source file: mills3.txt Date: Fri, 9 Jan 1998 19:32:06 +0100 Subject: 22TET From: Carl Lumma >>"Special Way?" You are a Yes fan, you must be a Genesis fan too :) Actually, I haven't really heard much Genesis. Since the 70's were before my time, I kinda miss the flow of stuff. I got the YES and ELP down, tho... >>Wow, I strongly disagree. Virtually every music theorist who does not care >>about tuning issues understands harmony in terms of the diatonic scale and >>tonal functions. "Functional harmony" -- ring a bell? Unfortunately, >>virtually every alternative-tuning theorist does not so understand harmony. >>This is unfortunate. It seems that "music theory" is divided into two >>schools, and Carl is perhaps only familiar with the alternative-tuning >>school. There is much of value in the other school, Carl. I guess what I meant is, the idea of generalizing the rules for use outside of the diatonic scale is new. I had one semester of High School and three semesters of Conservatory music theory, and one semester of Conservatory composition. The composition class was all about wacky stuff, since the prof was into wacky stuff. In the theory classes, we did learn the rules of functional harmony, and maybe there is an analog for most criteria in your "generalized" functional harmony... >>}The root of 2 part is understandable, considering that we need strong low >>}identies for our 7's to work. This seems contradictory to the rule that >>}the higher identities are more sensitive to mistuning, since there are more >>}low-numbered fractions near them. >> >>I don't see any contradiction. There are apparently two ideas at work here... A) That higher limit intervals are more sensistive to mis-tuning "since they are more apt to be confused with other intervals". B) What I call Tonality, and what Partch calls "Observation One". An effect created by the harmonic series, such that the number of pitches a given pitch will harmonize with is inversely proportional to the odd limit of the given pitch. Idea "A" seems to suggest that we temper the octave the most, the fifth the next most, and so on up. But we can't do this without wrecking tonality, since the low identities provide that, as explained in idea "B". Both A and B are really one idea, in that they're both caused by the way superparticular fractions get closer to eachother, so there's no "contradiction", true. But when tempering, it presents a trade-off situation, one which can't be well-addressed by just making the de-tuning inversely proportional to limit. >>}Paul's paper addresses this by making the standard deviation in log->>}frequency detuning inversely proportional to the limit of >>}the interval. >> >>That is not correct. I do offer this as an alternative model, but the first >>model, in which the standard deviation is constant for all intervals, is the >>one which yield the candidate tunings: I don't have the charts! But it says that the candidate tunings are the same, except that 22 & 26 are no longer better than 12 at the 5-limit. Nowhere do you say your candidates must be better than 12 at the 5-limit. >>}So the list of scales comes down to 22, 26, 27, and 31 tone equal >>temperament. >> >>Can you suggest a way to make the paper less confusing on this point? On the point of what scales are in the candidate list? >>}The example of the diminished 5th is given, but why it should be >>}considered a type of 5th, or why the P5 should not be considered a >>}7th is not made clear. >> >>Are you serious? Count scale steps. Anyone in a traditional theory class >>could answer this blinfolded; perhaps I presumed too much of the traditional >>theory background when writing this paper. 1) Conventional theory is full of holes. The tritone is spelled as an augmented forth in certain contexts, and as a diminished fifth in others, but it's the same pitch. All the enharmonics should be thrown out in 12. But that's just my opinion. Why you consider it a type of fifth in your example was, in any case, not clear to me. 2) The smallest step in 12 is the semitone. Counting semitones, the tritone and P5 do not share the same number of steps. Thus, the tritone is not a characteristic dissonance of the 5th in semitones. So what kind of step did you mean? I would expect a definition of "step" that would be good for all the temperaments the paper was looking at. Funny enough, I actually understand what you were trying to do with these characteristic dissonances in your criteria, but the lack of a robust definition from the beginning really hurt me. >>By the way, all the numbering and lettering in my paper is screwed up, thanks >>to Microsoft. I noticed that. But why is it Microsoft's fault? Did you used some automatic numbering scheme? I hate that trend in software nowadays! It's not already so easy that you can't do it yourself? >>I think 11TET would work much better for dissonant serialism than >>12TET, since 11TET is the most effective tuning for random >>dissonance, and 11TET is a subset of 22TET. You just had to get that in :~) >>}Far out avant garde music is certainly lacking of Special Way, if >>}not other things. >> >>And I do love far-out avant garde music like Henry Cow, when I'm in the mood for it. I love Phish when they go far out... I actually had written that there's a time and a place for far out stuff, but I deleted it for style considerations. Do you mean Henry Cowell? I got a great Cd with Set of Five, Four Combinations for Three Instruments, Hymn and Fuguing Tune #9, and Trio in Nine Short Movements. Great stuff. And Junta! >>I think 22TET can take you a lot farther out than 12TET. Some of the far out music being done at Conservatories nowadays (like what the prof of my composition class and his graduate students were doing) really doesn't tune at all... >>You did not address the rest of my paper, such as my demonstration that 22TET >>is virtually the best tuning for the decatonic scales. It may seem circular >>since I found the scales in 22TET, but one might have hoped for a better way >>of tuning them, which (perhaps unfortunately) does not exist. When I looked at the clock, I knew it was time to send. >>Modulatory effects only possible in JI -- can you give a specific example? I'm not the best person to ask, since the only JI I've done is on the Cosmolyra (everything in root position) and on an conventional organ tuned justly (only one key). It just stands to reason that there are all kinds of effects, probably most of them un-discovered, based on the "anomalies" of JI. The Ben Johnston thing does come to mind, this quote of his appearing in one of your ancient posts... "I asked myself, suppose in writing some of the really very intense slow movements that he did, Beethoven had not had the tempered scale to work with, but had instead just intonation, what might that music have been? Then I tried to write that piece, as an exercise for myself. It's actually part of a piece--I liked it well enough that I put it into a piece. But at first of course it's just to find out. The richness is one thing, the flexibility is another. The typical Beethoven progression, for example will cause you to drop by a microtonal amount every time the progression repeats itself. What I did was to work it out that the pitch would drop, gradually, until the whole thing drops almost a half step, and then I force it to come back up by working out progressions that would make it do that. That is, of course, quite different from what Beethoven was dealing with." I've not heard the results of his efforts. Probably the greatest expert on modulation in JI who ever lived was Partch, and I'm sure I've heard effects not possible in temperament in Rotate the Body and Delusion. Bill Alves has said... >Functionally, it may make sense to Paul to interpret the 22TET intervals as >he has done above. However, I definitely hear the 10 step interval as being >much more like an 11/8 (551 cents) than a 27/20. The 11/8 can have a >beautiful blue-note quality, which I myself have exploited on occassion. >The 19 step interval is almost dead-on a 20/11 (1035 cents), whose >inversion, by the way, is a 5/4 down from the 11/8. Here I publish an analysis of 22TET from a larger work of mine, still in progress. It lists low-numbered rational approximations of 22TET intervals, all within 7 cents deviation or better. The number of steps is listed on the left, with the cents between that step and the tonic, rounded to the nearest whole cent, appearing to the right.... 1- 55 2- 109 3- 164 11/10 (1) 4- 218 17/15 (1) 5- 273 7/6 (6) 6- 327 23/19 (4) 7- 382 5/4 (4) 8- 436 9/7 (1) 9- 491 10- 545 11/8 (6) 11- 600 31/22 (6) 12- 655 13- 709 3/2 (7) 14- 764 15- 818 16- 873 17- 927 18- 982 19- 1036 20- 1091 15/8 (3) 21- 1145 31/16 (0) 22- 1200 ..Even more to the right appear the rational approximations, with intervals listed only once (as otonality or utonality), so as to reveal the structure of the scale, if there is any. The cent deviations are in () to the most right, and are of course the same for the inversions. I used my own judgement for balancing cent deviation, prime limit, and odd limit when picking the rational approximations. Bill Alves assertion that the 11/10 is a 5/4 down from the 11/8 is confirmed by, and consistently represented in the above scheme. Please Note: How the intervals are heard depends on context!!! I'm just throwing in my analysis here because sometimes a particular context seems to be assumed when giving rational approximations for the 22TET intervals... Carl SMTPOriginator: tuning@eartha.mills.edu From: Prent Rodgers Subject: MIDI/Audio wish list PostedDate: 09-01-98 20:36:03 SendTo: CN=coul1358/OU=AT/O=EZH ReplyTo: tuning@eartha.mills.edu $MessageStorage: 0 $UpdatedBy: CN=notesrv2/OU=Server/O=EZH,CN=coul1358/OU=AT/O=EZH,CN=Manuel op de Coul/OU=AT/O=EZH RouteServers: CN=notesrv2/OU=Server/O=EZH,CN=notesrv1/OU=Server/O=EZH RouteTimes: 09-01-98 20:35:37-09-01-98 20:35:38,09-01-98 20:35:30-09-01-98 20:35:30 DeliveredDate: 09-01-98 20:35:30 Categories: $Revisions: Received: from ns.ezh.nl ([137.174.112.59]) by notesrv2.ezh.nl (Lotus SMTP MTA SMTP v4.6 (462.2 9-3-1997)) with SMTP id C1256587.006B9E9C; Fri, 9 Jan 1998 20:35:58 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA17601; Fri, 9 Jan 1998 20:36:03 +0100 Date: Fri, 9 Jan 1998 20:36:03 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA17551 Received: (qmail 14386 invoked from network); 9 Jan 1998 11:35:52 -0800 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 9 Jan 1998 11:35:52 -0800 Message-Id: <5030050002886855000002L552*@MHS> Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu