Decimal: Difference between revisions
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'''Decimal''' is an [[exotemperament]] in the [[dicot family]], [[semaphoresmic clan]], and [[jubilismic clan]] of [[regular temperament|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 [[tempering out|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 {{nowrap|(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. | |||
For technical data, see [[Dicot family #Decimal]]. | |||
== Interval chain == | |||
In the following table, odd harmonics 1–9 and their inverses are in '''bold'''. | |||
{| class="wikitable center-1 right-2 right-4" | |||
! rowspan="2" | # | |||
! colspan="2" | Period 0 | |||
! colspan="2" | 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 | |||
|} | |||
<nowiki/>* 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 == | |||
{| class="wikitable mw-collapsible mw-collapsed" | |||
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit prime-optimized tunings | |||
|- | |||
! rowspan="2" | | |||
! colspan="3" | Euclidean | |||
|- | |||
! Constrained | |||
! Constrained & skewed | |||
! Destretched | |||
|- | |||
! Tenney | |||
| CTE: ~7/4 = 955.608{{c}} | |||
| CWE: ~7/4 = 950.957{{c}} | |||
| POTE: ~7/4 = 948.443{{c}} | |||
|} | |||
[[Category:Decimal| ]] <!-- main article --> | |||
[[Category:Rank-2 temperaments]] | |||
[[Category:Exotemperaments]] | |||
[[Category:Jubilismic clan]] | |||
[[Category:Dicot family]] | |||
[[Category:Semaphoresmic clan]] | |||
Latest revision as of 13:36, 22 July 2025
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.
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 ¢ |