Equal Temperament Tuning SystemThe Equal Temperament Tuning System is a solution for the various discrepancies that are inherent in the Just Intonation Tuning System. These discrepancies are commonly referred to as commas such as the Pythagorean comma and the Syntonic comma.
There are Equal Temperaments that convert the open Spiral of Fifths into a closed Circle of Fifths (narrowest to widest):
- Pure Octave ET (12 Narrowed Fifths = 7 Pure Octaves).
- Pure 12th ET (12 Narrowed Fifths = 7 Widened Octaves).
- Pure 5th ET (12 Pure Fifths = 7 Widened Octaves).
Every Equal Temperament has its own unique set of partials and subpartials.
The scale of every Equal Temperament is symmetrical. This causes each partial to be the reciprocal of its corresponding subpartial in every Equal Temperament. I have illustrated this by using Pure 12th ET as an example below.
Partials and Subpartials of Pure 12th Equal Temperament5th Partial = 3
28/19.
4th Partial = 3
24/19.
3rd Partial = 3
19/19 = 3
1 = 3.
2nd Partial = 3
12/19.
1st Partial = 1st Subpartial = 3
0/19 = 1.
2nd Subpartial = 3
−12/19 = 1 / 312/19.3rd Subpartial = 3
−19/19 = 3−1 = 1 / 3.4th Subpartial = 3
−24/19 = 1 / 324/19.5th Subpartial = 3
−28/19 = 1 / 328/19.This analysis indicates that Partial = 1 / Subpartial for every corresponding partial and subpartial of every Equal Temperament.
Deriving the intervals and subintervals of an Equal Temperament by using its specific series of partials and subpartialsThe specific series of partials and subpartials of each Equal Temperament can be used in the same way as the harmonic and subharmonic series to derive each interval and subinterval of each Equal Temperament.
2nd Partial / 1st Partial = 3
12/19 / 3
0/19 = 3
12/19 − 0/19 = 3
12/19 = Pure 12th ET Octave.
3rd Partial / 2nd Partial = 3
19/19 / 3
12/19 = 3
19/19 − 12/19 = 3
7/19 = Pure 12th ET Fifth.
4th Partial / 3rd Partial = 3
24/19 / 3
19/19 = 3
24/19 − 19/19 = 35/19 = Pure 12th ET Fourth.
5th Partial / 4th Partial = 3
28/19 / 3
24/19 = 3
28/19 − 24/19 = 34/19 = Pure 12th ET Major Third.
2nd Subpartial / 1st Subpartial = 3
−12/19 / 3
0/19 = 3
−12/19 − 0/19 = 3
−12/19 = Pure 12th ET Sub-Octave.
3rd Subpartial / 2nd Subpartial = 3
−19/19 / 3
−12/19 = 3
−19/19 − (−12/19) = 3
−19/19 + 12/19 = 3
−7/19 = Pure 12th ET Sub-Fifth.
4th Subpartial / 3rd Subpartial = 3
−24/19 / 3
−19/19 = 3
−24/19 − (−19/19) = 3−24/19 + 19/19 = 3−5/19 = Pure 12th ET Sub-Fourth.
5th Subpartial / 4th Subpartial = 3
−28/19 / 3
−24/19 = 3−28/19 − (−24/19) = 3−28/19 + 24/19 = 3−4/19 = Pure 12th ET Sub-Major Third.
Equal Temperaments are in tune with their own partial and subpartial series
Pure 12th ET Major Third = A4-C#5.
A4 = 440 Hz.
C#5 = (440 × 34/19) Hz.
4th Partial = 324/19.
5th Partial = 328/19.
Beat Rate = Difference between 4th partial of C#5 and 5th partial of A4 = 324/19 × (440 × 34/19) Hz − 328/19 × 440 Hz = 0 Hz.
Equal Temperaments are out of tune with the harmonic and subharmonic series
Pure 12th ET Major Third = A4-C#5.
A4 = 440 Hz.
C#5 = (440 × 34/19) Hz.
4th Harmonic = 4.
5th Harmonic = 5.
Beat Rate = Difference between 4th harmonic of C#5 and 5th harmonic of A4 = 4 × (440 × 34/19) Hz − 5 × 440 Hz = 18 Hz.
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Roshan Kakiya
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