source file: m1455.txt Date: Mon, 22 Jun 1998 14:14:36 -0400 Subject: Re: Commas and Consistency From: "Paul H. Erlich" This message is in MIME format. Since your mail reader does not understand this format, some or all of this message may not be legible. ------ =_NextPart_000_01BD9E09.A133B37A Content-Type: text/plain Jonathan Walker, Your formulae for commas in equal temperaments are equivalent to Paul Rapoport's in various published articles. The consistency issue which Graham Breed is referring to involves the traditional definition of the syntonic comma as the difference between an octave and a major sixth up, and three perfect fifths up. Many equal temperaments have a better approximation to the major sixth than the best approximation of a major third plus the best approximation of a perfect fourth. Therefore two distinct syntonic commas can be defined for such equal temperaments. 5-limit consistency means that the best approximations of the perfect fifth/fourth, the major third, and the major sixth are all compatible. Given 5-limit consistency, then, there can be only one syntonic comma. ETs that are not 5-limit consistent include 11, 17, 20, and 21. 64tET is interesting in that the triads defined as you or Paul Rapoport would define them are not even second-best, they are third-best (i.e., altering either the major third or the perfect fifth yields a minor third [or major sixth] that is so much better that the entire triad is improved). Paul Erlich ------ =_NextPart_000_01BD9E09.A133B37A Content-Type: application/ms-tnef Content-Transfer-Encoding: base64 eJ8+IiwSAQaQCAAEAAAAAAABAAEAAQeQBgAIAAAA5AQAAAAAAADoAAEIgAcAGAAAAElQTS5NaWNy b3NvZnQgTWFpbC5Ob3RlADEIAQWAAwAOAAAAzgcGABYADgAOACQAAQAyAQEggAMADgAAAM4HBgAW AA4ADgArAAEAOQEBCYABACEAAAAzQkVBMkNCOEUyMDlEMjExQUNEMjAwODA1RkJFM0MwNQBBBwEE gAEAGwAAAFJlOiBDb21tYXMgYW5kIENvbnNpc3RlbmN5AHYJAQ2ABAACAAAAAgACAAEDkAYAbAgA AC4AAAADADYAAAAAAAMABIAIIAYAAAAAAMAAAAAAAABGAAAAAFKFAAC3DQAAHgAFgAggBgAAAAAA wAAAAAAAAEYAAAAAVIUAAAEAAAAEAAAAOC4wAAMABoAIIAYAAAAAAMAAAAAAAABGAAAAAAGFAAAA AAAACwAAgAggBgAAAAAAwAAAAAAAAEYAAAAAA4UAAAAAAAALAA+ACCAGAAAAAADAAAAAAAAARgAA AAAOhQAAAAAAAAMAEIAIIAYAAAAAAMAAAAAAAABGAAAAABCFAAAAAAAAAwARgAggBgAAAAAAwAAA AAAAAEYAAAAAEYUAAAAAAAADABSACCAGAAAAAADAAAAAAAAARgAAAAAYhQAAAAAAAB4AJIAIIAYA AAAAAMAAAAAAAABGAAAAADaFAAABAAAAAQAAAAAAAAAeACWACCAGAAAAAADAAAAAAAAARgAAAAA3 hQAAAQAAAAEAAAAAAAAAHgAmgAggBgAAAAAAwAAAAAAAAEYAAAAAOIUAAAEAAAABAAAAAAAAAAIB CRABAAAAWQMAAFUDAABFBQAATFpGdVeEHuwDAAoAcmNwZzEyNdIyAPszNgHoIAKkA+MJAgBjaArA c2V0MDYgBxMCgH0KgAjIIDubCW8OMDUCgAqBdWMAUL8LAwzQAcEI0ABBC2BuDhBIMDMzC6YgSgIg YVZ0EOADoFcHQGsEkCxXCqIKhAqAWQhhIAIQclRtdQtgZRiSIAWgbYcAwAQgC4AgZXF1B0CgIHRl bXAEkGEHgJ8CMAQgCsAZEBoRaXYHQMMa8RpgbyBQYRjgB/DYYXBvHNAAICcZwxvAZwUQCGAEIHB1 AmAEAGgbCYAbMXQN4Bvgcy4gLlQecBlhAIFzGnBuY455GdAEEApQIHdoDeAYaCBHGsAQ4G0gQtcJ 0R6QBAAgCXBmBJAFENcVoBwiC4B2BvB2B5EW0OUZEHQawGRpHtAWoQMgRwEBC4Aj8yBvZiNjc/55 AjACIA3gGWQbMCNUI+CvASAEkCABGRBiESB3CeF7GzAlAWMBkCMwKBEekSAtAMBqBbEAkHgW0CB1 /HAsKLMW0AnRHgAEkCIwfyhgGJAGkBbQBCAp4B8wTf8AcCAwGh8EIBDgKJInkhpw6wXAHMBwA2B4 B3AWwCTi/xwxI3IpKhbTI3InoB/gLk33JSEpBhbQaQsgHgAKQCNU/zD/MgUqxwhhFtAfMyIRBbD9 I5F3HEAj4B/gC4ArESWc/QQgYwORJ6AkZB6BGTIgcKcg4SxPHyE1LR5AbSPw/x+LB4AGIhbRHBEz nyQCBCD7JSUqyy81xCoAL2gy0yoG/y+MG0IHQAMgGXEKsB7QAmD6ZR8wRxuwJ/E7fyAgQJPebkCT G1E4tQIgbCAwAiAzJX4fMEVUPPUbQm5vPwVARK8FQDdhCkABACAxcjEqADE3KgAB0CoEMuIxHzA2 NHRIsCHSC4B/LhE9sSJyGeE9FyOwBzBk9wQgOSYZsXkIYCUQBcAca/sgoAhgbB6QOSQjYiFgSUbO ZURiERAfkWQtPaJAkw8gMBtCMtNTgyAoaS7bRBAqAWwuESJyZSPwNlH/QK5QEj8fT9AIkFFgGyEp ENsLgDKnWwWxKSldPQQh4f5zHEAY0CDhLeU9FxrxMvAfTqVNAxqQA2AjMGQpLvMXmhxjRXIeQBDQ FTEBQAsXoxHRAGFAAAAAAwAmAAAAAAADAC4AAAAAAAsAAgABAAAAHgBwAAEAAAATAAAAVFVOSU5H IGRpZ2VzdCAxNDU0AAACAXEAAQAAABsAAAABvZ31Hui4LOeQCeIR0qzSAIBfvjwFAASzW0AAQAA5 AKDCAp0Jnr0BAwDxPwkEAAAeADFAAQAAAAYAAABQQVVMRQAAAAMAGkAAAAAAHgAwQAEAAAAGAAAA UEFVTEUAAAADABlAAAAAAAMA/T/kBAAAAwCAEP////8CAUcAAQAAADMAAABjPVVTO2E9IDtwPUFj YWRpYW4tQXNzZXQ7bD1NQVJTLTk4MDYyMjE4MTQzNlotOTAyNQAAAgH5PwEAAABLAAAAAAAAANyn QMjAQhAatLkIACsv4YIBAAAAAAAAAC9PPUFDQURJQU4tQVNTRVQvT1U9QUFNL0NOPVJFQ0lQSUVO VFMvQ049UEFVTEUAAB4A+D8BAAAADwAAAFBhdWwgSC4gRXJsaWNoAAAeADhAAQAAAAYAAABQQVVM RQAAAAIB+z8BAAAASwAAAAAAAADcp0DIwEIQGrS5CAArL+GCAQAAAAAAAAAvTz1BQ0FESUFOLUFT U0VUL09VPUFBTS9DTj1SRUNJUElFTlRTL0NOPVBBVUxFAAAeAPo/AQAAAA8AAABQYXVsIEguIEVy bGljaAAAHgA5QAEAAAAGAAAAUEFVTEUAAABAAAcwRtECnQmevQFAAAgwerMzoQmevQEeAD0AAQAA AAUAAABSZTogAAAAAB4AHQ4BAAAAFwAAAENvbW1hcyBhbmQgQ29uc2lzdGVuY3kAAB4ANRABAAAA LgAAADxCRTFCRkQwNzM5RDhEMTExQUNDODAwODA1RkJFM0MwNTE0OTVFNkBNQVJTPgAAAAsAKQAA AAAACwAjAAAAAAADAAYQA43eaAMABxDAAwAAAwAQEAAAAAADABEQAAAAAB4ACBABAAAAZQAAAEpP TkFUSEFOV0FMS0VSLFlPVVJGT1JNVUxBRUZPUkNPTU1BU0lORVFVQUxURU1QRVJBTUVOVFNBUkVF UVVJVkFMRU5UVE9QQVVMUkFQT1BPUlRTSU5WQVJJT1VTUFVCTElTSEUAAAAAAgF/AAEAAAAuAAAA PEJFMUJGRDA3MzlEOEQxMTFBQ0M4MDA4MDVGQkUzQzA1MTQ5NUU2QE1BUlM+AAAAo9o= ------ =_NextPart_000_01BD9E09.A133B37A--