Decimal: Difference between revisions

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-'''Decimal''' is a temperament in both the [[dicot]] and [[semaphore]] families of temperaments. It is also [[hemipyth]], meaning that it has approximations of √2, √3, and √(3/2). It is special in that it is also an [[exotemperament]].
+'''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]].
+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.
-As with many exotemperaments, it is not itself particularly useful, but it has structural value.
+For technical data, see [[Dicot family #Decimal]].
 == Interval chain ==
-In the following table, odd harmonics and subharmonics 1–7 are labeled in '''bold'''.
+In the following table, odd harmonics 1–9 and their inverses are in '''bold'''.
-{| class="wikitable center-1 center-2 center-3"
+{| class="wikitable center-1 right-2 right-4"
 ! 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
 | '''1/1'''
+| 600.0
 | 7/5, 10/7
 |-
 | 1
-| '''8/7''', 7/6
+| 351.0
-| '''8/5''', 5/3
+| '''5/4''', 6/5
+| 951.0
+| '''7/4''', 12/7
 |-
 | 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]].
-[[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:Rank-2 temperaments]]
+[[Category:Exotemperaments]]
+[[Category:Jubilismic clan]]
 [[Category:Dicot family]]
-[[Category:Slendro clan]]
+[[Category:Semaphoresmic clan]]
-[[Category:Jubilismic clan]]