Decimal: Difference between revisions

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| MOS scales = [[4L 2s]], [[4L 6s]], [[10L 4s]]

| Mapping = 2; 2 1 1

| Ploidacot = diploid dicot

| Pergen = (P8/2, P4/2)

| Odd limit 1 = 7 | Mistuning 1 = 35.3 | Complexity 1 = 6

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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.

Decimal serves as a structural archetype for a 10-tone 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), and the 6:7:8 and 1/(6:7:8) chords as another ~~major-minor~~ pair, neutralized in decimal via ~~tempering~~ of 49/48.

Decimal serves as a structural archetype for a 10-tone system that views the [[4:5:6]] and [[10:12:15|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 [[21:24:28|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 does not neutralize these 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, though its structure is more complex than that of pajara, but easily extends to the [[11-limit]]. Both of these temperaments also temper out the marvel comma, [[225/224]].

@@ Line 7: / Line 7: @@
 | MOS scales = [[4L 2s]], [[4L 6s]], [[10L 4s]]
 | Mapping = 2; 2 1 1
+| Ploidacot = diploid dicot
 | Pergen = (P8/2, P4/2)
 | Odd limit 1 = 7 | Mistuning 1 = 35.3 | Complexity 1 = 6
@@ Line 15: / Line 16: @@
 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.
-Decimal serves as a structural archetype for a 10-tone 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), and the 6:7:8 and 1/(6:7:8) chords as another major-minor pair, neutralized in decimal via tempering of 49/48.
+Decimal serves as a structural archetype for a 10-tone system that views the [[4:5:6]] and [[10:12:15|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 [[21:24:28|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 does not neutralize these 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, though its structure is more complex than that of pajara, but easily extends to the [[11-limit]]. Both of these temperaments also temper out the marvel comma, [[225/224]].