Tuning, Timbre, Spectrum, Scale

Description

cf. this Music StackExchange answer:

William Sethares, creator of "xenotality" and "exotonality," wrote a book called Tuning, Timbre, Spectrum, Scale. According to Dave Benson in Music: A Mathematical Offering p. 490:

"The basic thesis of this book is the idea, first put forward by John Pierce, that the harmonic spectrum or timbre of an instrument determines the most appropriate scales and temperaments to be used, and therefore we could start with a prescribed scale and design a harmonic spectrum for which it would be appropriate. The author also introduces adaptive tunings, where the pitches of notes are allowed to be modified by a few cents from one chord to the next, in order to keep the chords better in tune. The book comes with a CD of examples."

The part I made bold is thus a theory of composing music that would involve Fourier analysis. Starting with the Fourier coefficients you want (i.e., "the prescribed scale"), the composer can get an exact "harmonic spectrum for which it would be appropriate" by taking the inverse Fourier transform.

acoustics of minor third bells

Table2. 2. Each note consists of three partials. If the sequence is played ascending, then the ?rst virtual pitch tends to be perceived, whereas if played descending, the second, lower virtual pitch tends to be heard. Only one virtual pitch is audible at a time. This can be heard in sound examples [S: 6] and [S: 7]. Note First Second Third Virtual Pitch Virtual Pitch partial partial partial ascending descending 1 600 800 1000 200. 0 158. 9 2 620 820 1020 205. 2 163. 0 3 640 840 1040 210. 4 167. 1 4 660 860 1060 215. 6 171. 2 5 680 880 1080 220. 9 175. 3 6 700 900 1100 226. 1 179. 4 7 720 920 1120 231. 3 183. 6 8 740 940 1140 236. 6 187. 7 9 760 960 1160 241. 8 191. 8 10 780 980 1180 247. 0 195. 9 11 800 1000 1200 252. 2 200. 0 Pitch and virtual pitch are properties of a single sound. For instance, a chord played by the violin, viola, and cello of a string quartet is not usually thoughtofashavingapitch;rather,pitchisassociatedwitheachinstrumental tone separately. Thus, determining the pitch or pitches of a complex sound source requires that it ?rst be partitioned into separate perceptual entities. Only when a cluster of partials fuse into a single sound can it be assigned a pitch. When listening analytically, for instance, there may be more “notes” presentthaninthesamesoundwhenlisteningholistically.