Carlos Alpha
Carlos Alpha is a tuning system conceptualized as a chain of steps between a 15edo and 16edo semitone in size, which reach a fifth after nine steps, 5/4 after five steps, and 8/7 after three steps. Originally, Carlos Alpha was conceptualized as multiple of these chains offset by octaves, but is also often used as a non-octave tuning system.
In regular temperament theory, Carlos Alpha corresponds to the valentine temperament, which has an octave as a period and a semitone of about 78 cents as a generator. It also easily admits an extension to the 2.3.5.7.11.17.23 subgroup to a high degree of accuracy.
It can be seen as equalizing the steps between 20:21:22:23:24:25, which suggests a perhaps somewhat inaccurate extension to the full 23-limit which equalizes 19:20:21:22:23:24:25:26; this is the most accurately tuned around 31edo. Valentine tempers out the slendrisma and can thus be seen as an extension splitting each step of slendric into three parts. In this way, it is a counterpart to miracle, which splits the slendric generator into two parts.
| Note | Up from the unison | Interpretation | Down from the octave | Interpretation |
|---|---|---|---|---|
| 0 | 0c | 1/1 | 1200c | 2/1 |
| 1 | 78c | 21/20, 25/24 | 1122c | 48/25, 40/21 |
| 2 | 156c | 11/10, 12/11 | 1044c | 20/11, 11/6 |
| 3 | 234c | 8/7 | 966c | 7/4 |
| 4 | 312c | 6/5 | 888c | 5/3 |
| 5 | 390c | 5/4 | 810c | 8/5 |
| 6 | 468c | 21/16 | 732c | 32/21 |
| 7 | 546c | 11/8 | 654c | 16/11 |
| 8 | 624c | 23/16 | 576c | 32/23 |
| 9 | 702c | 3/2 | 498c | 4/3 |
| 10 | 780c | 11/7 | 420c | 14/11 |
| 11 | 858c | 18/11 | 342c | 11/9 |
| 12 | 936c | 12/7 | 264c | 7/6 |
| 13 | 1014c | 9/5 | 186c | 10/9 |
| 14 | 1092c | 15/8, 32/17 | 108c | 16/15, 17/16 |
| 15 | 1170c | 96/49 | 30c | 49/48 |
As a non-octave temperament, it is essentially 9edf, in which case the second column of this table can be ignored and an in-tune 4/1 may be found at 31 generators up; 9edf can be seen as a 4.3.5.(8/7).11 temperament.
