Schismatic family: Difference between revisions

Line 56:

Garibaldi tempers out the [[garischisma]], equating the [[septimal comma]] with both the [[syntonic comma]] and the [[Pythagorean comma]]. The 7/4 is found at -14 fifths, represented by the double diminished octave (C-Cbb), or down-minor seventh (C-vBb) with the down-arrow representing the comma step. It necessitates a sharper fifth than pure.

Garibaldi tempers out the [[garischisma]], equating the [[septimal comma]] with both the [[syntonic comma]] and the [[Pythagorean comma]]. The 7/4 is found at -14 fifths, represented by the double diminished octave (C-Cbb), or down-minor seventh (C-vBb) with the down-arrow representing the comma step. It necessitates a sharper fifth than pure. Its [[S-expression]]-based comma list is {[[5120/5103|S8/S9]], [[225/224|S15]]}.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 225/224, 3125/3087

~~[[S-expression]]-based comma list: {[[5120/5103|S8/S9]], [[225/224|S15]]}~~

Line 83:

Line 81:

* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [700.000, 703.448] (7\12 to 17\29)

* 7- and 9-odd-limit [[diamond tradeoff]]: ~3/2 = [701.711, 702.915]

* 7- and 9-odd-limit diamond monotone and tradeoff: ~3/2 = [701.711, 702.915]

{{Optimal ET sequence|legend=1| 12, 29, 41, 53, 94, 241c, 335cd, 576ccd }}

Line 107:

Line 104:

* 11-odd-limit diamond monotone: ~3/2 = [701.887, 702.439] (31\53 to 24\41)

* 11-odd-limit diamond tradeoff: ~3/2 = [701.711, 702.915]

* 11-odd-limit diamond monotone and tradeoff: ~3/2 = [701.887, 702.439]

{{Optimal ET sequence|legend=1| 41, 53, 94, 229c, 323c, 417cce }}

Line 124:

Line 120:

Minimax tuning:

* 13- and 15-odd-limit: ~3/2 = {{monzo| 19/34 0 0 -1/34 0 1/34 }}

: ~~Eigenmonzo~~ (unchanged-interval) basis: 2.13/7

: eigenmonzo (unchanged-interval) basis: 2.13/7

Tuning ranges:

* 13- and 15-odd-limit diamond monotone: ~3/2 = [701.887, 702.439] (31\53 to 24\41)

* 13-odd-limit diamond tradeoff: ~3/2 = [701.711, 703.597]

* 15-odd-limit diamond tradeoff: ~3/2 = [701.676, 703.597]

* 13- and 15-odd-limit diamond monotone and tradeoff: ~3/2 = [701.887, 702.439]

{{Optimal ET sequence|legend=1| 41, 53, 94, 429ccdeef, 523ccdeef }}

Line 230:

Line 225:

* 11-odd-limit diamond monotone: ~3/2 = [700.000, 703.448] (7\12 to 17\29)

* 11-odd-limit diamond tradeoff: ~3/2 = [701.711, 704.377]

* 11-odd-limit diamond monotone and tradeoff: ~3/2 = [701.711, 703.448]

Line 253:

Line 247:

* 13-odd-limit diamond tradeoff: ~3/2 = [701.711, 704.377]

* 15-odd-limit diamond tradeoff: ~3/2 = [701.676, 704.377]

* 13- and 15-odd-limit diamond monotone and tradeoff: ~3/2 = [702.439, 703.448]

{{Optimal ET sequence|legend=1| 12f, 29, 41, 152cdf, 193cdf, 234cdf }}

Line 365:

Line 358:

Minimax tuning:

* 13- and 15-odd-limit: ~3/2 = {{monzo| 19/32 1/16 0 0 -1/32 }}

: ~~Eigenmonzo~~ (unchanged-interval) basis: 2.11/9

: eigenmonzo (unchanged-interval) basis: 2.11/9

@@ Line 56: / Line 56: @@
 {{Main| Garibaldi temperament }}
-Garibaldi tempers out the [[garischisma]], equating the [[septimal comma]] with both the [[syntonic comma]] and the [[Pythagorean comma]]. The 7/4 is found at -14 fifths, represented by the double diminished octave (C-Cbb), or down-minor seventh (C-vBb) with the down-arrow representing the comma step. It necessitates a sharper fifth than pure.
+Garibaldi tempers out the [[garischisma]], equating the [[septimal comma]] with both the [[syntonic comma]] and the [[Pythagorean comma]]. The 7/4 is found at -14 fifths, represented by the double diminished octave (C-Cbb), or down-minor seventh (C-vBb) with the down-arrow representing the comma step. It necessitates a sharper fifth than pure. Its [[S-expression]]-based comma list is {[[5120/5103|S8/S9]], [[225/224|S15]]}.
 [[Subgroup]]: 2.3.5.7
 [[Comma list]]: 225/224, 3125/3087
-[[S-expression]]-based comma list: {[[5120/5103|S8/S9]], [[225/224|S15]]}
 {{Mapping|legend=1| 1 0 15 25 | 0 1 -8 -14 }}
@@ Line 83: / Line 81: @@
 * 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [700.000, 703.448] (7\12 to 17\29)
 * 7- and 9-odd-limit [[diamond tradeoff]]: ~3/2 = [701.711, 702.915]
-* 7- and 9-odd-limit diamond monotone and tradeoff: ~3/2 = [701.711, 702.915]
 {{Optimal ET sequence|legend=1| 12, 29, 41, 53, 94, 241c, 335cd, 576ccd }}
@@ Line 107: / Line 104: @@
 * 11-odd-limit diamond monotone: ~3/2 = [701.887, 702.439] (31\53 to 24\41)
 * 11-odd-limit diamond tradeoff: ~3/2 = [701.711, 702.915]
-* 11-odd-limit diamond monotone and tradeoff: ~3/2 = [701.887, 702.439]
 {{Optimal ET sequence|legend=1| 41, 53, 94, 229c, 323c, 417cce }}
@@ Line 124: / Line 120: @@
 Minimax tuning:
 * 13- and 15-odd-limit: ~3/2 = {{monzo| 19/34 0 0 -1/34 0 1/34 }}
-: Eigenmonzo (unchanged-interval) basis: 2.13/7
+: eigenmonzo (unchanged-interval) basis: 2.13/7
 Tuning ranges:
 * 13- and 15-odd-limit diamond monotone: ~3/2 = [701.887, 702.439] (31\53 to 24\41)
 * 13-odd-limit diamond tradeoff: ~3/2 = [701.711, 703.597]
 * 15-odd-limit diamond tradeoff: ~3/2 = [701.676, 703.597]
-* 13- and 15-odd-limit diamond monotone and tradeoff: ~3/2 = [701.887, 702.439]
 {{Optimal ET sequence|legend=1| 41, 53, 94, 429ccdeef, 523ccdeef }}
@@ Line 230: / Line 225: @@
 * 11-odd-limit diamond monotone: ~3/2 = [700.000, 703.448] (7\12 to 17\29)
 * 11-odd-limit diamond tradeoff: ~3/2 = [701.711, 704.377]
-* 11-odd-limit diamond monotone and tradeoff: ~3/2 = [701.711, 703.448]
 {{Optimal ET sequence|legend=1| 12, 29, 41, 217ce, 258ce }}
@@ Line 253: / Line 247: @@
 * 13-odd-limit diamond tradeoff: ~3/2 = [701.711, 704.377]
 * 15-odd-limit diamond tradeoff: ~3/2 = [701.676, 704.377]
-* 13- and 15-odd-limit diamond monotone and tradeoff: ~3/2 = [702.439, 703.448]
 {{Optimal ET sequence|legend=1| 12f, 29, 41, 152cdf, 193cdf, 234cdf }}
@@ Line 365: / Line 358: @@
 Minimax tuning:
 * 13- and 15-odd-limit: ~3/2 = {{monzo| 19/32 1/16 0 0 -1/32 }}
-: Eigenmonzo (unchanged-interval) basis: 2.11/9
+: eigenmonzo (unchanged-interval) basis: 2.11/9
 {{Optimal ET sequence|legend=1| 12f, 41ef, 53, 118d, 171de }}