source file: mills2.txt Date: Fri, 10 Nov 1995 10:56:58 -0800 From: "John H. Chalmers" From: mclaren Subject: A generalized approach to harmony in non-just non-equal-tempered scales --- "Observe constantly that all things take place by change, and accustom yourself to consider that the nature of the universe loves nothing so much as to change the things which are to make new things like them. For everthing that exists is in a manner the seed of that which will be." [Marcus Aurelius, "Meditations"] Bill Sethares and Your Humble E-Mail Correspondent have both struggled for more than a year with the vertical implications of non-just non-equal-tempered scales. While Bill's spectral mapping procedure offers a way of adroitly controlling *sensory* consonance and dissonance in n-j n-e-t scales, this is quite a different matter from the tonal substructure implicit within the scale. Douglas Keislar's PhD thesis makes this clear. By rendering a scale like (say) 13/oct more consonant on the level of individual partials interacting within the critical band, we do *not* produce any greater sense of overall "tonality" for the listener. Even with smoothly beatless chords, 13-TET *still* sounds profoundly anti-tonal and non-cadential. Even though a I-IV-V-I may be made beatless through the miracle of modern digital signal processing, it still doesn't sound like "the way the scale wants to behave." And thus other compositional strategies must be employed, different from those dragged out of 12-TET and press-ganged into service. In short, "I don't think we're in Kansas anymore, Toto." And outside of 12-TET, you'd better take that into account...or your compositions will sound like very badly out-of-tune 12-TET leftovers. These complexities are sufficient to give pause to a composer contemplating a work in 19- or 53-tone equal, or (say) 13-limit or 31-limit JI; but when it comes to non-just non-equal-tempered tunings, what's a xenharmonic composer to do? The first and most important point in generating vertical structures in non-just non-equal-scales is to recognize that some general procedures *do* carry over from other tunings, albeit in highly modified form. John Chalmers has elaborated a classical method for scale construction which he calls "tritriadic" scale generation. The basic idea (based on a very ancient principle) is that scales have typically been generated by taking the Tonic, forming a triad; then a Mediant, and forming a triad, and finally a Dominant, and forming a triad. The set of tones formed by the union of all the pitches in the triads has conventionally produced the scales characteristic of Western music. John's inspiration was to vary the frequency ratios of the T, D and M chords to generate variant scales. (John Chalmers' tritriadic techniques have been unjustly neglected as a topic for this tuning forum; if something doesn't change, clearly I shall have to author several future posts on the subject.) Interestingly, something akin to this procedure can be employed with non-just non-equal-tempered tunings. One of the more obvious n-j n-e-t scales is the set of modes of the ideal vibrating cylinder free at both ends. These are given by Lord Rayleigh (1896) in his "Theory of Acoustics," Vol. 2, pg. 25, refining the result obtained by Hoppe in 1871: The frequency f of each partial is proportional to sqrt[[s^2]*[(s^2 -1)^2]/(s^2 + 1)] Setting s successivey to 2, 3, 4... and referring each tone to the fundamental of the inharmonic series, the partial frequencies are: f0 = sqrt(4*9/5) = 2.68328/2.68328 = 1.0 f1 = sqrt(9*64/10) = 7.58946/2.68328 = 2.82842 f2 = sqrt(16*225/17) = 14.55197/2.68328 = 5.4320 f3 = sqrt(25*576/26) = 23.5339/2.68328 = 8.77057 f4 = sqrt(36*1225/37) = 34.5329/2.68328 = 12.8696 f5 = sqrt(49*2304/50) = 47.5173/2.68328 = 17.7086 f6 = sqrt(64*3969/65) = 62.5135/2.68328 = 23.297422 f7 = sqrt(81*6400/82) = 79.510699/2.68328 = 29.631905 f8 = sqrt(100*9801/101) = 98.508682/2.68328 = 36.71204 f9 = sqrt(121*14400/120) = 120.00417/2.68328 = 44.72294 f10 = sqrt(144*20449/143) = 143.49913/2.68328 = 53.47899 &c. Reducing these values to cents gives Pitch 1 = 0 cents Pitch 2 = 1799.99 cents Pitch 3 = 2926.97 cents Pitch 4 = 3759.204 cents Pitch 5 = 4423.074 cents Pitch 6 = 4975.653 cents Pitch 7 = 5450.5181 cents Pitch 8 = 5866.8954 cents Pitch 9 = 6237.8177 cents Pitch 10 = 6579.5317 cents Pitch 11 = 6889.0807 cents &c. The primary consonant vertical structure in this system will be the complex Pitch 1 + Pitch 2 + Pitch 3. This is the *least* compact vertical structure available; notice that, because of the wide spacing between members of this inharmonic series, Plomp & Levelt's findings tell us that this primary vertical structure in this n-j n-e-t tuning will be consonant (provided that the timbre is made up of partials tuned to this scale) because no two partials will fall within the same critical band. However, there are other consonant vertical structures than the primary: for example, members 2, 3 and 4 of this inharmonic series could be used as a chord. This would produce: Secondary vertical structure: Pitch 1 = 0 cents Pitch 2 = 1126.98 cents Pitch 3 = 1959.214 cents A third-order vertical structure comes from members 3, 4 and 5 of the inharmonic series: Pitch 1 = 0 cents Pitch 2 = 832.234 cents Pitch 3 = 1496.104 cents And a fourth-order vertical structure comes from members 4, 5 and 6 of the inharmonic series: Pitch 1 = 0 cents Pitch 2 = 663.87 cents Pitch 3 = 1216.449 cents A fifth-order vertical structure comes from member 5, 6 and 7 of the inharmonic series: Pitch 1 = 0 cents Pitch 2 = 552.579 cents Pitch 3 = 1027.4441 cents A sixth-order vertical structure comes from members 6, 7 and 8 of the inharmonic series: Pitch 1 = 0 cents Pitch 2 = 474.858 cents Pitch 3 = 891.2424 cents A 7th-order vertical structure comes from members 7, 8 and 9 of the inharmonic series: Pitch 1 = 0 cents Pitch 2 = 416.3773 cents Pitch 3 = 787.2995 cents And an 8th-order vertical structures comes from members 8, 9 and 10 of the inharmonic series: Pitch 1 = 0 cents Pitch 2 = 370.9223 cents Pitch 3 = 712.6363 cents This is a slightly stretched neutral triad with both the fifth and the third somewhat sharper than the values of Erv Wilson's hypermeantone scale. A 9th-order vertical structures comes from members 8, 9 and 10 of the inharmonic series: Pitch 1 = 0 cents Pitch 2 = 341.714 cents Pitch 3 = 651.263 cents This vertical structure is not consonant because the distance between Pitch 2 and Pitch 3 is less than a critical bandwidth (290.5 cents) in the midrange of human hearing. (At extremely high frequencies this vertical complex would sound consonant, primarily because the upper partials lie above the range of human hearing.) Subsequent nth-order vertical complexes will clearly be less consonant. This gives us a set of harmonies which can be transposed to different steps of the scale to produce inharmonic progressions. (Again, we assume the partials are matched to the tuning.) Several points of note: First, John Chalmers' tritriadic scale generation techniques can be employed with the 8th-order consonant vertical structure. It will produce modes significantly different from those familiar from the harmonic series. Second, the inharmonic series considered here requires us to travel farther up to find a familiar vertical structure than does the ordinary harmonic series. In the classical Western case, the 4th-order vertical structure using harmonic series members 4, 5 and 6 forms the basis of Western harmony. Here, the 8th-order vertical structures using inharmonic series members 8, 9 and 10 form the basis of n-j n-e-t harmony in this particular inharmonic series. Third, different inharmonic series can easily be generated by modifying the equation for the modes of a vibrating tube. For instance, instead of The frequency f of each partial being proportional to sqrt[[s^2]*[(s^2 -1)^2]/(s^2 + 1)] , we could set f proportional to sqrt[[(s + 1)^2]*[(s^2 -1)^2]/s^2] or sqrt[[(s + 3)^2]*[(s^2 -1)^2]/(s+1)^2] or sqrt[[(s + 5)^2]*[(s^2 -3)^2]/(s+1)^2] or cube root of[[(s + 1)^2]*[(s^2 -1)^2]/s^2] or cube root of [[(s + 1)^3]*[(s^2 -1)^2]/s^3] or sqrt[[(s - 1)^2]*[s^3]/s^2] and so on. Clearly there are an infinite number of equations describing non-just non-equal-tempered scales, which can be obtained merely by varying the equation for the modes of a vibrating cylinder. A larger question is: What physical oscillator geometry corresponds to a given arbitrary equation? This is an extraordinarily difficult problem. It may be insoluble. While the inverse problem--given an arbitrary oscillator geometry, can the equation describing the modes of the system be found?--can at least be attacked numerically (if all else fails), the problem of obtaining an oscillator geometry from an inspection of the equations describing the modes of a cylinder may not have a single-valued solution. That is, different physical oscillatory system may produce the same frequency spectrum. This has been proven true in several cases, particularly the case of different drum geometries, and may be true for all physical oscillators. (For discussion of a general mathematical proof of this proposition, see Gordon, C., Webb D., and S. Wolpert, "One Cannot Hear the Shape Of A Drum," Bulletin of the American Mathematical Society (New Series), July 1992, Vol. 27, No. 1, pp.134-138) Thus far, we have examined only the overtone- equivalent members of the inharmonic series. What about subinharmonic vertical structures? That is the subject of the next post. --mclaren Received: from eartha.mills.edu [144.91.3.20] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Fri, 10 Nov 1995 23:33 +0100 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id NAA16690; Fri, 10 Nov 1995 13:33:38 -0800 Date: Fri, 10 Nov 1995 13:33:38 -0800 Message-Id: <9511101332.aa06606@cyber.cyber.net> Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu