source file: m1347.txt Date: Sat, 7 Mar 1998 12:26:14 -0600 (CST) Subject: Re: Meantone From: mr88cet@texas.net (Gary Morrison) >My ignorant question: what does "1/4 comma" mean? What's a comma? Here's the history: The earlier "pythagorean" major scale was defined by stacking exact 3:2-frequency-ratio perfect fifths: one downward from the root to define the subdominant (e.g., F in the key of C), and then five upward to define, successively, the dominant (G), supertonic (D), submediant (A), mediant (E), and leading tone (B). If you do that, you end up with an E defined as an 81:64 ratio (3:2 to the fourth power, divided by 4 to drop it down into the octave where we started) relative to the root. 81:64 however, beats against the more intuitive 5:4 (i.e., 80:64) major third. The pitch difference between the two is the syntonic comma, a ratio of 81:80. The premises behind quarter-comma meantone are that: 1. We want the mediant (the major third above the tonic) to be 5:4. 2. We want all (true) perfect fifths to be the same size. 3. We are willing to slightly compromise the ideality of the perfect fifth to make 1 and 2 possible. So the quarter-comma meantone solution is to build a scale in a circle of identical fifths that are not the ideal 3:2 ratio, but flat of that ideal by 1/4 of that 81:80 comma. That way, when you stack up each of four of those quarter-comma flat fifths, you drop the pitch by 1/4 comma for the G (in the key of C), 2/4 comma flat for the D, 3/4 flat for the A, and then when you get to the E, you are 4/4 (i.e., 1) of a comma flat of that 81:64 major third, which is (by definition) the more ideal 5:4 major third. All meantone tunings are based upon a circle of consistent-sized fifths (or more often a "broken" circle of fifths). The difference between the various meantone tunings is the size (i.e., tempering) of that fifth. That circle frequently starts on Eb, going on to Bb, F, C, G, etc., on to G#. The circle is frequently broken at the diminished sixth between G# and Eb. If that diminished sixth were treated as if it were a perfect fifth (e.g., as if it were G# to D#, or Ab to Eb, rather than G# to Eb), then it would be perceived as the "wolf" fifth, because it's WAAAAYY out of tune from the ideal 3:2 perfect fifth. The other solution is to continue the circle of fifths, allowing D# to be a different pitch from Eb, or, going the other way, Ab to be a different pitch from G#. It turns out that, purely by mathematical luck, if you continue that to a total of 31 fifths (knocking the pitches down by an octave as needed to keep them within the original octave's span) you end up very close to where you started - only about 6 cents off. (A cent is a unit of measurement for very small differences in pitch, one cent being 100th of the usual 12TET half-step.) That's another way of saying the 31TET is extremely close to the completed quarter-comma meantone system. The coincidence between third-comma meantone and 19TET is even closer still. Third-comma meantone uses a fifth tempered down by 1/3 of a comma so that the submediant (e.g. A in the key of C) has a 5:3 ratio relative to the tonic rather than 27:16 as you would get if you were to use exact 3:2 perfect fifths. It turns out that if you stack up 19 third-comma-flat fifths (dropping pitches down by octaves as needed to keep the pitches in same the octave as where you started), you end up less than 1 cent from where you started!