c 25 1:4
33/28 +1.1c 17 5:2
33/29 +7.0c 13 1:2
33/31 +8.2c 6 0:1
33/32 +3.3c 3 3:0
34/19 +7.4c 60 0:10
34/21 +0.8c 50 2:8
34/23 -6.7c 41 5:6
34/25 -1.0c 32 2:5
34/27 -0.9c 24 0:4
34/29 -8.0c 17 5:2
34/31 -6.7c 10 4:1
34/33 +1.7c 3 3:0
35/18 +1.2c 69 3:11
35/19 +7.6c 63 3:10
35/22 +3.8c 48 0:8
35/23 -6.5c 44 2:7
35/24 +3.2c 39 3:6
35/26 -2.1c 31 1:5
35/27 -0.7c 27 3:4
35/29 -7.8c 20 2:3
35/31 -6.6c 13 1:2
35/32 +5.1c 9 3:1
35/33 +1.9c 6 0:1
35/34 +0.2c 3 3:0
36/19 +6.4c 66 0:11
36/23 -7.7c 47 5:7
36/25 -2.1c 38 2:6
36/29 +7.7c 22 4:3
36/31 -7.8c 16 4:2
36/35 -1.2c 3 3:0
37/19 +3.8c 69 3:11
37/20 -1.6c 64 4:10
37/21 -2.8c 59 5:9
37/22 +0.0c 54 0:9
37/23 +6.4c 49 1:8
37/24 -0.6c 45 3:7
37/25 -4.6c 41 5:6
37/26 -5.9c 37 1:6
37/27 -4.5c 33 3:5
37/28 -0.8c 29 5:4
37/29 +5.1c 25 1:4
37/30 -3.6c 22 4:3
37/31 +6.3c 18 0:3
37/32 +1.3c 15 3:2
37/33 -1.9c 12 0:2
37/34 -3.6c 9 3:1
37/35 -3.8c 6 0:1
37/36 -2.6c 3 3:0
38/21 -6.6c 62 2:10
38/23 +2.6c 52 4:8
38/25 +8.2c 43 1:7
38/27 +8.3c 35 5:5
38/29 +1.3c 28 4:4
38/31 +2.5c 21 3:3
38/33 -5.8c 15 3:2
38/35 -7.6c 9 3:1
38/37 -3.8c 3 3:0
39/20 +6.2c 69 3:11
39/22 +7.8c 59 5:9
39/23 -2.5c 55 1:9
39/25 +3.2c 46 4:7
39/28 +7.0c 34 4:5
39/29 -3.8c 31 1:5
39/31 -2.6c 24 0:4
39/32 -7.5c 21 3:3
39/34 +4.2c 14 2:2
39/35 +4.0c 11 5:1
39/37 +7.8c 5 5:0
39/38 -5.0c 3 3:0
40/21 -1.1c 67 1:11
40/23 +8.0c 57 3:9
40/27 -2.9c 41 5:6
40/29 +6.7c 33 3:5
40/31 +7.9c 26 2:4
40/33 -0.3c 20 2:3
40/37 +1.6c 8 2:1
40/39 -6.2c 3 3:0
41/21 +8.3c 69 3:11
41/22 -5.6c 65 5:10
41/23 +0.8c 60 0:10
41/24 -6.2c 56 2:9
41/25 +6.4c 51 3:8
41/26 +5.2c 47 5:7
41/27 +6.5c 43 1:7
41/28 -6.4c 40 4:6
41/29 -0.5c 36 0:6
41/30 +7.5c 32 2:5
41/31 +0.7c 29 5:4
41/32 -4.3c 26 2:4
41/33 -7.5c 23 5:3
41/34 +7.4c 19 1:3
41/35 +7.3c 16 4:2
41/36 -8.2c 14 2:2
41/37 -5.6c 11 5:1
41/38 -1.8c 8 2:1
41/39 +3.2c 5 5:0
41/40 -7.3c 3 3:0
42/23 -7.5c 63 3:10
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64/55 -4.3c 16 4:2
64/57 +0.5c 12 0:2
64/59 +7.5c 8 2:1
64/61 -0.2c 5 5:0
64/63 -6.1c 2 2:0

--- In tuning@y..., Rick Tagawa <ricktagawa@e...> wrote:
> Following the piano stretching thread it dawns on me that the 13th
limit can admitted into the 72-tET
> by the umbrella of "stretched" tuning. By substituting Note 51
which is 9.4¢ too sharp for note 50
> which is 7.2¢ too flat . . . then we have a 13th which may not
be "ambiguous" at all?????

I'm not following you. 72-tET is consistent through the 17-limit,
which includes 13. That assumes "note 50". By changing to "note 51",
the potential for ambiguity can only increase.

Regarding octave-equivalence, I recently posted on the tuning-math
list the optimal stretching for 72-tET using an integer limit of 12.
If you'd like me to repeat this calculation for an integer limit of
13 or 14 or 15 or 16 or 17 or 18, ask me on the tuning-math list.
>
> Or perhaps you could use note 50 for the 13th whenever it occurs
below middle C and 51 whenever you
> want to use it above middle C?

Ouch -- 1217¢ octaves -- would be painful with some timbres.

--- In tuning@y..., Rick Tagawa <ricktagawa@e...> wrote:
> Following the piano stretching thread it dawns on me that > I would
normally have finished posting Dave Canright's calculations of just
intervals on my webpage

There's a problem with these calculations. Most of these ratios are
too complex to hear as such when played as dyads (in fact many are
identical in 72-tET), and would only become acoustically significant
in larger chords. But even some of the simpler dyads fail to combine
into triads in the way that they should. For example:

> 13/8 +7.2c 50 2:8

> 19/13 +7.0c 39 3:6

> 19/16 -2.5c 18 0:3

19/16 being 18 steps means 19/8 is 18 + 72 = 90 steps.

Meanwhile, stacking a 19/13 on top of a 13/8 would result in an outer
interval of 50 + 39 = 89 steps.

But stacking a 19/13 on top of a 13/8 should result in an outer ratio
of 19/8. (19/13 * 13/8 = 19/8).

This is what I mean when I say that 72-tET is only consistent through
an odd limit of 17.

So my advice (with highest respect for Canright's music and
theoretical work) is to chuck the part of the table from the first
instance of the number 19, to the end. You're not going to be able to
hear those intervals as dyads anyway, and they won't combine
correctly in larger chords.

>
>
> So my advice (with highest respect for Canright's music and
> theoretical work) is to chuck the part of the table from the first
> instance of the number 19, to the end. You're not going to be able to
> hear those intervals as dyads anyway, and they won't combine
> correctly in larger chords.
>

Just wanted to clarify the fact that Dave's tables were created to help me with my mapping of
various intervals possible with the 72-tET and I doubt if he's used the 72-tET himself. I just
don't want anyone to get the false impression that Dave is in any way endorsing what he calls
"ambiguous" approximations of just intervals.

As far as the 13th is concerned, Paul's clarafication is really helpful. However, I was thinking
about an isolated and very occasional use of the 13th if at all. In the past, I've tended to
approach the 13th and 11th much like a peak for a mountain climber, trying to feature it but once
in a piece. So whereas, I can now see the problem of integrating the 13th as part of the fabric
of music using 72-tET, I think I may be able to find a way to use it on occasion, say 2+ octaves
above middle C supported by a fundamental 2+ below middle C.

RT

--- In tuning@y..., Rick Tagawa <ricktagawa@e...> wrote:
> However, I was thinking
> about an isolated and very occasional use of the 13th if at all.
In the past, I've tended to
> approach the 13th and 11th much like a peak for a mountain climber,
trying to feature it but once
> in a piece. So whereas, I can now see the problem of integrating
the 13th as part of the fabric
> of music using 72-tET, I think I may be able to find a way to use
it on occasion, say 2+ octaves
> above middle C supported by a fundamental 2+ below middle C.
>
Sounds like that would work very well! I'd still use the 50/72
approximation for the 13th harmonic.

I look forward to hearing your music!