source file: m1399.txt Date: Tue, 28 Apr 1998 18:20:33 -0400 Subject: TUNING digest 1398: JI Tuning Resolution From: monz@juno.com (Joseph L Monzo) > 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 _____________________________________________________________________ You don't need to buy Internet access to use free Internet e-mail. Get completely free e-mail from Juno at http://www.juno.com Or call Juno at (800) 654-JUNO [654-5866]