TUNING digest 1398: JI Tuning Resolution

> Some time back I did some audio experiments
> to determine the extent to which I could actually
> discriminate beating in just intervals. Listening
> to rather loud synthesized sawtooth waves through
> headphones at about 440Hz, I found that the limit
> of my discrimination was at the 19 limit -- for example
> I could manage to tune a 19/13 by eliminating beats,
> but not a 21/13. So to my ear, a 21/13 a few tenths
> of a cent sharp or flat makes no distinguishable
> difference in terms of consonance, but does
> (just barely) for a 19/13.

This is very interesting to me, as I have already
hypothesized in my book that, in most musical
contexts (though certainly not in all -- see below),
19 is a kind of limit to the use of higher primes in
just-intonation, whether exactly-tuned or only
implied by tempered or lower-prime-limit just pitches.

In traditional 12-eq harmonic theory, with extended
chords built upwards in 3rds, if the near-quartertone
identities of 11 and 13 are assumed to be implied
by the "flat 5th / sharp 11th" and "flat 13th",
respectively, the rational implications of the chord-
members of a major chord line up pretty much as
follows:

27/16 13th
13/8 flat 13th

11/8 sharp 11th
4/3 11th (21/16 is another possibility)

19/16 sharp 9th (7/6 is another possibility)*
9/8 9th
17/16 flat 9th

15/8 major 7th
7/4 minor 7th

25/16 augmented 5th
3/2 5th
11/8 diminished 5th (23/16 ???)

5/4 3rd

1/1 root

* See the discussion about these two particular
intervals between Paul Erlich and and myself
re: Hendrix Chord [Tuning Digest #1376], as to
whether 19 is admissible as a rational implication
of the "sharp 9th".

Thus, the prime limit here falls around 17, 19
or 23.

The question marks beside 23/16 refer exactly to
the possible inability of primes above 19 to normally
be relevant. There is no problem with admitting
the composite [i.e., non-prime] harmonic identities
9 (= 3^2) and 15 (=3^1 * 5^1), as they both fall
within the 19-limit as lower odd-numbered identities.

If we include the 3-limit 27 (= 3^3) and the 5-limit
25 (=5^2), and possibly the 7-limit 21 (= 3^1 * 7^1),
we have gone all the way to a 27-limit in terms of
odd identities (excepting 23), but are still within the
19-limit in terms of prime factors.

However, if the next two prime factors are added
as harmonics, 29/16 falls about a quartertone
between the two 7ths, and 31/16 about a quartertone
between the "major 7th" and "octave". Traditional
theory already has a couple of 7ths to choose from
in chord-building, so these ratios fall somewhat
outside traditional harmonic concepts. So to me,
the fuzzy area lands squarely on 23.


> Figuring that the 19th partial of A440 will beat
> at 0.5 Hz if detuned by 0.1 cent, that becomes
> my desired accuracy for the goal of avoiding
> audible dissonance in a conventional pitch range...

This also provides corroboration to my observation
that, unless the music moves extremely slowly or
is designed specifically to illustrate very slight
differences in tuning (or probably both), a discrepancy
of up to even 1 cent is excusable.

> ...I can imagine musics that would want precision
> beyond that for particular effects such as very
> sustained chords with rock-solid lack of phase
> shifting...

Indeed, the extremely accurate Rayna synthesizer
utilized by La Monte Young in his entirely static
(at least on the surface) "Dream House" installations
is what finally enabled him to explore the effect of
much higher primes (up to 283 in the one currently
running).

> ...but that's another issue than dissonance.

Indeed again, the title of Kyle Gann's chapter on
Young's tunings in the book "Sound and Light"
(about La Monte and Marian) is called "The Outer
Edge of Consonance".

Joseph L. Monzo
monz@juno.com

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Oh, one other curious thing about brasses, is that their waveforms
(i.e., in the time domain) generally have a tall spike in them. That
probably means that the partials are pretty much in phase with one another.
Sums of cosine waves at whatever amplitude will produce such a spike.

So it's very easy for a brass instrument to peak a VU meter even if
their audible volume isn't as loud as would seem necessary to peak the VU.