|
|
| (26 intermediate revisions by the same user not shown) |
| Line 1: |
Line 1: |
| | |
| | {{Flat}}{{natural}}{{sharp}} |
| | |
| | ♭♮♯ |
| | |
| | <span class="bravura"></span> |
| | <span class="bravura"></span> |
| | <span class="bravura"></span> |
| | |
| | <span class="heji">n</span> |
| | |
| | == Template: Temperament data == |
| {{User:FloraC/Temperament data | | {{User:FloraC/Temperament data |
| | Subgroup = 2.3.5.7 | | | Subgroup = 2.3.5.7 |
| Line 140: |
Line 152: |
| |} | | |} |
|
| |
|
| == Tunings to be reviewed == | | == Tunings == |
| {| class="wikitable mw-collapsible mw-collapsed" | | {| class="wikitable mw-collapsible mw-collapsed" |
| |+style=white-space:nowrap | 5-limit Prime-Optimized Tunings | | |+style=white-space:nowrap | 5-limit Prime-Optimized Tunings |
| |- | | |- |
| ! Weight-skew\Order !! Chebyshevian !! Euclidean !! Manhattan | | ! rowspan="2" | |
| |- | | ! rowspan="2" | Manhattan |
| | Tenney || CTC<br>~3/2 = 696.5784¢<br>Eigenmonzo basis: 2.5 (1/4-comma tuning) || CTE<br>~3/2 = 697.2143¢ || CTOP<br>~3/2 = 698.0197¢
| | ! colspan="2" | Euclidean |
| | ! rowspan="2" | Chebyshevian |
| |- | | |- |
| | Tenney-Weil || CTWC<br>~3/2 = 696.5784¢<br>Eigenmonzo basis: 2.5 (1/4-comma tuning) || CTWE<br>~3/2 = 696.6512¢ || CTWOP<br>~3/2 = 696.5784¢<br>Eigenmonzo basis: 2.5 (1/4-comma tuning)
| | ! Unskewed |
| | ! Skewed |
| |- | | |- |
| | Equilateral || CEC<br>~3/2 = 696.5784¢<br>Eigenmonzo basis: 2.5 (1/4-comma tuning) || CEE<br>~3/2 = 696.8947¢<br>Eigenmonzo basis: 2.1875 (4/17-comma tuning) || CEOP<br>~3/2 = 697.6537¢<br>Eigenmonzo basis: 2.15 (1/5-comma tuning)
| | ! Equilateral |
| | | CEOP: ~3/2 = 697.654¢<br>(1/5-comma) |
| | | CEE: ~3/2 = 696.895¢<br>(4/17-comma) |
| | | CSEE: ~3/2 = 696.453¢<br>(11/43-comma) |
| | | CEC: ~3/2 = 696.578¢<br>(1/4-comma) |
| |- | | |- |
| | Equilateral-Weil || CEWC<br>~3/2 = 696.5784¢<br>Eigenmonzo basis: 2.5 (1/4-comma tuning) || CEWE<br>~3/2 = 696.4534¢<br>Eigenmonzo basis: 2.48828125/3 (11/43-comma tuning) || CEWOP<br>~3/2 = 696.5784¢<br>Eigenmonzo basis: 2.5 (1/4-comma tuning) | | ! Tenney |
| | | CTOP: ~3/2 = 698.020¢ |
| | | CTE: ~3/2 = 697.214¢ |
| | | CWE: ~3/2 = 696.651¢ |
| | | CTC: ~3/2 = 696.578¢<br>(1/4-comma) |
| |- | | |- |
| | Benedetti || CBC<br>~3/2 = 696.5784¢<br>Eigenmonzo basis: 2.5 (1/4-comma tuning) || CBE<br>~3/2 = 697.3738¢<br>Eigenmonzo basis: 2.{{monzo| 0 25 36 }} (36/169-comma tuning) || CBOP<br>~3/2 = 698.1598¢<br>Eigenmonzo basis: 2.30375 (3/17-comma tuning)
| | ! Benedetti, <br>Wilson |
| |- | | | CBOP: ~3/2 = 698.160¢<br>(3/17-comma) |
| | Benedetti-Weil || CBWC<br>~3/2 = 696.5784¢<br>Eigenmonzo basis: 2.5 (1/4-comma tuning) || CBWE<br>~3/2 = 696.7868¢<br>Eigenmonzo basis: 2.{{monzo| 0 5 31 }} (31/129-comma tuning) || CBWOP<br>~3/2 = 696.5784¢<br>Eigenmonzo basis: 2.5 (1/4-comma tuning)
| | | CBE: ~3/2 = 697.374¢<br>(36/169-comma) |
| | | CSBE: ~3/2 = 696.787¢<br>(31/129-comma) |
| | | CBC: ~3/2 = 696.578¢<br>(1/4-comma) |
| |} | | |} |
|
| |
|
| Line 162: |
Line 186: |
| |+style=white-space:nowrap | 7-limit Prime-Optimized Tunings | | |+style=white-space:nowrap | 7-limit Prime-Optimized Tunings |
| |- | | |- |
| ! Weight-skew\Order !! Chebyshevian !! Euclidean !! Manhattan | | ! rowspan="2" | |
| | ! rowspan="2" | Manhattan |
| | ! colspan="2" | Euclidean |
| | ! rowspan="2" | Chebyshevian |
| |- | | |- |
| | Tenney || CTC<br>~3/2 = 696.8826¢<br>Eigenmonzo basis: 2.7 || CTE<br>~3/2 = 696.9521¢ || CTOP<br>~3/2 = 696.6458¢
| | ! Unskewed |
| | ! Skewed |
| |- | | |- |
| | Tenney-Weil || CTWC<br>~3/2 = 696.8826¢<br>Eigenmonzo basis: 2.7 || CTWE<br>~3/2 = 696.6562¢ || CTWOP<br>~3/2 = 696.8826¢<br>Eigenmonzo basis: 2.7 | | ! Equilateral |
| | | CEOP: ~3/2 = 697.344¢ |
| | | CEE: ~3/2 = 696.884¢ |
| | | CSEE: ~3/2 = 696.725¢ |
| | | CEC: ~3/2 = 696.883¢ |
| |- | | |- |
| | Equilateral || CEC<br>~3/2 = 696.8826¢<br>Eigenmonzo basis: 2.7 || CEE<br>~3/2 = 696.8843¢<br>Eigenmonzo basis: 2.{{monzo| 0 1 4 10 }} || CEOP<br>~3/2 = 697.3437¢<br>Eigenmonzo basis: 2.21 | | ! Tenney |
| | | CTOP: ~3/2 = 696.646¢ |
| | | CTE: ~3/2 = 696.952¢ |
| | | CWE: ~3/2 = 696.656¢ |
| | | CTC: ~3/2 = 696.883¢ |
| |- | | |- |
| | Equilateral-Weil || CEWC<br>~3/2 = 696.8826¢<br>Eigenmonzo basis: 2.7 || CEWE<br>~3/2 = 696.7248¢<br>Eigenmonzo basis: 2.4117715/9 || CEWOP<br>~3/2 = 696.8826¢<br>Eigenmonzo basis: 2.7
| | ! Benedetti, <br>Wilson |
| |-
| | | CBOP: ~3/2 = 697.842¢ |
| | Benedetti || CBC<br>~3/2 = 696.8826¢<br>Eigenmonzo basis: 2.7 || CBE<br>~3/2 = 697.0147¢<br>Eigenmonzo basis: 2.{{monzo| 0 1225 1764 2250 }} || CBOP<br>~3/2 = 697.8422¢<br>Eigenmonzo basis: 2.750141
| | | CBE: ~3/2 = 697.015¢ |
| |- | | | CSBE: ~3/2 = 696.631¢ |
| | Benedetti-Weil || CBWC<br>~3/2 = 696.8826¢<br>Eigenmonzo basis: 2.7 || CBWE<br>~3/2 = 696.6306¢<br>Eigenmonzo basis: 2.{{monzo| 0 -3290 3171 7215 }} || CBWOP<br>~3/2 = 696.8826¢<br>Eigenmonzo basis: 2.7
| | | CBC: ~3/2 = 696.883¢ |
| |} | | |} |
|
| |
|
| == Temperament pages == | | == Temperament pages == |
| Note: | | Note: |
| # Order: [[subgroup]], [[comma list]], ([[sval]]) [[mapping]], (sval) mapping generators, gencom mapping, [[gencom]], lattice basis, [[wedgie]], [[optimal tuning]] ([[CTE]]), [[minimax tuning]], [[tuning ranges]], [[algebraic generator]], [[Optimal ET sequence]], [[badness]], [[complexity spectrum]], and others. | | # Order: [[subgroup]], [[comma list]], ([[sval]]) [[mapping]], (sval) mapping generators, gencom mapping, [[gencom]], lattice basis, [[wedgie]], [[optimal tuning]]s ([[CTE]], [[CWE]]/[[POTE]]), [[minimax tuning]], [[tuning ranges]], [[algebraic generator]], [[optimal ET sequence]], [[badness]], [[complexity spectrum]], and others. |
| # Comma list should show the simplest commas sufficient to define the temperament, stated in [[Normal lists #Normal interval list]]. | | # Comma list should show the simplest commas sufficient to define the temperament, stated in [[Normal lists #Normal interval list]]. |
| # Mapping generators should show all the ratios as used in the mapping, including the period. | | # Mapping generators should show all the ratios as used in the mapping, including the period. |
| # Since minimax tunings are based on tonality diamond, it should explicitly state the ''odd'' limit, or a diamond function of ratios. | | # Since minimax tunings are based on tonality diamond, it should explicitly state the ''odd'' limit, or a diamond function of ratios. |
| # For subgroup temperaments, "mapping" becomes "sval mapping", add "gencom mapping" and "gencom". If TE is TE is TE (''sic''), simply show "CTE", otherwise show "subgroup CTE" or "inharmonic CTE" instead. | | # For subgroup temperaments, "mapping" becomes "sval mapping", add "gencom mapping" and "gencom". If TE is TE is TE (''sic''), simply show "CTE/CWE", otherwise show "subgroup/inharmonic CTE/CWE" instead. |
|
| |
|
| Get a family for: | | Get a family for: |
| Line 232: |
Line 268: |
| * <s>Kleismic family</s> | | * <s>Kleismic family</s> |
| * <s>Kleismic rank three family</s> | | * <s>Kleismic rank three family</s> |
| * [[Hemimean clan]] | | * <s>Würschmidt family</s> |
| * [[Hemimean family]]
| |
| * <s>Unicorn family</s> | | * <s>Unicorn family</s> |
| * <s>Shibboleth family</s> | | * <s>Shibboleth family</s> |
| Line 243: |
Line 278: |
| * <s>Sycamore family</s> | | * <s>Sycamore family</s> |
| * <s>Semicomma family</s> | | * <s>Semicomma family</s> |
| * [[Orwellismic temperaments]] | | * <s>Orwellismic temperaments</s> |
| * <s>Orwellismic family</s> | | * <s>Orwellismic family</s> |
| * [[Hemimage temperaments]] | | * <s>Hemimean clan</s> |
| * [[Hemimage family]] | | * <s>Hemimean family</s> |
| * [[Porwell temperaments]] | | * <s>Hemifamity temperaments</s> |
| | * <s>Hemifamity family</s> |
| | * <s>Porwell temperaments</s> |
| * <s>Porwell family</s> | | * <s>Porwell family</s> |
| * [[Hemifamity temperaments]] | | * <s>Hemimage temperaments</s> |
| * [[Hemifamity family]] | | * <s>Hemimage family</s> |
| * [[Porcupine family]] | | * <s>Porcupine family</s> |
| * [[Porcupine rank three family]] | | * <s>Porcupine rank three family</s> |
| * [[Tetracot family]] | | * <s>Tetracot family</s> |
| * [[Diaschismic family]] | | * <s>Diaschismic family</s> |
| * [[Diaschismic rank three family]] | | * <s>Diaschismic rank three family</s> |
| * [[Compton family]] | | * <s>Compton family</s> |
| * [[Amity family]] | | * <s>Amity family</s> |
| * [[Misty family]] | | * <s>Misty family</s> |
| * [[Undim family]] | | * <s>Undim family</s> |
| * [[Lehmerismic temperaments]] | | * <s>Lehmerismic temperaments</s> |
| * [[Kalismic temperaments]] | | * <s>Kalismic temperaments</s> |
| * [[Ragismic microtemperaments]] | | * <s>Ragismic microtemperaments</s> |
| * <s>Ragismic family</s> | | * <s>Ragismic family</s> |
| * [[Landscape microtemperaments]] | | * <s>Landscape microtemperaments</s> |
| * [[Landscape family]] | | * <s>Landscape family</s> |
| * [[Dimcomp family]] | | * <s>Dimcomp family</s> |
| * [[Mirkwai clan]] | | * <s>Mirkwai clan</s> |
| * [[Mirkwai family]] | | * <s>Mirkwai family</s> |
| * [[Breedsmic temperaments]] | | * <s>Quince clan</s> |
| * [[Breed family]] | | * <s>Breedsmic temperaments</s> |
| * [[Escapade family]] | | * <s>Breed family</s> |
| * [[Gravity family]] | | * <s>Escapade family</s> |
| * [[Cataharry temperaments]] | | * <s>Gravity family</s> |
| * [[Cataharry family]] | | * <s>Cataharry temperaments</s> |
| * [[Varunismic temperaments]] | | * <s>Cataharry family</s> |
| | * <s>Varunismic temperaments</s> |
| * <s>Rastmic rank three clan</s> | | * <s>Rastmic rank three clan</s> |
| * <s>Biyatismic clan</s> | | * <s>Biyatismic clan</s> |
| * [[Ptolemismic clan]] | | * <s>Ptolemismic clan</s> |
| * <s>Valinorsmic clan</s> | | * <s>Valinorsmic clan</s> |
| * <s>Pentacircle clan</s> | | * <s>Pentacircle clan</s> |
| * [[Keenanismic temperaments]] | | * <s>Keenanismic temperaments</s> |
| * [[Werckismic temperaments]] | | * <s>Werckismic temperaments</s> |
| * [[Swetismic temperaments]] | | * <s>Swetismic temperaments</s> |
| * [[Wizmic microtemperaments]] | | * <s>Wizmic microtemperaments</s> |
| * [[Metric microtemperaments]] | | * <s>Metric microtemperaments</s> |
| | * <s>Horwell temperaments</s> |
| | * <s>Horwell family</s> |
| * <s>Luna family</s> | | * <s>Luna family</s> |
| * [[Vulture family]] | | * <s>Vulture family</s> |
| * [[Tricot family]] | | * <s>Tricot family</s> |
| * [[Minortonic family]] | | * <s>Minortonic family</s> |
| * [[Gammic family]] | | * <s>Gammic family</s> |
| * [[Garischismic clan]] | | * <s>Vishnuzmic family</s> |
| * [[Canousmic temperaments]] | | * <s>Garischismic clan</s> |
| * [[Canou family]] | | * <s>Canousmic temperaments</s> |
| * [[Semicanousmic clan]] | | * <s>Canou family</s> |
| * [[Semiporwellismic clan]] | | * <s>Semicanousmic clan</s> |
| * [[Olympic clan]] | | * <s>Semiporwellismic clan</s> |
| * [[Alphaxenic rank three clan]] | | * <s>Olympic clan</s> |
| * [[Mirwomo temperaments]] | | * <s>Alphaxenic rank three clan</s> |
| * [[Horwell temperaments]] | | * <s>Mirwomo temperaments</s> |
| * [[Horwell family]] | | * <s>Gariboh clan</s> |
| * [[Gariboh clan]] | | * <s>Gariboh family</s> |
| * [[Gariboh family]] | | * <s>Octagar temperaments</s> |
| * [[Octagar temperaments]] | | * <s>Octagar family</s> |
| * [[Octagar family]] | | * <s>Wesley family</s> |
| | * <s>Apotome family</s> |
| | * [[Mabila family]] |
| | * [[Maquila family]] |
| | * [[Maja family]] |
| | * [[Ditonmic family]] |
| | * <s>Greenwoodmic temperaments</s> |
| | * <s>Avicennmic temperaments</s> |
|
| |
|
| == Regular temperament properties == | | == Regular temperament properties == |
| Line 367: |
Line 414: |
| ; 12-tone | | ; 12-tone |
| : [[1L 11s]], [[2L 10s]], [[3L 9s]], [[4L 8s]], <s>5L 7s</s>, <s>6L 6s</s>, <s>7L 5s</s>, [[8L 4s]], [[9L 3s]], [[10L 2s]], [[11L 1s]] | | : [[1L 11s]], [[2L 10s]], [[3L 9s]], [[4L 8s]], <s>5L 7s</s>, <s>6L 6s</s>, <s>7L 5s</s>, [[8L 4s]], [[9L 3s]], [[10L 2s]], [[11L 1s]] |
|
| |
| (Bounded by branch depth = 7)
| |
|
| |
| {| class="wikitable center-all"
| |
| ! colspan="7" | Generator
| |
| ! Cents
| |
| ! L
| |
| ! s
| |
| ! L/s
| |
| ! Comments
| |
| |-
| |
| | 4\7 || || || || || || || 685.714 || 1 || 1 || 1.000 ||
| |
| |-
| |
| | || || || || || || 27\47 || 689.362 || 7 || 6 || 1.167 ||
| |
| |-
| |
| | || || || || || 23\40 || || 690.000 || 6 || 5 || 1.200 ||
| |
| |-
| |
| | || || || || || || 42\73 || 690.411 || 11 || 9 || 1.222 ||
| |
| |-
| |
| | || || || || 19\33 || || || 690.909 || 5 || 4 || 1.250 ||
| |
| |-
| |
| | || || || || || || 53\92 || 691.304 || 14 || 11 || 1.273 ||
| |
| |-
| |
| | || || || || || 34\59 || || 691.525 || 9 || 7 || 1.286 ||
| |
| |-
| |
| | || || || || || || 49\85 || 691.765 || 13 || 10 || 1.300 ||
| |
| |-
| |
| | || || || 15\26 || || || || 692.308 || 4 || 3 || 1.333 ||
| |
| |-
| |
| | || || || || || || 56\97 || 692.784 || 15 || 11 || 1.364 ||
| |
| |-
| |
| | || || || || || 41\71 || || 692.958 || 11 || 8 || 1.375 ||
| |
| |-
| |
| | || || || || || || 67\116 || 693.103 || 18 || 13 || 1.385 ||
| |
| |-
| |
| | || || || || 26\45 || || || 693.333 || 7 || 5 || 1.400 ||
| |
| |-
| |
| | || || || || || || 63\109 || 693.578 || 17 || 12 || 1.417 ||
| |
| |-
| |
| | || || || || || 37\64 || || 693.750 || 10 || 7 || 1.429 ||
| |
| |-
| |
| | || || || || || || 48\83 || 693.976 || 13 || 9 || 1.444 ||
| |
| |-
| |
| | || || 11\19 || || || || || 694.737 || 3 || 2 || 1.500 || L/s = 3/2
| |
| |-
| |
| | || || || || || || 51\88 || 695.455 || 14 || 9 || 1.556 ||
| |
| |-
| |
| | || || || || || 40\69 || || 695.652 || 11 || 7 || 1.571 ||
| |
| |-
| |
| | || || || || || || 69\119 || 695.798 || 19 || 12 || 1.583 ||
| |
| |-
| |
| | || || || || 29\50 || || || 696.000 || 8 || 5 || 1.600 ||
| |
| |-
| |
| | || || || || || || 66\131 || 696.183 || 21 || |13 || 1.615 || Golden meantone
| |
| |-
| |
| | || || || || || 47\81 || || 696.296 || 13 || 8 || 1.625 ||
| |
| |-
| |
| | || || || || || || 65\112 || 696.429 || 18 || 11 || 1.636 ||
| |
| |-
| |
| | || || || 18\31 || || || || 696.774 || 5 || 3 || 1.667 || Meantone is in this region
| |
| |-
| |
| | || || || || || || 61\105 || 697.143 || 17 || 10 || 1.700 ||
| |
| |-
| |
| | || || || || || 43\74 || || 697.297 || 12 || 7 || 1.714 ||
| |
| |-
| |
| | || || || || || || 68\117 || 697.436 || 19 || 11 || 1.727 ||
| |
| |-
| |
| | || || || || 25\43 || || || 697.674 || 7 || 4 || 1.750 ||
| |
| |-
| |
| | || || || || || || 57\98 || 697.959 || 16 || 9 || 1.778 ||
| |
| |-
| |
| | || || || || ||32\55 || || 698.182 || 9 || 5 || 1.800 ||
| |
| |-
| |
| | || || || || || || 39\67 || 698.507 || 11 || 6 || 1.833 ||
| |
| |-
| |
| | || 7\12 || || || || || || 700.000 || 2 || 1 || 2.000 || Basic diatonic<br>(Generators smaller than this are proper)
| |
| |-
| |
| | || || || || || || 38\65 || 701.539 || 11 || 5 || 2.200 ||
| |
| |-
| |
| | || || || || || 31\53 || || 701.887 || 9 || 4 || 2.250 ||
| |
| |-
| |
| | || || || || || || 55\94 || 702.128 || 16 || 7 || 2.286 ||
| |
| |-
| |
| | || || || || 24\41 || || || 702.409 || 7 || 3 || 2.333 ||
| |
| |-
| |
| | || || || || || || 65\111 || 702.703 || 19 || 8 || 2.375 ||
| |
| |-
| |
| | || || || || || 41\70 || || 702.857 || 12 || 5 || 2.400 ||
| |
| |-
| |
| | || || || || || || 58\99 || 703.030 || 17 || 7 || 2.428 ||
| |
| |-
| |
| | || || || 17\29 || || || || 703.448 || 5 || 2 || 2.500 ||
| |
| |-
| |
| | || || || || || || 61\104 || 703.846 || 18 || 7 || 2.571 ||
| |
| |-
| |
| | || || || || || 44\75 || || 704.000 || 13 || 5 || 2.600 ||
| |
| |-
| |
| | || || || || || || 71\121 || 704.132 || 21 || 8 || 2.625 || Golden neogothic
| |
| |-
| |
| | || || || || 27\46 || || || 704.348 || 8 || 3 || 2.667 ||
| |
| |-
| |
| | || || || || || || 64\109 || 704.587 || 19 || 7 || 2.714 ||
| |
| |-
| |
| | || || || || || 37\63 || || 704.762 || 11 || 4 || 2.750 ||
| |
| |-
| |
| | || || || || || || 47\80 || 705.000 || 14 || 5 || 2.800 ||
| |
| |-
| |
| | || || 10\17 || || || || || 705.882 || 3 || 1 || 3.000 || L/s = 3/1
| |
| |-
| |
| | || || || || || || 43\73 || 706.849 || 13 || 4 || 3.250 ||
| |
| |-
| |
| | || || || || || 33\56 || || 707.143 || 10 || 3 || 3.333 ||
| |
| |-
| |
| | || || || || || || 56\95 || 707.368 || 17 || 5 || 3.400 ||
| |
| |-
| |
| | || || || || 23\39 || || || 707.692 || 7 || 2 || 3.500 ||
| |
| |-
| |
| | || || || || || || 59\100 || 708.000 || 18 || 5 || 3.600 ||
| |
| |-
| |
| | || || || || || 36\61 || || 708.197 || 11 || 3 || 3.667 ||
| |
| |-
| |
| | || || || || || || 49\83 || 708.434 || 15 || 4 || 3.750 ||
| |
| |-
| |
| | || || || 13\22 || || || || 709.091 || 4 || 1 || 4.000 || Archy is in this region
| |
| |-
| |
| | || || || || || || 42\71 || 709.859 || 13 || 3 || 4.333 ||
| |
| |-
| |
| | || || || || || 29\49 || || 710.204 || 9 || 2 || 4.500 ||
| |
| |-
| |
| | || || || || || || 45\76 || 710.526 || 14 || 3 || 4.667 ||
| |
| |-
| |
| | || || || || 16\27 || || || 711.111 || 5 || 1 || 5.000 ||
| |
| |-
| |
| | || || || || || || 35\59 || 711.864 || 11 || 2 || 5.500 ||
| |
| |-
| |
| | || || || || || 19\32 || || 712.500 || 6 || 1 || 6.000 ||
| |
| |-
| |
| | || || || || || || 22\37 || 713.514 || 7 || 1 || 7.000 ||
| |
| |-
| |
| | 3\5 || || || || || || || 720.000 || 1 || 0 || → inf ||
| |
| |}
| |