source file: mills2.txt Date: Thu, 5 Sep 1996 13:42:41 -0700 Subject: From McLaren From: John Chalmers From: mclaren Subject: disagreements about notation -- Paul Rapoport posted a reply to some of my (probably ignorant)=20 comments on his article in Xenharmonikon 16, "The Notation of Equal Temperaments," published in 1995, pp. 61-85. Paul's article dealt with the intractable issue of designing coherent and systematic notations for the full range of equal temperaments. No small accomplishment, that. Paul has succeeded in producing a notationally coherent set of set of conventions that will theoretically allow the systematic notation of any equal temperament. Pauls' achievement is considerable, and the article valuable, for three reasons: [1] Paul appears to be the first theorist who has put together an integrated & coherent notation which can apply not just to a limited range of ETs, but to all of 'em; [2] He has generalized Blackwood's procedures (as when B'wood used every other note from 26 to notate 13) to regularize the notation of equal temperaments in which either the limma =3D the apotome, or in which the neutral mode is the=20 predominant melodic paradigm; [3] Paul's approach is open-ended enough and flexible enough to admit of considerable expansion, potentially to other systems than those he considers in his article. All together, Paul's article "The Notation of Equal Temperaments" is among the most impressive and theoretically dextrous=20 written on the subject. That said, permit me to respond to some of Paul's wise and insightful comments: Paul posts: "1. Brian says that I don't. The basing of a notation system on fifths is all I did, whether or not those fifths are good or bad, or whether an ET has fifths at all. For those that don't (e.g. 13-tET), I proposed a general solution to notate them within the same framework." This is an excellent point, and it cleverly dodges the question of what's an "implicit presumption." My experience is that when you notate music on a five-line staff (never mind the names of the notes or exact shape of the accidentals), the immediate presumption created in the minds of the average musician is: "Oh, this is familiar! Here's a perfect fifth, and here's a major second, and...I guess there are some odd accidentals, but this is really just like 12, isn't it?" No, 13-TET doesn't sound a bit like 12. The problem with notating highly xenharmonic equal temperaments like 13 or 23 on the five-line staff is that the 5-line staff itself carries baggage. It automatically seduces the unwary observer into assuming that you've got a perfect fifth, major 3rd, &c. This is an=20 *implicit presumption.* Use the five-line staff *at all*=20 and you can't escape creating this illusion. Paul posts: "2. The article shows how to do this. Of course many may prefer the method for ETs with recognizable fifths; see point 1." =20 Paul & I disagree over what is meant by "usefully notated." Clearly, Paul has a different idea of "useful" than do I. To me, "useful"=20 notation is that which most accurately reflects the *sound* of the equal temperament and its fundamental melodic/harmonic paradigms. Paul Rapoport may mean something different-- for instance, he might mean "the notation that most clearly brings out certain abstract structures present in the tuning." =20 In any case, let me give you an example: in 21-TET a *basic* melodic mode is the neutral mode. Here you're in catch-22 if you stick with the five-line staff and conventional note-names-- because if you write the neutral mode 1 4 7 10 13 16 19 (scale-steps numbered 1 to 21) as C D E F G A B, you will be bamboozling the person who reads the notation inasmuch as s/he will assume there's a semitone twixt E and F: but in the 7-note neutral mode of 21 all successive notes have the same interval, 171.428 cents. On the other hand, if you flat the E or sharp the F, your reader will assume that the distance D-Eb or F#-G is smaller than the other steps. Again, having settled on a five-line staff and conventional note-names, these implicit assumptions are *unavoidable* regardless of which particular=20 exotic symbols are used instead of # and b...and thus these implicit assumptions (created unavoidably in the reader) render such=20 notation less than useful (again, in my opinion). I should mention that Paul Rapoport knows *vastly* more than I do=20 about notating the equal temperaments. So he'll surely answer what to me seem like serious objections. My guess is that I'm overlooking something obvious and fairly deep here.=20 Paul posts: "3. Many uses of ETs do exactly that, especially the ones that allow recognizable tonal or modal progressions. This does not mean that they grew from JI historically, which is a different and differently arguable point." =20 The key words here are "many" and "allow themselves." =20 Paul is absolutely correct in implying that some xenharmonic composers use ETs as quasi-JI arrays--Ezra Sims comes to mind, some of the music of Fokker, etc. But nowadays with retunable synths, most modern xenharmonic composers who use equal temperaments generally use 'em because they like the way each ET "sounds" or because each ET enjoys structural or melodic or harmonic properties which render it musically unique and impressive. In short, nowadays, if you want JI, you just press a button and retune your synth to JI. So this "quasi-JI" use of ETs has dropped drastically. Also, with the advent of true JI, composers can hear just how far from JI most equal temperaments sound. Most ETs sound unique--not like JI.=20 Now that all of us can hear that, there's been a steadily growing realization that each ETs is musically unique not for this or that aproximation of this or that JI array but because of its unique internal structural and modal properties... And this awareness on the part of composers and listeners alike has steadily eroded the tendency to hear any particular ET as a JI analog. (53 and 72 remain exceptions in this regard, depending on the ratios.) Paul's objection here may well relate more to theoretical than to compositional concerns. Certainly listeners react to music in ETs when we perform them in terms of: "Wow, 19 is really smooth but 17 is brilliant and steely," rather than, "Wow, 19 sounds much more just than 17 does." Paul posts: "4. Brian spends some time discussing why small intervals, e.g. 1/17 or 1/31 octave don't sound like JI, for no reason I can determine." =20 The reason is simple. Throw in a bunch of chromatic 1/31 or 1/17 octave intervals in succession and the impression of a quasi-JI construct disappears. Now that we can get all 31 or 53 or 72 or whatever tones per octave at once, my experience is that=20 xen composers/performers are much more willing to sound radically microtonal chromatic passages. Neil Haverstick is a shining example. "Hit 'em with tiny intervals" seems to be his watchword, and it's a knockout. Listen to Neil's superb "Birdwalk" if you want to be rocked back in your seat. This is very different from the "bad old days" back when the Prophet 5 rev 3 was all you had, and you had to choose "12 out of" 31 or "12 out of" 53, etc. Paul's comments in his numbered rebuttals 3 and 4 seem more appropriate to 12-out-of-X usage than=20 all-notes-of-X-at-once microtonal usage. =20 Again, for the obvious reason that lots of chromatic successive=20 notes destroy any sense of "JI-ness" *even* in a highly JI-like=20 ET like 53. Paul posts: "5. Not quite; he tried to make them as tonal/modal as possible." Paul is very close to being accurate here, but his statement is just shy of 100% correct. In fact Blackwood tried=20 to=20 make the ETs as tonal/modal as possible *in the Pythagorean framework*-- there are tonal/modal constructs entirely *non-Pythagorean.* Viz., the neutral mode used by the Kwaiker of Guatamala, the Thais, etc. Blackwood's usage forces the ETs into either major or minor=20 melodic modes derived from Pythagorean theories and constructs. This doesn't work musically for more than half the equal temperaments between 5 and 33 tones per octave. Incidentally, I mentioned that upwards of 40% of=20 Blackwood's claims about the ETs were incorrect. =20 Here's the rundown: B claims that 21-TET has no recognizable diatonic mode. [Blackwood, Research Notes, NEH Grant R0-29376-78-0642, 1978-81, pg. 9] This is verifiably wrong, as you can hear for yourself. Tune up 21 & sound: MAJOR IN 21-TET: 1 5 8 10 13 17 20 MINOR IN 21-TET: 1 5 7 10 13 16 19 (notes numbered numerically from 1 to 21, Carillo-style) Your ears will end the debate. Blackwood on page 124 of his NEH Grant Report claims that in 16-TET "the effect of CEG is that of a minor triad with acceptable thirds, but a singularly peculiar perfect fifth." [Blackwood, Research Notes, NEH Grant R0-29376-78-0642,=20 1978-81, pg. 124] *Nonsense.* Tune up the triad 1-5-10 (starting at A440 Hz, these are frequencies 400 Hz - 546.417 Hz - 649.803 Hz). Does this triad sound as though it has anything *remotely* akin to a perfect fifth? In fact the triad sounds like discordant diminished chord of some kind. No perfect fifth is perceptible, Blackwood is simply wrong. But don't take my word for it--again, let your ears decide. In discussing 17-TET, Blackwood states that there is no consonant triad available. Wrong. The neutral triad formed by 1-6-11 (again, these are scale steps numbered from 1 to 17, Carillo-style) sounds just fine. =20 Blackwood claims that 19 has a recognizable diatonic scale: to me it sounds unacceptably distorted. The 2/19 of a whole-tone leading tone is the Achilles heel of this mode--it just doesn't work at all, and the 9-tone gapped mode 1 4 7 8 9 12 15 18 19 works infinitely better. But let your ears decide.=20 Lastly, Blackwood claims that 18-TET allows the construction of various chords which he writes as V, V7, etc. This is contrary to what my ears hear, to what common sense tells us. Rather than creating a sense of tonality, diminished seventh chords *destroy* a sense of tonality. Thus, for Blackwood to claim that (for example) "the V/II harmnies are both complex altered chords with roots missing" (as on pg. 213 of his grant report) and that "the chord in bar 96 is..a minor dominant ninth with a lowered fifth and lowered seventh" and that this creates any sense of tonal harmonic progression... Preposterous. When you lower the fifth as much as the 18-TET scale does, it makes no sense to speak of "dominant chords with lowered fifths." The thing doesn't sound even remotely like a dominant anything, it sounds like a diminished chord and it's discordant as hell. Listen to an 18-TET 4:5:6 triad and you'll hear instantly that there is no recognizable fifth, there is no dominant, there is no V chord, there is no sense of tonality. Blackwood has simply let the theory and the numbers carry him away to a conclusion contrary to what the ears hear. Again, don't take my word for it--play the 18-TET progression I-IV-V-I and let your ears decide whether this creates any sense of tonality or not: 12=091=096=0912 6=0913=0918=096 1=098=0912=091 I-----IV----V----I (Pitches of 18-TET notated from 1 to 18; vertically stacked numbers indicate notes sounded together.) Can you hear any kind of V chord? Is this a functional dominant? So adding up the total, we find that Easley Blackwood has made statements verifiably incorrect (tune up=20 your synth, don't take my word for it) about 16, 17, 18, 19 and 21 tone equal temperament. That's 5 out of the 12 equal temperaments twixt 12 and 24: a 40% error rate. =20 Paul posts: "6. It does, but no one I know would want to perform or analyze from. Brian goes on to claim that a point which he dreamed up himself." Paul might imagine that I dreamed up this point, but the fact remains that when faced with anything other than an ultra-conservative=20 completely common-practice-period notation, Paul's interest appears to drop drastically. I've seen it happen in person. If I play Paul a notated piece, he shows real interest--but play him music notated in a MIDI file and he's apt to say, as he did to me face-to-face, "I can't make anything of this without a score." (1992, personal communication, in person.) This seems a strong bias against accepting or listening critically to music not notated in a quasi-conventional way...but then again, perhaps I misinterpret. =20 As to the question of a MIDI file or numerical notation being a notation that "no one I know would want to perform or analyze from," well, Paul knows me, and I've no problems performing or analyzing from such a notation. So the statement falls short of absolutely strict accuracy. Actually, in my experience numbers are *far* simpler than using note names.=20 Bill Schottstaedt and Julian Carillo are two obvious examples of=20 superb composers who prefer this kind of notation, and you could make a good argument that all Csound-using computer composers analyze, perform and use numerical notation all the time. So when Paul says "no one I know" prefers using numerical rather than quasi-ceonventional 5-line-staff-based notation, perhaps it's a=20 slight overstatement... Paul posts: "7. <"Essentially no one attends or gives live acoustic concerts any more."> Statements like this lead people on this forum to ignore Brian." This statement is numerically accurate. The question here is: what is meant by "essentially no one"? =20 Let's do the math. Compare the total number of hours of recorded music played last year with the total number of hours of=20 live acoustic concerts played last year. (Rock concerts don't count since they are nowadays karaoke to a DAT.) If the latter number is C and the former is B, then B/C is indeed essentially zero. Let's consider just one concrete example: DMX is a digital music channel broadcasting 24 hours a day sans commercials. You pay 10 bucks a month to your cable company for it, you get 30 channels, and it's CD-quality sound. 24 hours/day*365 days/year *30 channels * 30 million subscribers (at last count, 1995) =3D 7.884 BILLION person-hours of listening to recorded music. How does this number of person-hours compare to the total amount of live acoustic music person-hours listened to in 1995? Let's take a guess...say, 200 people per live acoustic concert, maybe 1000 concerts per day across the U.S., say 2 hours per concert, total =3D 200*1000*365*2 =3D 189,800 person-hours of live acoustic concert. Now divide live/recorded person-hours of listening: 189,800/(7.884 exp 9) =3D 0.000024 Isn't this "essentially zero"? Paul is being silly here. He's quibbling. This is as near zero as makes no difference. Quibbles about how closely 0.000024 approaches zero are obviously superfluous--obviously virtually the music listened to by virtually all the people in the U.S. is recorded music. Clearly the obvious fact is that by and large 99.999% of the population no longer attends live concerts of acoustic music. Essentially all=20 music is nowadays heard off CDs, off cassettes, on the radio, or some other recorded form. To deny this is to discredit yourself by denying the facts.=20 Paul posts: "8. Sorry, no sale. Keeping to that principle of notation would imply that the only structure in 25-tET is groups of pentatonic scales. But 25 has a quite usable major 3rd and natural 7th, which staying with C D E G A won't reveal adequately." =20 The issue in this particular case is what is meant by "less=20 than useful." Clearly my meaning is different from Paul's.=20 My definition of "useful" in this case involved bringing to the surface the fundamental pentatonic mode of pitches which remain exactly the same in 5-TET through 45-TET. Paul meant something else... no doubt we're both right. Paul posts: "9. It is true, however. Several articles show why. But it is possible both to use and relieve this fact in a systematic way. My article does this, but Brian doesn't seem to recognize that." Paul is doubtless correct that I don't recognize his point. I'll have to re-read his article, clearly. Mea culpa. Paul posts: "10. <22-tET has nothing in common with traditional tunings.> Does anyone agree with this?" This has already been dealt with by Paul Erlich, who also disagrees with Paul. My claim is overstated: 22-TET has a musically recognizable semitone in common with highly=20 Pythagorean tunings, and it shares a near-just 4:5:6. Over-emphasis is often necessary to get my point across. Murmur politely and average folks will step on your face as they walk over you without noticing--shout and they'll pay attention. (Shout the truth as they'll grab for a noose, but that's a different matter...) Paul posts: "11. My solutions for these and others like them reveal exactly what he says is lacking. For example, there is no major or minor third in 14, and no perfect fifth in 13: this emerges from what I suggest be done to notate these ETs." =20 The issue here is whether the fact that the musical nature of=20 these ETs eventually "emerges" at the end of a lengthy process of elaborate notation is sufficient to do musical justice to these exotic ETs. Paul clearly feels it is, and I clearly don't. Reasonable people can disagree. Paul posts: "12. <14-tET is nothing more than two sets of 7-equal.> This comment is similar to Brian's comment about 25 being nothing more than 5 sets of 5-equal, and equally incorrect. Blackwood's 14-note etude and mine in 25 should be enough to show that." =20 Actually, Paul and I are both correct here. My intended subtext is that melodically the neutral mode is the primary melodic structure in 14, not a diatonic major or diatonic minor scale. In that context my statement is correct; Paul's point is presumably that other musically modes exist in 14 (and 25). His statement is also correct. Paul posts: "13. <35-tET is a nightmare to notate.> Anyone who reads my article is invited to determine that this is not so. If anyone wishes, I'll apply the method I've evolved to that, since I didn't illustrate it in my article. It is a curious case, but by no means dreadful." Paul is speaking notationally--I'm speaking musically. 35 is notationaly dire because it boasts two "perfect fifths" about equally far from the just 3/2. The ear thus tends to identify one or the other as "the" perfect fifth depending on context--if you use a primarily 7-TET context, the 685.714 cent fifth sounds "right" and if you use the 5-TET context, the 720-cent fifths sounds "right." My experience is that no known notation captures this musical reality. On the other hand Paul's notation surely has subtleties and advantages I've not yet appreciated..I'll have=20 to study his Xenharmonikon 16 article again. Paul posts: "14. This is too vague for me. Besides, sharps and flats aren't defined in terms of semitones, but at least initially as 7 perfect fifths (up or down), in a Pythagorean sense, anyhow. What interests me most are cases where the fifths are so far from just that the sharp and flat end up denoting either fairly large or fairly small intervals." On reflection, Paul is right. My language in this regard is too vague. Paul posts: "15. <"The use of sharps and flats in 19 is willf =D4 perverse."> There are many counterexamples to this categorical denial. Of all ETs, 19-tET is closest to 12." This is musically not true in my experience. 22 sounds to my ears much closer to 12 than 19, because the 1/11 octave interval sounds much closer to a semitone than does a 2/19 octave interval.=20 19 is a third-tone scale, with whole tones and thirds of tones. Thus, the leading tone in 19 must be either 2/19 a (too flat) or 1/19 a (too sharp); by comparison 1/11 octave works just fine as a=20 leading tone in 22. The other issue here is that notating 19 with #, b, 5-line staff & traditional note names *badly* confuses xenharmonists, because the 12-TET conditioning is so strong. I've seen it happen. Because everything is notated with *exactly* the same symbols as used=20 in 12-TET, xenharmonists stumble and stumble again when trying to interpret this...yes...this perverse notation for 19-TET. =20 Oddball accidentals or different note names (Yasser's suggestion) really seem infinitely less confusing to me and to most folks I've worked with in playing 19-TET pieces. Finally, Paul posts: "It's a discussion about matters of notation, which I see in a=20 broader context than one of convenience or necessity: a context=20 which may reveal structures and at the same time be useful for all the things notation is intended for." Wise and sensible. Incidentally, Paul, John Chalmers told me that you studied with Easley Blackwood. If this is incorrect, mea culpa.=20 --mclaren Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Fri, 6 Sep 1996 01:46 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA23125; Fri, 6 Sep 1996 01:47:58 +0200 Received: from eartha.mills.edu by ns (smtpxd); id XA23353 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id QAA00600; Thu, 5 Sep 1996 16:47:56 -0700 Date: Thu, 5 Sep 1996 16:47:56 -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