Stoney's "Theoretical Possibilities for Equally Tempered Systems"

Of the several published evaluations of equal temperaments, most evaluate
approximations to various just intervals without regard to whether these
approximations would contradict one another in the context of triads (or any
larger chords). This is the consistency issue I've brought up many times
before. As an example, Max Meyer (and more recently, Joe Monzo) have
suggested 24-tone equal temperament (quartertones) as a way of achieving
better 7-limit harmony. The 7:4 is only 19 cents sharp of 4 3/4 tones, 7:5
is only 17 cents flat of 3 tones, and 7:6 is only 17 cents sharp of 1 1/2
tones. But this will not lead to better 7-limit tetrads. For example, adding
the 4 3/4-tone interval above the root of a 4:5:6 (major) triad (tuned as in
12-equal) leads to a representation of 7:5 which is an unacceptable 33 cents
flat. Using 12-equal, the 7:5 is only 17 cents flat, but the 7:6 is 33, and
the 7:4 is 31, cents sharp.

Wendy Carlos, Yunik & Swift, and Ezra Sims are among those who don't appear
to have realized the implications of inconsistency. When I met Sims and we
discussed the issue, he admitted that I raised a good point but told me that
triads went out a long time ago. I don't know what he meant, since much of
his music uses more than two voices. On the other hand, William Stoney's
evaluation starts by constructing a complete harmonic chord (3, 5, 7, and
15-limit) in each equal temperament, and calculates the approximation
quality of each of the intervals in that chord. Since the complete chord is
constructed first, there is no possibility for the approximations evaluated
to be inconsistent with one another. This means, of course, that in equal
temperaments that would normally be considered inconsistent in the given
limit, some intervals are approximated with an error larger than half a
step.

Stoney's method of constructing the complete chord is interesting. He
considers the approximations of three consecutive harmonics, starting and
ending with even ones, to be "proportionally spaced" if the middle tone is
closer (in cents) to the higher tone than to the lower. "Neutral spacing"
occurs if the middle tone is halfway between the upper and lower tones.
"Inverse spacing" occurs if the middle tone is closer to the lower tone than
to the upper tone. Stoney rejects the possibility of inverse spacing, and
subject to that constraint he essentially uses the closest approximation (in
cents) to each harmonic. For example, he evaluates 12-tone equal temperament
as follows:

"
ORDNO Cents Value HARFAC Deviation
0 0.000 8 0.000
1 100.000
2 200.000 9 -3.910
3 300.000
4 400.000 10 13.687
5 500.000
6 600.000 11 48.682
7 700.000 12 -1.955
8 800.000
9 900.000 13 59.473
10 1000.000 14 31.175
11 1100.000 15 11.732
12 1200.000 16 0.000

Deviations Summary:
8:12 -1.955
8:10:12 AVER 10.427 MAX 10:12 -15.641 MIN 8:12 -1.955
8:10:12:14 AVER 18.846 MAX 12:14 33.130 MIN 8:12 -1.955
8 THRU 15 AVER 27.111 MAX 9:13 63.383 MIN 8:12 -1.955
"

He places the 13th harmonic at 900 rather than 800 cents to acheive
proportional spacing, despite the 59-cent error involved. However, placing
it at 800 cents would lead to other large errors, such as a 13:11 than is 89
cents flat. Since he is including all intervals within the limit in his
evaluation, his "proportional spacing" criterion is (inadvertently) a good
method for penalizing tunings for inconsistency.

His conclusions:

"
1. The conventional 12-semitone temperament yields greatest purity of
intonation in the simpler intervals of the fifth and fourth, moderate
deviation for thirds and sixths, and the rather lagre deviation of 31 cents
for he seventh harmonic. The existence of the eleventh and thirteenth
harmonics is doubtful.
2. Any improvement over the 12-semitone temperament, for purposes of just
intonation, would require at least 19 degrees to the octave.
3. A 17-degree system yields good fifths (705.9 Cents) but too wide thirds
(423.5 Cents). A 19-degree system yields nearly just thirds; its fifths are
7 Cents narrower than just. A 22-degree system yields a reasonable
approximation of just intonation through the twelfth harmonic, and good
relative spacing but too sharp intonation (by 32 Cents) of the thirteenth
harmonic. The fifth is 7 cents wider than the just value.
[...]
In summary of the above, systems (in ascending order of average deviation
for HARFACS 8 through 15) of 72, 58, 53, 70, 65, and 41 degrees yield low
fifth deviation of 2 Cents or less, very good relative spacing, and
acceptable approximation of harmonics 8 through 15. A temperament of 31
degrees has merit, but the fifth deviation of -5 Cents is relatively large.
The most promising low-order systems are those of 24, 22, and 19 degrees.
For these last three systems practical experimentation over an adequate
period of time and utilizing suitable designed instruments would be required
in order to assess their respective deficiencies and merits.
"

The article was published in _The Computer and Music_, edited by Harry B.
Lincoln, Cornell University Press, 1970.

On Wed, 5 May 1999, Paul H. Erlich wrote:
> Wendy Carlos, Yunik & Swift, and Ezra Sims are among those who don't appear
> to have realized the implications of inconsistency. When I met Sims and we
> discussed the issue, he admitted that I raised a good point but told me that
> triads went out a long time ago. I don't know what he meant, since much of
> his music uses more than two voices.

Maybe he thought that by "triad" you meant traditional 4:5:6 triads only?

[mondo snippo]
> In summary of the above, systems (in ascending order of average deviation
> for HARFACS 8 through 15) of 72, 58, 53, 70, 65, and 41 degrees yield low
> fifth deviation of 2 Cents or less, very good relative spacing, and
> acceptable approximation of harmonics 8 through 15. A temperament of 31
> degrees has merit, but the fifth deviation of -5 Cents is relatively large.
> The most promising low-order systems are those of 24, 22, and 19 degrees.

Interesting to compare these results with a scan through the
consistency-level/error tables at my site.

--pH <manynote@lib-rary.wustl.edu> http://library.wustl.edu/~manynote
O
/\ "Hey--do you think I need to lose some weight?"
-\-\-- o
NOTE: dehyphenate node to remove spamblock. <*>

>> Wendy Carlos, Yunik & Swift, and Ezra Sims are among those who don't appear
>> to have realized the implications of inconsistency. When I met Sims and we
>> discussed the issue, he admitted that I raised a good point but told me that
>> triads went out a long time ago. I don't know what he meant, since much of
>> his music uses more than two voices.

Paul Hahn wrote,

>Maybe he thought that by "triad" you meant >traditional 4:5:6 triads only?

Nope -- my example was the 8:13:19 triad in 72-tET.

Paul Hahn wrote,

>[mondo snippo]
>> In summary of the above, systems (in ascending order of average
deviation
>> for HARFACS 8 through 15) of 72, 58, 53, 70, 65, and 41 degrees yield low
>> fifth deviation of 2 Cents or less, very good relative spacing, and
>> acceptable approximation of harmonics 8 through 15. A temperament of 31
>> degrees has merit, but the fifth deviation of -5 Cents is relatively
large.
>> The most promising low-order systems are those of 24, 22, and 19 degrees.
>
>Interesting to compare these results with a scan through the
>consistency-level/error tables at my site.

The agreement is not too wonderful; do you see a corroboration or a
refutation? (Stoney only went up to 72.)

On Thu, 6 May 1999, Brett Barbaro wrote:
> Paul Hahn wrote,
>>Interesting to compare [Stoney's] results with a scan through the
>>consistency-level/error tables at my site.
>
> The agreement is not too wonderful; do you see a corroboration or a
> refutation? (Stoney only went up to 72.)

I'd call it rough corroboration, with a few (potentially intriguing)
differences.

--pH <manynote@lib-rary.wustl.edu> http://library.wustl.edu/~manynote
O
/\ "Hey--do you think I need to lose some weight?"
-\-\-- o
NOTE: dehyphenate node to remove spamblock. <*>