On Wed, 21 Oct 1998, M. Schulter wrote: > Possibly less well known, the septimal schisma of 33554432:33480783 or > about 3.80 cents is a bridge to a quasi-7-based world of > "superefficient" cadences and third-tone steps. It is the difference > between the Pythagorean comma and the septimal comma (64:63, about > 27.26 cents) which separates basic Pythagorean intervals from their > 7-based counterparts.
Oddly enough, I was just poking around in some old tuning list messages today, and guess what I found:
| Date: Mon, 13 Mar 95 08:52:11 -0800 | From: Manuel Op de Coul | Reply-To: tuning@eartha.mills.edu | To: manynote@library.wustl.edu | Subject: Other harmonic 7th comma | | Bosanquet has written that 14 fifths downwards (the Pythagorean double | diminished octave) is very close to the harmonic seventh. Is the | comma belonging to it, 33554432/33480783 = 2^25 * 3^-14 * 7^-1 = | 3.8041 cents ever called Bosanquet's comma, does anyone know? | | Manuel Op de Coul coul@ezh.nl
Another interval that might be a candidate for the name "septimal schisma" is the 2401/2400, about .72 cents. It is the difference between the 50/49 and the 49/48, both intervals which result fairly directly from septimal voice-leading.
--pH http://library.wustl.edu/~manynote O /\ "Foul? What the hell for?" -\-\-- o "Because you are chalking your cue with the 3-ball."
The interval 33554432/33480783 has also been named by Eduardo Sa'bat, Beta 2. Septimal schisma seems a good name to me. The bridges from Margo's post are easily found with Scala. It can take all the combinations of two intervals and check whether a given interval (some comma for example) is a sum or difference of them. The list of interval names intnam.par that is provided can be used for that. So do:
load intnam.par show combination 33554432/33480783
> An open question: might an extended Pythagorean > tuning with its contrasts of basic 3-based, > quasi-5-based, and quasi-7-based intervals in > some way have a kinship to the three genera > (diatonic, chromatic, enharmonic) or to Guido > d'Arezzo's three hexachords (soft, natural, and > hard)?
I wish I had read this while the discussion on it was hot, because I've been thinking along exactly these lines.
In my book, I demonstrate how it would have been possible for the ancient Greeks to formulate both the chromatic and enharmonic genera in their tuning of the famous "lyre with 12-strings" for which Timotheus was ridiculed, by retuning the strings continuously by 3:2 "5th"s and 4:3 "4th"s until the proper intervals were reached.
I'm not well-read on the following idea, but it's been documented that the Babylonians (c. 2000 BC) tuned by ear by means of consecutive 3:2 "5th"s and 4:3 "4th"s, and I think it's quite possible that all four of the great ancient centers of civilization -- China, Egypt, the Indus valley, and Mesopotamia -- could have either independently "discovered" extended 3-Limit tuning or adopted it thru trade from whichever developed it first. This in turn would have been incorporated into Greek practice, as the Greeks imported much of the scientific knowledge of all but China.
Knowing first-hand how far a curious mind will go with what he thinks is a great new idea, I think that when any brilliant ancient theorist first discovered the principle of generating new notes from a succession of 3:2s, it would be highly unlikely that he would stop after the 12th just because it sounded so close to the starting pitch. Indeed, the fact that it _was_ so close and yet not the same certainly would have impelled him on to see what other kinds of relationships there were. He probably wouldn't have stopped until he just plain got tired of doing all that calculating.
If he had the same kind of inquisitiveness that Darreg, Partch, and Schoenberg had, and many of us have, he would have written pieces that utilized all those unusual notes, to see just what that would sound like. So maybe there was some ancient Greek walking around with a toga that said "quasi-7-Limit Rules!".
- Joe Monzo monz@juno.com http://www.ixpres.com/interval/monzo/homepage.html
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