Harmonic series vs Inharmonicity - redefining octave in percussion instruments

 When an instrument is played, when put the sound into the spectrum analysis (Fast Fourier Transform), there are multiple tones sounding at the same time, which contributes to the timbre of each instrument. For most instruments, strings, winds, and voice, the frequencies of vibration follow the pattern of harmonic series. Different loudness of individual harmonics contributes to the unique timbre of each instrument. Harmonic series follows the pattern from the whole multiples of the fundamental frequency (f,2f,3f,4f,5f, etc).

However, the inharmonic percussion instruments do not follow the harmonic series, thus, this pattern is called inharmonicity. The timbre, as Sethares (2010) describes, is gong-like or bell-like. The spectrum analysis of some iconic orchestral sounds can be found in my previous article.

For percussion containing wood or metal bars with free ends, the pattern of harmonic partials is irregular. Sethares (2010) calculated the pattern of the first 6 partials as:

        f, 2.76f, 5.41f, 8.94f, 13.35f, and 18.65f

E.1

To rethink Partch's Genesis Scale with 43 pitches in each octave and his other tonality diamonds are dealing with harmonic series and undertone series (otonality and u tonality), apart from his adaptive viola, he made and composed a number of pieces for. percussion. The inharmonicity of percussion comes with harmonic series. What an interesting mismatch!

If I would like to create a temperament for this harmonic pattern, I would follow the process of Sethares (2010), drawing a dissonant curve first. The math is here. 

The dissonant curve of E.1

The graph above shows that the 605Hz and 432.8Hz are two major consonant spots, corresponding to the first and second partial, second and third partial respectively of from the harmonic pattern above. 

The traditional 2:1 octave, the first and second harmonic, follows the harmonic series, it is defined as the octave. Following this concept, the irregular pattern above, the first and second harmonic, as the "octave"  ratio, is the first demonstration in the video. The second demonstration is based on the ratio of second and third harmonics, (5.41/2.76=1.967) also the fifth interval in traditional harmonic series. Finally, the comparison of conventional 12 TET.

Although equal temperament often divided an octave 2.75 (220:604Hz), in this case, the dividing fifth 1.967 (220:432.8Hz ) sounds more arrived. There are two reasons, first, apart from the ratio of first and second harmonics, the average ratio of all harmonics is 1.96563483. Other harmonic partials balanced back the dissonant; the fifth is more consonant than the octave according to the dissonant calculation. Second, the octave of 2:1 ratio is deep-rooted in our mind, the 220:440Hz is an octave that affects the judgment of an octave. The 1.967 fifth ratio in the special tuning is close to 2 in the traditional octave as well.

Sethares, W. A. (2010) Tuning, timbre, spectrum, scale. 2nd ed. London: Springer.

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