source file: mills2.txt Date: Mon, 30 Sep 1996 12:19:30 -0700 Subject: The tonicity of tempered intervals and t From: PAULE ..................................continued from last week. According to Golstein[22], the precision with which freqency information is transmitted to the brain's central pitch processor is between 0.6% and 1.2% within a certain optimal frequency range. Musical tones normally have partials within this range, and since harmonic partials are integer multiples of thier fundamentals, they will yield the same ratio-interpretation for their fundamentals as would the fundamentals transposed into this range. (This is why we did not worry about overtones in the first place.) We can therefore model the tones with normal probability distributions in log-frequency space; the standard deviation (let's call it s) used can be between 0.6% and 1.2%, or greater to reflect various non-ideal conditions. The use of log-frequency space corresponds most closely with psychoacoustic research and musical practice. Now the "width" in 17, log(a(j))-log(b(j)) represents a section of log-freqeuncy space and thus a portion of the distribution lies within this range. The exact amount of probability contained withing this range is thus the certainty with which u=f(1)/f(2) is interpreted as p(j)/q(j), and is proportional to 1/(s*sqrt(2pi))*integral(exp(-.5((x-log(u))/s)^2),dx) in log-frequency space. Now the width in (19) vanishes for large N, but it remains proportional to 1/L(j), so (replacing the integral by a product since W is small compared to s) we define the certainty with which u is intepreted as p(j)/q(j) as C(u,p(j)/q(j))=1/(s*sqrt(2pi)L(j))*exp(-.5((log(p(j)/q(j))-log(u))/s)^2 ) (20) (20) can be extended to chords of more than two notes as follows: A chord of n notes contains n(n-1)/2 different intervals. But since the whole chord must be related to a single fundamental, any individual interval might not be in lowest terms, and the "limit" should now be the greatest of all the odd factors of the numbers used to represent the chord. Denote this limit by L(t); we therefore define NC(L(j,t))=NC(L(j))L(j)/L(t) We then have C(chord)=product over all intervals(C(L(j,t))^(2/n)) where the exponent 2/n is the number of independent degrees of freedom in tuning the chord, n-1, divided by the number of factors in the product, n(n-1)/2. This is done becuse multiplying probabilities of n-1 independent judgments is the probablity of the overall judgment; the exponent is thus a correction for the fact that the number of intervals increases faster than the number of notes. In other words, take the geometric mean certainty, and multiply it by itself a number of times appropriate to the true number of degrees of freesom. This effectively reduces the standard deviation s by a fatcor 1/(sqrt(n-1)), as one would guess from general statistical considerations. As for finding the roots of chords, Parncutt states,[23] The pitch pattern of a complex tone may be recognized even if parts of the pattern are missing, or extra elements are added. Similarly . . .the root of a chord may be perceived if notes corresponding to harmonics of the root are missing, or if notes not corresponding to harmonics are added. For example, the root of the C major triad is weakened, but not changed, if the note E (which corresponds to the fifth harmonic of C) is replaced by Eb (which doesn't correspond to any normally audible harmonic of C); the root (C) is maintained by the strong root implication of the fifth C-G. So a good guess would be that for a four-note chord, whichever of its two, three, or four-note subgroups has an interpretation of the gretest C-value is the subgroup that determines the root; the root itself is the note interpreted as a power of 2, since it is then octave-equivalent to the fundamental. If none of the notes of the chord are interpreted as a power of two, the chord seems unstable, as the sensation of the fundamental is quite secure. [13] Adapted from van Eck, C. L. van Panthaleon. 1981. _J. S. Bach's Critique of Pure Music_. Princo, Culemborg, The Netherlands, Appendix II. [14] Parncutt, Richard. 1989. _Harmony: A Psychoacoustical Approach_. Springer-Varlag, New York, p. 70. [15] Goldstein, J. L., 1973. "An optimum processor theory for the central formation of the pitch of complex tones," _J. Acoust. Soc. Amer._ Vol. 54 p. 1496. [16] Terhardt, E. 1974. "Pitch, consonance, and harmony." _J. Acoust. Soc. Amer._ Vol 55 p. 1061. [17] Wightmann, F. L. 1973. "The pattern-transformation model of pitch." J. Acoust. Soc. Amer. Vol. 54 p. 407. [18] If we were to fix f(q) and vary f(p), we would only have an upper bound on q, giving us an infinite series of fractions to consider. Fixing the center of the interval leads to Mann's criterion, p+q<=N. Mann, Chester D. 1990. _Analytic Study of Harmonic Intervals_. Tustin, Calif. Either way, we still end up with (7) below, so that the relative certainties of various interpretations of a given interval will remain the same. (5) is chosen because the series produced by it display a greater degree of octave equivalence. [19] Three proofs of this are given in Hardy, G. H. and Wright, E.M. 1960. _An Introduction to the Theory of Numbers_. Oxford University Press, London, Ch. 3. [20] See Mann, p. 163 for one possible justification. [21] Partch, p. 184 [22] Goldstein, p. 1499 [23] Parncutt, p. 70. Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Tue, 1 Oct 1996 02:00 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA06949; Tue, 1 Oct 1996 01:00:22 +0100 Received: from eartha.mills.edu by ns (smtpxd); id XA06580 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id RAA29662; Mon, 30 Sep 1996 17:00:19 -0700 Date: Mon, 30 Sep 1996 17:00:19 -0700 Message-Id: <960930235656_71670.2576_HHB57-4@CompuServe.COM> Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu