source file: m1443.txt Date: Wed, 10 Jun 1998 14:32:31 -0400 Subject: Reply to Daniel Wolf From: "Paul H. Erlich" >< But the simplest interval between the 9/8 and >Could you give us the condensed version of why 19/17 should be simpler th= >an >11/9? It shouldn't! Read again what I wrote. >I find an 11/9 much more easy to tune than a 19/17 whether with >oscillators or strings. Me too! >The two types of dissonance I refer to are (1) dissonances as classes of >intervals independent of the exact intonation and (2) dissonances perceiv= >ed >on the basis of the exact intonation. Let's consider the definitions of dissonance formulated by Helmholtz and Plomp & Levelt and used by Kameoka & Kuriyagawa and Sethares. For two pure tones in a typical register, there is no dissonance at the unison, dissonance rises dramatically at a difference of about 1/12 tone, maximal dissonance occurs at a difference of about 1/8 tone, and falls off gradually to zero at a difference of about a minor third. Total dissonance is defined as the sum over all pure tone pairs of the pairwise dissonances. Therefore, for typical harmonic timbres, the most dissonant interval of all is the unison detuned by 1/8 tone, since every partial of one note is 1/8 tone away from an equally loud partial of the other note. The second most dissonant interval is the octave detuned by 1/8 tone, since every partial of the higher note is 1/8 tone away from a somewhat quieter partial of the lower note. The third or fourth most dissonant interval is the perfect fifth detuned by 1/8 tone, since every other partial of the higher note is 1/8 tone from an almost as loud partial of the lower note. With inharmonic timbres such as found in the gamelan, this last interval is not necessarily dissonant at all, since there is nothing resembling a third harmonic partial in the tones (the octave is produced, to a small extent, by the ear and/or brain -- witness phenomena such as second-order beating). If one looks at the graphs of dissonance versus interval size for complex harmonic tones in any of these references, one will see the highest peaks lying a very short distance from the lowest troughs (the latter are at the simplest ratios). These peaks are very narrow because all the pairs of partials involved rise in dissonance as one approaches the peak, and they all fall off in dissonance as one retreats from it. So these are examples of your type (2) dissonance. Now consider an interval such as 11/9. The 6th partial of the lower tone is about 1/6 tone off the 5th partial of the higher tone, and the 5th partial of the lower tone is about 1/6 tone off the 4th partial of the higher tone. As one enlarges the interval, the former pair will fall off in dissonance, while the latter pair will intensify in dissonance until one is about 1/8 tone flat of a 5/4. Similarly there is both an increase and a decrease in dissonance as one decreases the interval to about 1/8 tone sharp of a 6/5. Since the total dissonance is the sum of the dissonances of the pairs of partials, there is a bit of a plateau of total dissonance around 11/9. So this is an example of your type (1) dissonance. This plateau is not as high as the peaks that flank the simplest ratios. If the partials through the 11th are audible (as on a piano), one can tune the 11/9 exactly by eliminating the beating. However, the beatings between the 6th and 5th partials of the lower note and the 5th and 4th partials of the upper nore will be more powerful and therefore the 11/9 will represent a very minor local minimum of dissonance at best. One can also consider the virtual pitch/harmonic entropy component of dissonance, which leads to similar conclusions. I will discuss this if you wish.