Decimal
Decimal is an exotemperament in the dicot family, semaphoresmic clan, and jubilismic clan of temperaments. It is a weak extension of dicot, the 5-limit temperament tempering out 25/24, splitting the octave in two parts, each representing 7/5~10/7. It is also the prototypical fully hemipyth temperament, with sqrt(2) representing 7/5~10/7, sqrt(3) representing 7/4~12/7, sqrt(3/2) representng 5/4~6/5, and sqrt(4/3) representing 7/6~8/7, with a pergen of (P8/2, P4/2), splitting all Pythagorean intervals in two.
| Decimal |
7-limit 21-odd-limit: 35.3 ¢
7-limit 21-odd-limit: 10 notes
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. This equates the 4:5:6 and 6:7:8 triads with their inverses, therefore also equating the 4:5:6:7 major tetrad with the 1/(12:10:8:7) minor tetrad. The neutral third and semifourth 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, … tones.
Decimal can serve as a structural archetype for a decatonic 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 distinguishes major and minor 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 to half of 25/24 by tempering out 2401/2400, though its structure is more complex than that of pajara. 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 ¢ |