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'''Decimal''' is an [[exotemperament]] in both the [[dicot]] and [[semaphore]] families of temperaments. It is also the prototypical fully [[hemipyth]] temperament, with approximations of √2 at [[7/5]], √3 at [[7/4]], √(3/2) at [[5/4]] and √(4/3) at [[8/7]], and [[pergen]] (P8/2, P4/2), splitting all Pythagorean intervals.  
'''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 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 a neutral third approximating √(3/2) and [[7/6]][[~]][[8/7]] to be a neutral semifourth approximating √(4/3). 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 ...
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]]
For technical data, see [[Dicot family #Decimal]].


As with many exotemperaments, it is not itself particularly useful, but it has structural value.
== Interval chain ==
In the following table, odd harmonics 1–9 and their inverses are in '''bold'''.  


== Interval chain ==
{| class="wikitable center-1 right-2 right-4"
In the following table, odd harmonics and subharmonics 1–7 are labeled in '''bold'''.
{| class="wikitable center-1 center-2 center-3"
! rowspan="2" | #
! rowspan="2" | #
! colspan="1" | Period 0
! colspan="2" | Period 0
! colspan="1" | Period 1
! colspan="2" | Period 1
|-
! Approx. Ratios
! Approx. Ratios
|-
| -2
| '''3/2'''
| 21/20, 15/14
|-
|-
| -1
! Cents*
| 12/7, '''7/4'''
! Approx. ratios
| 6/5, '''5/4'''
! Cents*
! Approx. ratios
|-
|-
| 0
| 0
| 0.0
| '''1/1'''
| '''1/1'''
| 600.0
| 7/5, 10/7
| 7/5, 10/7
|-
|-
| 1
| 1
| '''8/7''', 7/6
| 351.0
| '''8/5''', 5/3
| '''5/4''', 6/5
| 951.0
| '''7/4''', 12/7
|-
|-
| 2
| 2
| '''4/3'''
| 701.9
| 28/15, 40/21
| '''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]].
One can see that the 10-note mos of the decimal temperament contains the 7-odd-limit [[tonality diamond]].


[[Category:Temperaments]]
== 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:Decimal| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Exotemperaments]]
[[Category:Jubilismic clan]]
[[Category:Dicot family]]
[[Category:Dicot family]]
[[Category:Slendro clan]]
[[Category:Semaphoresmic clan]]
[[Category:Jubilismic 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

7-limit prime-optimized tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~7/4 = 955.608 ¢ CWE: ~7/4 = 950.957 ¢ POTE: ~7/4 = 948.443 ¢
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