Unequal Beating Temperament System - Young I
Introduction
One of my main aims is to establish beats-based versions of the cents-based mathematical models that are currently in use. The Equal Beating 5ths and 4ths Temperament System, with its Δ and s variables from Alfredo Capurso's Circular Harmonic System, serves as the Circle of Equally Beating 5ths and 4ths that is to be modified to form beats-based mathematical models of Unequal Temperament. The s variable acts as a beat rate modifier that alters the beat rate relationship of the 5ths and 4ths of the tuning chain within the Temperament Octave A3-A4. The Unequal Beating Temperament System has naturally emerged as a flexible, musically expressive system through the relaxation of the Equal Beating Temperament System's rigid requirement for 1 : 1 beat ratios. The Equal Beating Temperament System has more to do with the mathematical / scientific side of piano tuning, whereas the Unequal Beating Temperament System has more to do with its musical / artistic side. It must be stated that the Unequal Beating Temperament System retains equally beating 5ths and 4ths within the Temperament Octave A3-A4, albeit in a segmented fashion that contrasts with the uniform treatment of beat rates in the Equal Beating Temperament System. To put it another way, I am using the Circle of Unequally Beating 5ths and 4ths. To get the ball rolling, I have decided to create a beats-based mathematical model of one of my favourite Unequal Temperaments of all time - Thomas Young's First Temperament. This post serves as the conclusion of my research into the mathematics of tuning. Good luck to anyone who wishes to take on the monumental task of reworking all the cents-based mathematical models in existence into beats-based ones. This contribution of mine gets that project up and running. Enjoy!
Equation
((((((((((((((((((((((((440 × 2 + (Δ × 2)) / 3) × 4 + (Δ × 2)) / 3) × 2 + (Δ × 2)) / 3) × 4 + (Δ × 1)) / 3) × 2 + (Δ × 1)) / 3) × 4 + (Δ × 0)) / 3) × 4 + (Δ × 0)) / 3) × 2 + (Δ × 0)) / 3) × 4 + (Δ × 0)) / 3) × 2 + (Δ × 1)) / 3) × 4 + (Δ × 1)) / 3) × 2 + (Δ × 2)) / 3) = 220
Δ = 78683 / 83000
Temperament Octave
| Notes |
Frequencies (Hz) |
Young I Cents |
12-ET Cents |
Offsets (Cents) |
| A3 |
220.00 |
0.00 |
0.00 |
0.00 |
| A#3 |
233.73 |
104.82 |
100.00 |
+4.82 |
| B3 |
246.55 |
197.27 |
200.00 |
−2.73 |
| C4 |
262.36 |
304.83 |
300.00 |
+4.83 |
| C#4 |
277.02 |
398.96 |
400.00 |
−1.04 |
| D4 |
293.97 |
501.77 |
500.00 |
+1.77 |
| D#4 |
311.64 |
602.87 |
600.00 |
+2.87 |
| E4 |
329.05 |
696.97 |
700.00 |
−3.03 |
| F4 |
350.12 |
804.43 |
800.00 |
+4.43 |
| F#4 |
369.35 |
897.00 |
900.00 |
−3.00 |
| G4 |
392.59 |
1002.61 |
1000.00 |
+2.61 |
| G#4 |
415.52 |
1100.91 |
1100.00 |
+0.91 |
| A4 |
440.00 |
1200.00 |
1200.00 |
0.00 |
| Intervals |
Notes |
Sizes (Cents) |
Beat Rates (Hz) |
| Narrow 5th |
D4-A4 |
698.23 |
1.90 |
| Wide 4th |
D4-G4 |
500.83 |
1.90 |
| Narrow 5th |
C4-G4 |
697.78 |
1.90 |
| Wide 4th |
C4-F4 |
499.61 |
0.95 |
| Narrow 5th |
A#3-F4 |
699.61 |
0.95 |
| 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 |
699.73 |
0.95 |
| Wide 4th |
B3-E4 |
499.71 |
0.95 |
| Narrow 5th |
A3-E4 |
696.97 |
1.90 |
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Roshan Kakiya
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