Pianotech

Cents

C	0.00
C#	100.05
D	200.09
D#	300.14
E	400.18
F	500.23
F#	600.27
G	700.32
G#	800.36
A	900.41
A#	1000.45
B	1100.50
C	1200.55

Frequencies in Hz

A0	27.47
A#0	29.10
B0	30.83
C1	32.66
C#1	34.61
D1	36.67
D#1	38.85
E1	41.16
F1	43.61
F#1	46.20
G1	48.95
G#1	51.86
A1	54.95
A#1	58.22
B1	61.68
C2	65.35
C#2	69.24
D2	73.36
D#2	77.72
E2	82.34
F2	87.24
F#2	92.43
G2	97.93
G#2	103.76
A2	109.93
A#2	116.47
B2	123.40
C3	130.74
C#3	138.52
D3	146.76
D#3	155.49
E3	164.74
F3	174.54
F#3	184.92
G3	195.93
G#3	207.58
A3	219.93
A#3	233.01
B3	246.88
C4	261.56
C#4	277.12
D4	293.61
D#4	311.08
E4	329.58
F4	349.19
F#4	369.97
G4	391.97
G#4	415.29
A4	440.00
A#4	466.18
B4	493.91
C5	523.29
C#5	554.42
D5	587.41
D#5	622.35
E5	659.38
F5	698.60
F#5	740.16
G5	784.20
G#5	830.85
A5	880.28
A#5	932.65
B5	988.13
C6	1046.91
C#6	1109.20
D6	1175.18
D#6	1245.10
E6	1319.17
F6	1397.65
F#6	1480.79
G6	1568.89
G#6	1662.22
A6	1761.11
A#6	1865.88
B6	1976.88
C7	2094.49
C#7	2219.09
D7	2351.11
D#7	2490.98
E7	2639.17
F7	2796.17
F#7	2962.52
G7	3138.76
G#7	3325.49
A7	3523.33
A#7	3732.94
B7	3955.01
C8	4190.30

Stack of 24 Pure Twelfths

Stack of 24 Pure Twelfths = (3 / 1)²⁴.

1200 × log₂[(3 / 1)²⁴] = 45646.920 cents. 



Stack of 19 Pure Fifteenths

Stack of 19 Pure Fifteenths = (4 / 1)¹⁹.

1200 × log₂[(4 / 1)¹⁹] = 45600.000 cents.



Double Pythagorean Comma

The Double Pythagorean Comma is the discrepancy between the Stack of 24 Pure Twelfths and the Stack of 19 Pure Fifteenths:

Double Pythagorean Comma = Stack of 24 Pure Twelfths / Stack of 19 Pure Fifteenths = (3 / 1)²⁴  /  (4 / 1)¹⁹ = (531441 / 524288)².

1200 × log₂[(531441 / 524288)²] = 46.920 cents.



Pure 26th Equal Temperament: 24 Narrowed Twelfths and 19 Widened Fifteenths

Number of Twelfths : Number of Fifteenths = 24 : 19.


Pure Twelfth Narrowed by 1 / 24  ×  24 / (24 + 19) Double Pythagorean Comma = Pure Twelfth Narrowed by 1 / 24  ×  24 / 43 Double Pythagorean Comma = Pure Twelfth Narrowed by 1 / 43 Double Pythagorean Comma = 3 / 1  /  [(531441 / 524288)²]^{1 / 43} = 3 / 1  ×  [(531441 / 524288)²]^{− 1 / 43}.

1200 × log₂{3 / 1  ×  [(531441 / 524288)²]^{− 1 / 43}} = 1900.864 cents.


Pure Fifteenth Widened by 1 / 19  ×  19 / (24 + 19) Double Pythagorean Comma = Pure Fifteenth Widened by 1 / 19  ×  19 / 43 Double Pythagorean Comma = Pure Fifteenth Widened by 1 / 43 Double Pythagorean Comma = 4 / 1  ×  [(531441 / 524288)²]^{1 / 43}.

1200 × log₂{4 / 1  ×  [(531441 / 524288)²]^{1 / 43}} = 2401.091 cents.


Pure Twenty Sixth = 3 / 1  ×  [(531441 / 524288)²]^{− 1 / 43}    ×    4 / 1  ×  [(531441 / 524288)²]^{1 / 43} = 3 / 1  ×  4 / 1 = 12 / 1.

1200 × log₂(12 / 1) = 4301.955 cents.



What is the amount of the Double Pythagorean Comma by which the Stack of 24 Pure Twelfths should be narrowed?

24 / (24 + 19) Double Pythagorean Comma = 24 / 43 Double Pythagorean Comma.


What is the amount of the Double Pythagorean Comma by which each Pure Twelfth in the Stack of 24 Pure Twelfths should be narrowed?

1 / 24  ×  24 / 43 Double Pythagorean Comma = 1 / 43 Double Pythagorean Comma.


What is the amount of the Double Pythagorean Comma by which the Stack of 19 Pure Fifteenths should be widened?

19 / (24 + 19) Double Pythagorean Comma = 19 / 43 Double Pythagorean Comma.


What is the amount of the Double Pythagorean Comma by which each Pure Fifteenth in the Stack of 19 Pure Fifteenths should be widened?

1 / 19  ×  19 / 43 Double Pythagorean Comma = 1 / 43 Double Pythagorean Comma.


What happens if a Pure Twelfth Narrowed by 1 / 43 Double Pythagorean Comma is combined with a Pure Fifteenth Widened by 1 / 43 Double Pythagorean Comma?

The Pure Twenty Sixth emerges.


What is the temperament that has been produced?

Pure 26th Equal Temperament.



Download Word Document:

https://my.ptg.org/HigherLogic/System/DownloadDocumentFile.ashx?DocumentFileKey=d48c6a8d-aab5-fb22-ae6d-549a069e5b3d&forceDialog=0


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Roshan Kakiya
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