source file: mills3.txt Date: Thu, 18 Dec 1997 02:23:17 +0100 Subject: Re G. Gibson in TD1265 and on the sensitivity of intervals to mistuning From: "Paul H. Erlich" >}In the more usual timbres at least, the 7:6 interval beats too much to >}be consonant, affected as it is by the unison. The 7:5 is obscured by >}the fifth and fourth. There is a slightly stronger case for the 7:4, but >}note that the first partial of the lower tone beats with the fundamental >}of the upper tone (8:7). > >Dissonance and consonance are merely relative acoustical qualities; how they >are used in music is a human cultural decision. To speak of them in such >absolute terms elevates one particular style of music to the status of divine >revelation. Medieval and Chinese music used 3-limit dyads as the basic >harmonic consonances. Later modal and then tonal Western music used 5-limit >triads as the baisc consonances; dyads were now excluded from the texture as >they sound "incomplete" in this context. It is not difficult to find examples >in Wagner, Stravinsky, and jazz where the basic consonances are 7-limit >tetrads, by virtue of all other harmonies being even more dissonant. Triads >sound incomplete in this sort of context. Some features of this sort of style >are harmonic movement by a half-octave, representing a 7:5 but forming a >cycle which quickly closes; and chromatic melodic segments which are more >than just passing figures. However, in such a style it has been difficult to >impart a sense of logic and inevitability that is present in much tonal >music. This is because > >1) The (7-limit) consonant and (even more) dissonant chords do not arise from >a fixed pattern of intervals within a scale -- in other words there is no >grammatical support for the consonant/dissonant distinctions. >2) There is no pattern of intervals which differentiates a given note as the >"tonic." >3) 7-limit chords are quite out-of-tune in 12-tet. > >My decatonic scales in 22-tet solve these and other problems. > >}Your phrase "vanishing of the syntonic comma" is unfortunate. This comma >}does vanish in the 22-tone equal, but at a value (218 cents) that leads >}to excruciatingly dissonant 'major thirds' (436 cents), minor thirds >}(273 cents), major sixths (927 cents) and minor sixths (764 cents) as >}soon as one modulates more than a few keys. > >The syntonic comma arises musically in two ways. Either it is the difference >between a 5:4 ("major third") and four 3:2s ("perfect fifths"), or it is the >difference between a 5:3 ("major sixth") and three 3:2s, octaves ignored. >These consonances must first be present in the tuning system and musical >style before the determination of the size of the comma can take place. In >22TET, there is no doubt that if the 5:4 and 5:3 are used as consonances, >they are represented by 382 and 872 cents respectively, as the just values >are 386 and 884. The 3:2 is represented by 709 cents, a value that is very >slightly better than that of 19TET relative to just intonation. In mod 1200 >arithmetic, we find: 4*709-382 = 55, and 3*709-872 = 55. Both definitions >give a syntonic comma of 55 cents, or one degree, in 22TET. > >One cannot set the syntonic comma to a fixed value (0) and then use these >relations to derive the size of the basic consonances. The consonances come >first. > >Since the syntonic comma does not vanish in 22TET, the whole diatonic system >fails, a point you don't need to keep making, if you're paying any attention >my replies. I don't disagree with you and never did. The terms "third," >"fifth," "semitone," etc. are therefore not strictly appropriate in >discussing 22TET. However, if you do choose to use them, it should be with >respect with some reasonable distortion of the diatonic scale, for example a >major scale like the Indian ma-grama 4 3 2 4 3 4 2, where five triads are >consonant, rather than a fifth-generated creature like 4 4 1 4 4 4 1, whre no >triads are consonant. Actually, the three "minor" triads in this latter scale >are close to just 6:7:9 chords, which many people quite like. But let's leave >these diatonic non-issues behind. > >}But even if one remains >}within the same key, many intervals within the tonal fabric become >}dissonant so long as fifths of 709 cents are preserved. To those who >}treasure each new whining, commatic dissonance like the goose's golden >}egg, and hate consonance with a passion, this temperament is indeed a >}gift from the gods. > >Let me make one caveat to the claim that if the syntonic comma does not >vanish, the diatonic system fails. There is an important period in Western >musical history where a major scale with a comma defect was actually used. >This was in the late medieval period, when consonant thirds and sixths were >gaining acceptance, but Pythagorean temperament was still sine qua non. The >procedure was to tune a Pythagorean series starting with, say, Gb, to obtain >the scale C Db D Eb E F Gb G Ab A Bb B C in Pythagorean intonation. Now >examine the "major" scale starting on D: D E Gb G A B Db D. In cents it is 0 >204 384 598 702 882 1086 1200. The I, iii, IV, V and vi triads are only 2 >cents off just intonation! The ii triad is quite out-of-tune, however, and >would have to be avided in such music, and modulation is not possible. But >this was a period before triadic harmony had not yet established itself, >though the diatonic scale had long reigned supreme and continued to do so. >Melodically the difference between the two sizes of whole tone is just a bit >too small to be disturbing, and so it seems likely that this scale could have >been the ma-grama of ancient India -- in terms of a 22-tone Pythagorean >tuning, the structure of this scale is indeed 4 3 2 4 3 4 2. Indian music >uses a melody against a tonic drone, so the out-of tune intervals between the >second and fourth and between the second and sixth never occur harmonically. > >}22-tone equal is to observe that the three consonant cycles (those of >}the fifth, major third and minor third with their inversions) are >}incommensurable. > >Introduce the septimal harmonies and the cycles become "commensurable" again, >if I may continue your misuse of the word. See my paper. > >}Of >}course the centre of 22-tone equal in India is nowadays supposed to be >}the South. > >Where did you get this idea? > >}Tuning now to a different issue, you appear to assume that because the >}historic Arab theorists - most of them more concerned to interpret >}ancient Greek theory than the practice of their own musicians - >}prescribe or record a tuning, all are Neanderthals who do not fall down >}before this tuning in awe. I certainly do not hold Arabic music in >}contempt. > >As I recall, you began on this list by attempting some selective >interpretation of Arab theorists -- anything that supports 1/3-tones, you >like. I am speaking simply as a musician who listens. I listen closely -- at >blues jams I will often echo a singer or other soloist note-for-note >including all microtonal inflections and slides. In high school I trained >myself with a computer to infallibly recognize randomly played 31-equal >intervals, though I must admit to falling short of Johnny Reinhard's >1200-equal abilities. The simple fact is that Arabic music often divides a >minor third into two equal parts. > >}I am aware of the cents in a (12-tone equal) tone. Difference of opinion >}need not give rise to random unpleasantness. > >Just curious, what is this in reference to? > >}I had much rather be called a 'popularist' than an elitist. > >Yes, I thought it a more accurate description. > >}I do not deny that non-Western traditions use intervals different from >}those commonly used in the West. Thank God that they do! >}I do however categorically deny that any singers whatever (allowance >}made for the occasional alien with 31-tone equal ears) >}can reliably reproduce intervals narrower than 55-60 cents in melody. > >Partch and several fine interpreters of his music have produced many >counterexamples, now available on CD. > >}I wish to clarify my statements regarding the tone of 22-tone equal. > >}This temperament does not merge 10:9 (182 cents) and 9:8 (204 cents) in >}the sense that it does not merge the two at a value intermediate between >}the two( as _do_ the harmonically usable temperaments) > >Correction: diatonically useable temperaments > >}but it does merge >}them at a value quite divergent from both, namely at 218 cents. So in >}one sense it does not merge (or compromise between) the two species of >}tone, but in another sense it does merge them, at a very unfortunate >}excentric value. > >How you can say it merges them is beyond me. The "9:8" arises from >consonances as two perfect fifths, and the "10:9" arises as a major third >minus two perfect fifths. Thus the difference between them is, by definition, >a syntonic comma. The syntonic comma, we have seen, is represented by 55 >cents in 22-equal. The "10:9" is represented by 164 cents, and the "9:8" by >218 cents, in 22-equal. > >}All harmonically usable temperaments > >Again, this should say "diatonically usable" > >}: 12- 19- 31- 43- 50- 55-tone equal, >}etc merge the two species of tone at a value intermediate between 10:9 & >}9:8. This permits the cycles of the fifth, major third and minor third >}to be integrated within a single coherent system. These temperaments >}divide the tone into 2,3,5,7,8 & 9 equal parts respectively. > >}It is interesting that no usable temperament divides the tone into four >}equal parts. Both 22- and 24-tone equal do so, and both are dogs, though >}for very different reasons. There is also a gap in the sequence at 6. > >Why is 24TET unusable? > >And what about 26TET, where the syntonic comma truly does vanish? You feel >that the ratios of 3 (3:2, 4:3) are more sensitive to mistuning than the >ratios of 5 (5:4, 6:5, 5:3, 8:5), and therefore 26-equal would be out of the >question. However, Harry Partch was convinced that the opposite was true, and >rational approximation theory, with a suitable musical interpretation, quite >agrees with Partch. My ears find evidence for both points of view: at first >the prominent beating of out-of-tune ratios of 3 is more disturbing, but >ultimately it is not as detrimental to the musical meaning of the intervals >as an equal amount of mistuning in more complex ratios. This latter fact >explains why 19TET is much better than 12TET for 5-limit harmony; weighting >the ratios of 3 much more than the ratios of 5 would make 12TET look better. >One gets used to the prominent beating of the ratios of 3 in 19TET and 22TET, >and misses it when it's gone. The beating in 26TET is only a shade faster. >Many musical cultures use ratios of 3 in musically significant ways but >mistune them relative to JI by up to about 20 cents. The West mistunes ratios >of 5 by similar amounts in attempting to use 12TET. On balance, treating all >consonances as equally susceptible to mistuning may make the most sense, >especially in a context of complete harmonies, where the more complex ratios >can be more easily understood by virtue of the way they combine. I did so in >deriving 2-3 cents as the optimal amount of stretching for 19TET; unlike >Gregg, I used squared error as the measure of dissonance, the appropriateness >of which I recently discussed. SMTPOriginator: tuning@eartha.mills.edu From: Gregg Gibson Subject: Practical Bases of Just Intonation PostedDate: 18-12-97 03:19:01 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: 18-12-97 03:16:56-18-12-97 03:16:56,18-12-97 03:16:33-18-12-97 03:16:34 DeliveredDate: 18-12-97 03:16:34 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 C1256571.000C8764; Thu, 18 Dec 1997 03:18:47 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA18616; Thu, 18 Dec 1997 03:19:01 +0100 Date: Thu, 18 Dec 1997 03:19:01 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA18603 Received: (qmail 6742 invoked from network); 17 Dec 1997 18:18:58 -0800 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 17 Dec 1997 18:18:58 -0800 Message-Id: <3498EA05.140F@ww-interlink.net> Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu