Decimal
Lua error in Module:Infobox_regtemp at line 138: attempt to perform arithmetic on local 'generator_size' (a nil value). Decimal is an exotemperament in the dicot family, semaphoresmic clan, and jubilismic clan of temperaments. It is also the prototypical fully hemipyth temperament, with approximations of 7/5~10/7 at sqrt(2), 7/4~12/7 at sqrt(3), 5/4~6/5 at sqrt(3/2) and 7/6~8/7 at sqrt(4/3), and pergen (P8/2, P4/2), splitting all Pythagorean intervals.
More precisely, it is the 7-limit temperament that tempers out both 25/24, the classic chromatic semitone, and 49/48, the septimal diesis. These two intervals have a rather similar function separating close intervals and creating "major" and "minor" triads (either pental ones splitting the perfect fifth or septimal ones splitting the perfect fourth), and tempering them out allows 5/4~6/5 to be sqrt(3/2) a neutral third and 7/6~8/7 to be a sqrt(4/3) neutral semifourth. These can be equated (far more accurately) to 11/9 and 15/13 respectively, tempering out 243/242 and 676/675 and extending this temperament to the 13-limit. Since (25/24)/(49/48) = 50/49, it also tempers that out, splitting the octave in two equal parts. As both the generator and period are half that of the diatonic scale, this means it forms mos scales of 4, 6, 10, 14, 24, 38, … tones.
Decimal serves as a structural archetype for a 10-tone system that views the 4:5:6 and 1/(4:5:6) chords as a major–minor pair (which is equated in decimal temperament as 25/24 is tempered out), and the 6:7:8 and 1/(6:7:8) chords as another major–minor pair, neutralized in decimal via vanishing of 49/48.
A more accurate system based on 10 interval classes that does not neutralize these chords is pajara, where 50/49 remains tempered and 49/48 is equated to 25/24. An even more accurate one is miracle, which equates 50/49 with 49/48, though its structure is more complex than that of pajara, but easily extends to the 11-limit. Both of these temperaments also temper out the marvel comma, 225/224.
For technical data, see Dicot family #Decimal.
Interval chain
In the following table, odd harmonics 1–9 and their inverses are in bold.
| # | Period 0 | Period 1 | ||
|---|---|---|---|---|
| Cents* | Approx. ratios | Cents* | Approx. ratios | |
| 0 | 0.0 | 1/1 | 600.0 | 7/5, 10/7 |
| 1 | 351.0 | 5/4, 6/5 | 951.0 | 7/4, 12/7 |
| 2 | 701.9 | 3/2 | 101.9 | 15/14, 21/20 |
| 3 | 1052.9 | 9/5, 15/8 | 452.9 | 9/7, 21/16 |
| 4 | 203.8 | 9/8 | 803.8 | 45/28, 54/35 |
| 5 | 554.8 | 27/20, 45/32 | 1154.8 | 27/14, 63/32 |
* In 7-limit CWE tuning, octave reduced
One can see that the 10-note mos of the decimal temperament contains the 7-odd-limit tonality diamond.
Tunings
| Euclidean | |||
|---|---|---|---|
| Constrained | Constrained & skewed | Destretched | |
| Tenney | CTE: ~7/4 = 955.608 ¢ | CWE: ~7/4 = 950.957 ¢ | POTE: ~7/4 = 948.443 ¢ |