Unequal Beating Temperament System - Kirnberger III and Kellner
Kirnberger III
Equation
((((((((((((((((((((((((440 × 2 + (Δ × 3 × (21.506289597 / 23.460010385))) / 3) × 4 + (Δ × 3 × (21.506289597 / 23.460010385))) / 3) × 2 + (Δ × 3 × (21.506289597 / 23.460010385))) / 3) × 4 + (Δ × 0)) / 3) × 2 + (Δ × 0)) / 3) × 4 + (Δ × 0)) / 3) × 4 + (Δ × 0)) / 3) × 2 + (Δ × 0)) / 3) × 4 + (Δ × 1 × ((23.460010385 − 21.506289597) / (23.460010385 / 12)))) / 3) × 2 + (Δ × 0)) / 3) × 4 + (Δ × 0)) / 3) × 2 + (Δ × 3 × (21.506289597 / 23.460010385))) / 3) = 220
Δ = 0.966360326
Pythagorean Comma = 1200 × log2(531441 / 524288) = 23.460010385 cents
Syntonic Comma = 1200 × log2(81 / 80) = 21.506289597 cents
Schisma = Pythagorean Comma − Syntonic Comma = 1200 × log2(531441 / 524288) − 1200 × log2(81 / 80) = 23.460010385 cents − 21.506289597 cents = 1.953720788 cents
Temperament Octave
| Notes |
Frequencies (Hz) |
Kirnberger III Cents |
12-ET Cents |
Offsets (Cents) |
| A3 |
220.00 |
0.00 |
0.00 |
0.00 |
| A#3 |
233.78 |
105.19 |
100.00 |
+5.19 |
| B3 |
246.50 |
196.92 |
200.00 |
−3.08 |
| C4 |
263.00 |
309.10 |
300.00 |
+9.10 |
| C#4 |
277.07 |
399.33 |
400.00 |
−0.67 |
| D4 |
294.22 |
503.27 |
500.00 |
+3.27 |
| D#4 |
311.71 |
603.24 |
600.00 |
+3.24 |
| E4 |
328.67 |
694.97 |
700.00 |
−5.03 |
| F4 |
350.67 |
807.15 |
800.00 |
+7.15 |
| F#4 |
369.76 |
898.88 |
900.00 |
−1.12 |
| G4 |
393.18 |
1005.22 |
1000.00 |
+5.22 |
| G#4 |
415.61 |
1101.28 |
1100.00 |
+1.28 |
| A4 |
440.00 |
1200.00 |
1200.00 |
0.00 |
| Intervals |
Notes |
Sizes (Cents) |
Beat Rates (Hz) |
| Narrow 5th |
D4-A4 |
696.73 |
2.66 |
| Wide 4th |
D4-G4 |
501.95 |
2.66 |
| Narrow 5th |
C4-G4 |
696.11 |
2.66 |
| Pure 4th |
C4-F4 |
498.04 |
0.00 |
| Pure 5th |
A#3-F4 |
701.96 |
0.00 |
| Pure 4th |
A#3-D#4 |
498.04 |
0.00 |
| Pure 4th |
D#4-G#4 |
498.04 |
0.00 |
| Pure 5th |
C#4-G#4 |
701.96 |
0.00 |
| Wide 4th |
C#4-F#4 |
499.55 |
0.97 |
| Pure 5th |
B3-F#4 |
701.96 |
0.00 |
| Pure 4th |
B3-E4 |
498.04 |
0.00 |
| Narrow 5th |
A3-E4 |
694.97 |
2.66 |
Kellner
Equation
((((((((((((((((((((((((440 × 2 + (Δ × (2 + (2 / 5)))) / 3) × 4 + (Δ × (2 + (2 / 5)))) / 3) × 2 + (Δ × (2 + (2 / 5)))) / 3) × 4 + (Δ × 0)) / 3) × 2 + (Δ × 0)) / 3) × 4 + (Δ × 0)) / 3) × 4 + (Δ × 0)) / 3) × 2 + (Δ × 0)) / 3) × 4 + (Δ × 0)) / 3) × 2 + (Δ × (2 + (2 / 5)))) / 3) × 4 + (Δ × 0)) / 3) × 2 + (Δ × (2 + (2 / 5)))) / 3) = 220
Δ = 1967075 / 2134329
Temperament Octave
| Notes |
Frequencies (Hz) |
Kellner Cents |
12-ET Cents |
Offsets (Cents) |
| A3 |
220.00 |
0.00 |
0.00 |
0.00 |
| A#3 |
233.44 |
102.69 |
100.00 |
+2.69 |
| B3 |
246.67 |
198.10 |
200.00 |
−1.90 |
| C4 |
262.62 |
306.60 |
300.00 |
+6.60 |
| C#4 |
276.67 |
396.83 |
400.00 |
−3.17 |
| D4 |
294.07 |
502.39 |
500.00 |
+2.39 |
| D#4 |
311.26 |
600.74 |
600.00 |
+0.74 |
| E4 |
328.89 |
696.14 |
700.00 |
−3.86 |
| F4 |
350.17 |
804.65 |
800.00 |
+4.65 |
| F#4 |
368.90 |
894.87 |
900.00 |
−5.13 |
| G4 |
392.83 |
1003.69 |
1000.00 |
+3.69 |
| G#4 |
415.01 |
1098.78 |
1100.00 |
−1.22 |
| A4 |
440.00 |
1200.00 |
1200.00 |
0.00 |
| Intervals |
Notes |
Sizes (Cents) |
Beat Rates (Hz) |
| Narrow 5th |
D4-A4 |
697.61 |
2.21 |
| Wide 4th |
D4-G4 |
501.30 |
2.21 |
| Narrow 5th |
C4-G4 |
697.09 |
2.21 |
| Pure 4th |
C4-F4 |
498.04 |
0.00 |
| Pure 5th |
A#3-F4 |
701.96 |
0.00 |
| Pure 4th |
A#3-D#4 |
498.04 |
0.00 |
| Pure 4th |
D#4-G#4 |
498.04 |
0.00 |
| Pure 5th |
C#4-G#4 |
701.96 |
0.00 |
| Pure 4th |
C#4-F#4 |
498.04 |
0.00 |
| Narrow 5th |
B3-F#4 |
696.77 |
2.21 |
| Pure 4th |
B3-E4 |
498.04 |
0.00 |
| Narrow 5th |
A3-E4 |
696.14 |
2.21 |
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Roshan Kakiya
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