source file: mills2.txt Date: Tue, 4 Jun 1996 15:05:06 -0700 Subject: Commas From: PAULE I guess my bias on the "pythagorean thirds" issue is that while just intonation is beautiful, and open-ended tunings are bountiful, for music that goes by at a reasonable pace there is a lot to be gained from simple structures that exploit enharmonic "punning." For example, we can look at some common commas and the (equal-tempered) tuning systems that do and do not distinguish them: _5-limit_ 81/80 (four perfect fifths up, two octaves and a major third down) (syntotic comma) -- distinguished in: 15, 22, 27, 32, 33, 34 not distinguished in: 12, 19, 26, 31 (meantone systems) 128/125 (one octave up, three major thirds down) (diesis) -- distinguished in: 19, 22, 26, 31, 32, 34 not distinguished in: 12, 15, 27, 33 (divisible by 3) _7-limit_ 36/35 (two perfect fifths up, a harmonic seventh and a major third down) (septimal quarter tone) (syntotic comma plus septimal comma) -- distinguished in: 15, 19, 22, 26, 27, 31, 32, 34 not distinguished in: 12, 33 49/48 (two harmonic sevenths up, an octave and a perfect fifth down) (larger septimal sixth tone) -- distinguished in: 12, 22, 26, 27, 31, 32, 33 not distinguished in: 15, 19, 34 50/49 (an octave and two major thirds up, two harmonic sevenths down) (smaller septimal sixth tone) -- distinguished in: 15, 19, 27, 31, 33, 34 not distinguished in: 12, 22, 26, 32 64/63 (two octaves up, a harmonic seventh and two perfect fifths down) (septimal comma) -- distinguished in: 19, 26, 31, 33, 34 not distinguished in: 12, 15, 22, 27, 32 126/125 (a harmonic seventh and two perfect fifths up, an octave and three major thirds down) (diesis minus septimal comma) -- distinguished in: 26, 32, 33 not distinguished in: 12, 15, 19, 22, 27, 31, 34 225/224 (two perfect fifths and two major thirds up, a harmonic seventh and an octave down) (septimal quarter tone minus diesis) -- distinguished in: 15, 26, 27, 34 not distinguished in: 12, 19, 22, 31, 32, 33 245/243 (an octave, two harmonic sevenths and a major third up, five perfect fifths down) (larger septimal sixth tone minus syntotic comma) -- distinguished in: 12, 15, 26, 31, 32, 33, 34 not distinguished in: 19, 22, 27 The commas could have been defined in terms of different constituent intervals (such as minor thirds), and some of the results would have come out differently, but the main result is the same: all of the equal temperaments considered here are capable of some enharmonic punning, while representing other commas with a nonzero number of scale degrees. This number is not always positive -- you should be able to see that from the above. In each equal temperament, then, certain harmonic progressions will center around a smaller number of notes (a scale), while others will create an abundance of chromatic (or hyperchromatic) motion. Received: from eartha.mills.edu [144.91.3.20] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Wed, 5 Jun 1996 00:20 +0100 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id PAA28544; Tue, 4 Jun 1996 15:20:37 -0700 Date: Tue, 4 Jun 1996 15:20:37 -0700 Message-Id: Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu