source file: mills2.txt Date: Mon, 24 Feb 1997 09:47:37 -0800 Subject: Reply to Matt Nathan From: Manuel.Op.de.Coul@ezh.nl (Manuel Op de Coul) From: PAULE I think the diatonic scale long predated the discovery of 5-limit harmony in most cultures (India is a possible exception, though only 3-limit consonances are mentioned in the original texts). It was constructed from melodic considerations only, which means that ratios beyond the 3-limit are of negligible importance. Ascribing 5-limit ratios to the diatonic scale is in most cases, as I have said before, a historico-geographic fallacy, and a very Eurocentric one at that. It must be said, though it partially supports Matt's viewpoint, that many cultures (the West, India, Arabia) seem to have discovered the 5-limit relations through their SCHISMATIC, not SYNTONIC, approximations. It seems that Pythagorean tuning was extremely well developed long before any theoretical understanding of the 5-limit came about. So something resembling a just diatonic scale did arise in all of these cultures. In Pythagorean tuning, this is D E Gb G A Cb Db D. But this scale is already "corrupted" by harmonic considerations. The paradigm of similar tetrachords seems to be important for melodic comprehensibility. This in itself can lead to many scales as are found throughout the world, including some non-octave-repeating middle eastern scales. The diatonic scale simply takes this concept to its extreme, as tetrachordality manifests itself in every octave species. This requires a single size of whole step, though I must admit that the syntonic comma is too small to make the 5-limit JI diatonic scale very disturbing melodically. Increase the size of the syntonic comma while keeping excellent 5-limit approximations (try going from 34-equal to 22-equal to 15-equal) and the scale does sound melodically incorrect. Melodically, I much prefer the "Pythagorean" diatonic scale in 22-equal, where the thirds are tuned 7:6 and 9:7, to the 5-limit diatonic scale in the same tuning. Ptolemy's scales (much like his astronomical models) seem to have been derived from geometrical and mystical, not scientific (the appropriate science for music is psychoacoustics), considerations, so they don't count. I must qualify this by saying that Ptolemy did specifically mention the psychological desirability of the division of the major third into two equal parts, but felt that since all musical intervals must be measured rationally, this could only be satisfied approximately! In this sense, JI is but the imperfect temperament and something meantone-like is the Platonic ideal! The assertion that melody is aesthetically prior to harmony is of course a subjective one, but something I have learned the hard way. As a child I taught myself keyboard harmony, and equated composition of and meaning in music to the harmonic progressions. But unknowingly, I was already obeying melodic considerations in voice leading, etc. I once composed a progression on guitar which, I had no idea until I wrote it down, alternated between chords diatonic to the key and highly chromatic ones. The diatonic scale defines tonality, and shifts thereof, in Western music. Harmonic usage is, I believe, governed by harmonic-series or small-integer-ratio-type considerations, but the melodic aspects are at least as important for the overall effect of the music. Deriving the diatonic scale from the tonic, subdominant, and dominant triads is utterly inaccurate, in my opinion. There are too many pieces in major that focus on I-ii-I progressions, and pieces in minor (aeolian, to be more academic) that focus on i-VII-i progressions, for this to be an accurate explanation, for not even the tuning of the chords will be correctly specified by such an explanation. >(I read, I think in Helmoholtz, >about the comparative difficulty of solfege students to >stay in tune as a group when accompanied by the (et) >church organ and ease when unaccompanied and allowed >to sign in JI.) If they were not harmonizing but singing in unison, I seriously doubt that they gravitated towards a scale with unequal whole steps, though Helmholtz was clearly biased in favor of believing that they did! >Comma shift is probably a misnomer too (like "wandering tonic"). >I'd rather think of it not as a "shift" to a different version >of the same pitch class, but a movement to a new pitch class. I don't think such an analytic framework would hold up if applied to common-practice Western music. For starters, principles of thematic development are likely to be utterly destroyed. As for quantifying dissonance, there are many psychoacoustic factors at play and incorporating them all into a single model does not seem advisable since the different mathematical definitions of dissonance probably have different, but equally valid, manifestations in the field of musical perception. I recommend The Acoustical Society of America Volume 45 No 6: Cononance Theory Part I+II from Akio Kameoka and Mamoru Kuryagawa for understanding the "roughness" aspect of dissonance. Plomp and Levelt and Bill Sethares are good sources too. As for the "tonalness" aspect, which is mandated by the phenomena known variously as virtual pitch, residue pitch, complex tone pitch, fundamental tracking, and possibly periodicity pitch, I have been working on this for some time. I have been communicating particularly with James Kukula on this topic (James, are you there?). >Is your dissonance function not applicable to comparing >perfect representations of different intervals? (I guess >it would give equal dissonance, 0, for both?) I don't believe all just intervals, or even all just intervals within a certain odd-number limit, have zero dissonance. However, since the exact form of such a dissonance function is contreversial and certainly depends on what _kind_ of dissonance you're talking about, I simply made a highly plausible calculus approximation to only that part of such a function as is needed to optimize meantone tunings, etc. Other work (see above) is concerned with more general questions. >> >Will you explain this dissonance function with >> >figures please? It sounds interesting but I'm >> >not sure what you mean by derivative. >> >> ...calculus. >Interesting. Do you have a formula? The whole point is that it doesn't matter what the exact form dissonance function is! The formulas that result from this assumption is, as I tried to make clear in words, minimize (error of 3:1)^2+(error of 5:1)^2+(error of 5:3)^2 and minimize (3*error of 3:1)^2+(5*error of 5:1)^2+(5*error of 5:3)^2. The results of applying these to meantone tunings I have already posted in both cents and exact formulae. If you want me to work you through the calculus please try to reach me offline, I don't think we need to scare any more musicians off this list. Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Mon, 24 Feb 1997 19:02 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA17836; Mon, 24 Feb 1997 19:02:06 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA17839 Received: from by ella.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) id JAA11505; Mon, 24 Feb 1997 09:59:58 -0800 Date: Mon, 24 Feb 1997 09:59:58 -0800 Message-Id: <3311D6CE.63FD@ix.netcom.com> Errors-To: madole@mills.edu Reply-To: tuning@ella.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@ella.mills.edu