Re G. Gibson in TD1265 and on the sensitivity of intervals to mistuning

>}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
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