source file: m1442.txt Date: Tue, 9 Jun 1998 15:07:48 -0400 Subject: RE: Disappearing Commas From: "Paul H. Erlich" >Could anyone tell me - is there a unique theoretical limit at which a >comma (of different kinds?) may be said to disappear? I don't understand this--can you rephrase? >Are there >strict and not so strict definitions of which systems (or musical >sequences?) of which this can be said to pertain/occur? Is there a >single accepted definitive reference which defines the 'disappearance >of a comma'? (Helmholtz?) These are things I missed in my education. With regard to the syntonic comma and some of the "better" ETs, you probably can get quite a lot from references like Helmholtz. Otherwise, I must humbly suggest that my consistency stuff is key. This has been covered on the list before, contact me and I'll try to dig up the relevant TDs. The syntonic comma is traditionally defined both as the difference between a Pythagorean minor third and a just minor third, and as the difference between a Pythagorean major third and a just major third. When using the closest approximations to the consonant intervals (one constructs the Pythagorean intervals from chains of perfect fifths) in an ET not consistent within the 5-limit, these definitions may not agree, so the comma itself, and the question of its disappearance, may not be well-defined. Similarly, commas constructed from n-limit intervals are uniquely defined in all tunings consistent within the n-limit. Given consistency, the determination of the size of a comma in an ET is fairly trivial. In meantone tuning, the syntonic comma disappears by definition. That is, even though it may be possible to find a better perfect fifth above C than G, by extending the meantone tuning to 100 or so notes per octave, and using such a perfect fifth to define a non-vanishing syntonic comma, that's not really a kosher use of meantone. Other non-closed tuning systems may also have "canonical" representations for the consonances, and one should use these, not some distant improvements, to define the commas. Going back to meantone, one can make a case for defining the canonical 7/4 as the augmented sixth, and all septimal commas can be determined from there. But that is already in conflict with the interpretation of the dominant seventh chord as some form of 4:5:6:7. Going to higher primes (odd composites don't lead to any new commas) in meantone tuning or other non-closed systems is really questionable unless those systems were designed with those primes in mind.