source file: mills2.txt Date: Wed, 19 Jun 1996 12:33:45 -0700 Subject: Re: Post from Brian McLaren From: alves@osiris.ac.hmc.edu (Bill Alves) I would like to respond to some of Brian's good points about just intervals, but before getting into specifics, I would like to talk about the musical applicability of these studies. I would imagine that if you asked subjects to state their "preference" between minor seconds and major thirds, the thirds would be a near unanimous choice. Does this mean that minor seconds should never be used in music? Also, I don't think many people since the time of Helmholtz have claimed that small-number ratios are the only criterion of consonance. Certainly I have not. Consonance, in a musical sense as well as a strictly psychoacoustic sense (if there is such a thing) is a very complex phenomenon that depends on the ratio, on its relationship to the critical bandwidth, on the relative timbres, the absolute frequencies, the loudness, and, not least of all, the musical context. Finally, as John Chalmers has pointed out, these studies are not without their cultural biases. Though such biases difficult to test, given the current ubiquity of twelve-tone equal temperament through radio and television, I think that one is bound to show a "preference" for the familiar. Some evidence for this comes from the article Brian quoted earlier: Kessler, Hansen, and Shepard, "Tonal Schemata in the Perception of Music in Bali and the West" (Music Perception, Winter 1984 V2/2, 131-165). >[1] [Comparison of sqrt(3/2) to 45/32] I personally find 45/32 much too high a ratio to be called "consonant," and I don't really hear it as just. To take a fairer comparison, I do find 7/5 more consonant than a neutral third. >[2] Tune up 11/9 and compare it to 9/8. Which sounds more >consonant? This is true and illustrates my problem with a reliance on LCM analysis. Because 9/8 is small and, depending on the absolute frequency, may lie within the critical bandwidth, I find it relatively dissonant. I don't think that the relative simplicity of the ratios are as much of an issue. >[3] Tune up a second inversion 4:5:6 chord--that is, a just major >chord with a 4/3 in the bass, i.e., a 9:12:15 triad. Now compare it >with a 10:12:15 minor chord. If one resolves the numbers here you have 4/3 and 5/3 in the first and 6/5 and 3/2 in the second. Based only on the ratios, I find 5/3 and 6/5 equivalent -- they are inversions of each other. That leaves 4/3 and 3/2. Clearly 3/2 is the more stable because it lies lower in the harmonic series. >[4] Tune up the ratio 3:5:7 and then tune up the ratio 18:22:27. >Which one sounds more consonant? Here I have to disagree. While they both have distinctive sounds, I find the 3/5 and 5/7 intervals more consonant than the 11/9 interval of the second triad. >This also sidesteps the dilemma of the just perfect fourth. The dilemma of the perfect fourth is in part a historical artifact of producing counterpoint by counting intervals relative to the lowest sound voice (which for simplicity I'll call the bass). Thus the apparent consonance of the perfect fourth and perfect fifth sounding together if one only looks at them relative to the bass is taken care of by considering the fourth dissonant. As is obvious to anyone looking at all the interval combinations, the dissonance is not so much the fourth as the second between the two upper voices. The second reason for considering the fourth relative to the bass a "dissonance" is to explain the need for the second inversion triad to resolve. To me, this need for resolution is not so much because the 9:12:15 is "dissonant" as the simplest interval lies higher in the harmonic series than either the octave or the perfect fifth. Therefore the chord sounds less stable than one based on a 3/2. Also, some medieval musicians were not bothered by this relative instability. >Too, as Norman Cazden points out: >"Traditional rules of harmony and counterpoint explicitly >proscribe parallel successions of perfect intervals, though >with the unaccountable excpetion of that same troublesome >fourth when it is not in the lowest placement. Using the stylistic conventions of one musical culture during one period of history is a weak justification at best. The main reason behind the prohibition of parallel perfect intervals was to maintain the independence of the voices sought by European polyphonic composers of the 15th to 19th centuries. Parallel octaves causes the texture to suddenly thin because the octave lies so low in the harmonic series -- which is why we tend to consider men and women singing in parallel octaves virtually the same as monophony. Put another way, the upper voices are doubling a harmonic already present in the lower voices. So, do we prohibit all parallel intervals, since they must be found in the same harmonic series of some fundamental somewhere? Of course not. As we move further up the harmonic series the impression of the two voices operating "as a single unit" becomes less and less prominent. So where does one draw the line? Perfect unison? Perfect octave? Perfect fifth? Perfect fourth? Major third? Minor third? Well, composers of the 15th century chose the draw the line between the perfect fifth and the perfect fourth. Incidentally, it is interesting to note that 13th century motet composers quite intentionally avoided parallel thirds, though parallel fifths and fourths are often found. I have nothing against composers who prefer the musical usefulness of equal temperament or any other type of temperament. Nazir Jairazhboy, the Indian music master quoted by Brian earlier, points out that he finds that tempered intervals can give extra tension to a melody in need of resolution. As Brian has pointed out as well, the Javanese find perfect 2/1 octaves rather lifeless and usually try to compress or expand them. However, to say that there is some natural predilection for just, equal-tempered, or any other kind of scale is, I think, to misunderstand the relationship between nature and art. Certainly one would think Brian would understand this, having composed in JI himself, according to John. Bill ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^ Bill Alves email: alves@hmc.edu ^ ^ Harvey Mudd College URL: http://www2.hmc.edu/~alves/ ^ ^ 301 E. Twelfth St. (909)607-4170 (office) ^ ^ Claremont CA 91711 USA (909)621-8360 (fax) ^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Received: from eartha.mills.edu [144.91.3.20] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Wed, 19 Jun 1996 23:49 +0100 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id OAA04807; Wed, 19 Jun 1996 14:49:04 -0700 Date: Wed, 19 Jun 1996 14:49:04 -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