There has been a lot of tuning system development in 20th century, starting with the 43-tone Monophony of Harry Partch. Since a lot of the initial development focused on creating extended just intonation systems, it is worth taking a look at one of the oldest alternatives to Pythagorean and Equal Tempered tunings - Just Intonation.
Still in use and advocated by many today, Just Intonation is built on the premise of minimizing the ratio numbers in each interval. However, traditionally, this was possible only by restricting music to one tonality or setting up additional keys to handle additional notes. Advances in electronic music have made it possible to retune a synthesizer on the fly, thereby maintaining Just interval relationships, while modulating from key to key.
There is considerable evidence that singers and instrumentalists use Just intervals, when performing without the restriction of equal-tempered accompaniment. In concordance with the theory of relative consonance, originally proposed by Pythagoras, Just intervals are most “pure” or “consonant,” since they are built on the smallest ratios possible.
Here’s what Just intonation would look like, as applied to the twelve-tone scale. One would be forced to choose between the augmented fourth and the diminished fifth (as well as several other intervals), without the addition of an extra key:
| SEMITONE |
POSITION |
DIFFERS FROM EQUAL TEMPERAMENT |
| 1 (C) |
0 cents |
None (unison) |
| 2 (C# or Db) |
112 cents |
12 cents |
| 3 (D) |
182 cents |
- 18 cents |
| 4 (D# or Eb) |
316 cents |
16 cents |
| 5 (E or Fb) |
386 cents |
- 14 cents |
| 6 (F or E#) |
498 cents |
- 2 cents |
| 7 (F# or Gb) |
590/610cents |
-10/ 10cents(augmented fourth/diminished fifth) |
| 8 (G) |
702 cents |
2 cents |
| 9 (G# or Ab) |
814 cents |
14 cents |
| 10 (A) |
886 cents |
- 14 cents |
| 11 (A# or Bb) |
1018 cents |
18 cents |
| 12 (B or Cb) |
1088 cents |
- 12 cents |
| 13 (C) |
1200 cents |
None (octave) |
Note the large differences between Just and Equally-Tempered intervals. The tempered major thirds, especially, continue to be a source of torment for modern musicians. Click here to listen to a just major third, followed by a tempered major third, returning to a just major third.
Another interesting scale was developed more recently, by Wendy Carlos, who is probably the most influential figure in modern microtuning and electronic music. The Alpha scale, as she called it, consists of equal steps of 78 cents and was initially arrived at by splitting the minor third in four. Although this scale encompasses a large number of pure intervals (including the minor and major thirds, and the fifth), its most interesting characteristic is that it completely bypasses the octave, with the nearest cent values at 1170 cents before and 1248 cents after. Click here for an audio example.
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