Decimal: Difference between revisions

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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 both the [[dicot]] and [[semaphore]] families 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 ~~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 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, ….

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.

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! colspan="1" | Period 1

|-

! Approx. ~~Ratios~~

! Approx. ratios

! Approx. ~~Ratios~~

! Approx. ratios

|-

| -2

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[[Category:Rank-2 temperaments]]

[[Category:Exotemperaments]]

[[Category:Jubilismic clan]]

[[Category:Dicot family]]

[[Category:Semaphoresmic clan]]

~~[[Category:Jubilismic clan]]~~

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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 both the [[dicot]] and [[semaphore]] families 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 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 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, ….
-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.
@@ Line 14: / Line 14: @@
 ! colspan="1" | Period 1
 |-
-! Approx. Ratios
+! Approx. ratios
-! Approx. Ratios
+! Approx. ratios
 |-
 | -2
@@ Line 43: / Line 43: @@
 [[Category:Rank-2 temperaments]]
 [[Category:Exotemperaments]]
+[[Category:Jubilismic clan]]
 [[Category:Dicot family]]
 [[Category:Semaphoresmic clan]]
-[[Category:Jubilismic clan]]