I can show how an 85-key piano can be completely tuned with only fifths and fourths whose values have been derived from a simple circle of fifths, containing zeros and fractions, of a well temperament.I will be using Thomas Young's Second Temperament throughout the rest of this post. Here is the circle of fifths for this tuning upon which the rest of this post will be based:
The zeros indicate the fifths should be pure and the fractions indicate the amount by which the fifths should be tempered.A pure fifth is a fifth whose ratio is 3/2. A logarithm can be used to calculate its value in cents:Pure fifth = 1200 x log2(3/2) = 701.96 cents.−1/6 indicates the pure fifth should be reduced by a sixth of the Pythagorean comma. Pythagorean comma = 23.46 cents.1/6 of the Pythagorean comma = 23.46 / 6 = 3.91 cents.Tempered fifth = 701.96 - 3.91 = 698.05 cents.We can confirm that a chain of 12 fifths = a chain of 7 octaves:7 octaves = 1200 x 7 = 8400.12 fifths = 701.96 x 6 + 698.05 x 6 = 8400.12 fifths = 7 octaves. The Pythagorean comma has been eliminated.Fourths will also need to be used so their values must also be calculated.A pure fourth is a fourth whose ratio is 4/3. A logarithm can be used to calculate its value in cents:Pure fourth = 1200 x log2(4/3) = 498.04 cents.The inversion of a pure fifth is a pure fourth. The value of the pure fourth can also be calculated by subtracting the value of the pure fifth from the value of the octave:Pure fourth = 1200 - 701.96 = 498.04 cents.The value of the tempered fourth can be calculated by subtracting the value of the tempered fifth from the value of the octave:Tempered fourth = 1200 - 698.05 = 501.95 cents.Summary:
Pure fifth = 701.96 cents (offset from 12-TET = +1.96 cents).
Tempered fifth = 698.05 cents (offset from 12-TET = −1.95 cents).
Pure fourth = 498.04 cents (offset from 12-TET = −1.96 cents).
Tempered fourth = 501.95 cents (offset from 12-TET = +1.95 cents).The following procedure can be tested by using an electronic tuning device to tune an 85-key piano:1. Determine the first and last notes of the piano. They will be the same because 7 octaves are physically equal to 85 keys. The notes could be C1 and C8 or A0 and A7. I will be using C1 and C8 for this procedure. This means every octave should start and end with C for this procedure.2. Tune 1 octave from C[unison] to C[octave].3. Tune a chain of alternating tempered fifths and tempered fourths within this octave: C[unison]-G (fifth), G-D (fourth), D-A (fifth), A-E (fourth), E-B (fifth) and B-Gb (fourth).4. Tune a chain of alternating pure fourths and pure fifths within this octave: Gb-Db (fourth), Db-Ab (fifth), Ab-Eb (fourth), Eb-Bb (fifth) and Bb-F (fourth). F-C[octave] (fifth) should already be pure.Every interval within this octave should have been tuned now. The chain of fifths and fourths is still unbroken:C[unison]-G-D-A-E-B-Gb-Db-Ab-Eb-Bb-F-C[octave].5. C[octave] becomes C[unison] in the next octave. Repeat steps 2 - 5 until all 7 octaves have been tuned.Conclusion:
Every interval from C1 to C8 should have been tuned now. Only fifths and fourths have been used throughout this entire procedure. I have essentially derived all this information from this simple circle of fifths:
All the information above suggests that any 85-key piano can be tuned with any well temperament by using only fifths and fourths. I think this tuning method can be practical if inharmonicity is controlled.
Everything I have posted on this post should be thoroughly checked and tested.------------------------------
Roshan Kakiya
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