Inharmonicity causes the overtones of the fundamental frequency to become sharp.
Source:
http://daffy.uah.edu/piano/page4/page8/index.htmlInharmonicity has been accounted for to some extent by RK 12-TET because its intervals are slightly sharper than each overtone of the fundamental frequency:
1st harmonic (Ratio: 1/1) = 0.00 cents.
RK 12-TET = 0.00 cents.
2nd harmonic (Ratio: 2/1) = 1200.00 cents.
RK 12-TET = 1201.68 cents.
3rd harmonic (Ratio: 3/1) = 1901.96 cents.
RK 12-TET = 1902.65 cents.
4th harmonic (Ratio: 4/1) = 2400.00 cents.
RK 12-TET = 2403.35 cents.
------------------------------
Roshan Kakiya
------------------------------
Original Message:
Sent: 05-09-2019 22:34
From: Kent Swafford
Subject: Roshan Kakiya's 12-Tone Equal Temperament (RK 12-TET)
You include no rationale for the temperament. The fifth is tempered by half the amount that it is tempered in pure octave ET? Is that all? So?
Sent from my iPhone
Original Message------
Roshan Kakiya's 12-Tone Equal Temperament (RK 12-TET)
Fifth = 700.98 cents.
Octave = 1201.68 cents.
12 Fifths = 8411.73 cents.
7 Octaves = 8411.73 cents.
| Note |
Cents |
| C |
0.00 |
| C# |
100.14 |
| D |
200.28 |
| D# |
300.42 |
| E |
400.56 |
| F |
500.70 |
| F# |
600.84 |
| G |
700.98 |
| G# |
801.12 |
| A |
901.26 |
| A# |
1001.40 |
| B |
1101.54 |
| C |
1201.68 |
Pythagorean Comma
12 Pure Fifths = 8423.46 cents.
7 Pure Octaves = 8400.00 cents.
12 Pure Fifths ≠ 7 Pure Octaves.
Pythagorean Comma = 12 Pure Fifths − 7 Pure Octaves = 8423.46 cents − 8400.00 cents = 23.46 cents.
Solution: Meet in the Middle
Average = (8423.46 cents + 8400.00 cents) / 2 = 16823.46 cents / 2 = 8411.73 cents.
12 Tempered Fifths = 8411.73 cents.
7 Tempered Octaves = 8411.73 cents.
The Pythagorean Comma has been eliminated.
Verification: 12 Tempered Fifths − 7 Tempered Octaves = 8411.73 cents − 8411.73 cents = 0.00 cents.
------------------------------
Roshan Kakiya
------------------------------