53-TET - the higher resolution of Sensory Dissonance

The mathematical way to measure the degree of dissonance uses the formulas suggested by Sethares (2010). The formulas below calculate the dissonant level of one pair of sine tones with loudness considered. To calculate the two notes with harmonics, calculate all pairs of sine tones and sum all individual results. The 53-TET (see the previous article) can provide wider range of consonance and dissonace as well as the possibility in between.

Sethares (2010), Appendix E

a as loudness measure in sone

f as freq. where f2>f1

x* =0.24

b1=3.5, b2=5.75; s1=0.021, s2=19

This temperament is played by a synthesizer developed in Pure Data. With the same instrument and loudness, all possible intervals, within the temperament, start from 440Hz to an octave above, 880Hz. The dissonant unit is only a relative measurement.

The reason for using Equal Temperament of an octave instead of other tuning is that it can guarantee the octave is a perfect 2:1 ratio. It also echoes the sound design of the synthesizer. The harmonics of it are based on harmonic series (f,2f,3f,4f,5f...). The octave that ET provides is important, for example, the second and the fourth are octave relationships in between them, and the fundamental.

The harmonic design of the synth.

Back to two dissonant charts above, the reason to have the greatest consonant in an octave interval also implies that the harmonics also line up with octave. Instruments following harmonic series expect this synthesizer, strings, winds, and piano are the examples. However, the percussion instruments do not follow it, which are considered inharmonic instruments. Although western orchestral percussions are manufactured to slightly adjust to the harmonic series, it is still limited.

Here is the performance of the 53-TET midi keyboard.


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

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