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| {{Fractional-octave navigation|15}} | | {{Infobox fractional-octave|15}} |
| | [[15edo]] has some close approximations including [[22/21]] to one step and [[77/64]] to four steps. |
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| == Quindecic ==
| | Temperaments discussed elsewhere include [[blackwood family #Quindecic|quindecic]], [[keemic temperaments #Pentadecal|pentadecal]], [[Landscape microtemperaments #Slendscape|slendscape]], and [[Landscape microtemperaments #Phosphorus|phosphorus]]. |
| Quindecic preserves the [[11-limit]] structure of [[15edo]], with an independent generator for [[13/1|harmonic 13]].
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| [[Subgroup]]: 2.3.5.7.11.13 | | == Pentadecoid == |
| | In pentadecoid, the period (1\15) is given an interpretation of [[22/21]], and four of them represent the keenanismic minor third, [[77/64]] = ([[147/128]])⋅([[22/21]]). The temperament is named for an analog of its relating temperaments, [[decoid]] ({{nowrap| 130 & 140 }}) and [[quintosec family #Triacontoid|triacontoid]] ({{nowrap| 120 & 150 }}). Triacontoid is a weak extension of pentadecoid, as well as [[Landscape microtemperaments #Slendscape|slendscape]] ({{nowrap| 255 & 270 }}). |
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| [[Comma list]]: 28/27, 49/48, 55/54, 77/75 | | [[Subgroup]]: 2.3.7.11 |
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| [[Mapping]]: {{mapping| 15 24 35 42 52 0 | 0 0 0 0 0 1 }} | | [[Comma list]]: 1362944/1361367, 1771561/1769472 |
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| : mapping generators: ~22/21, ~13
| | {{Mapping|legend=1| 15 108 0 94 | 0 -2 1 -1 }} |
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| [[Optimal tuning]]s:
| | : mapping generators: ~22/21, ~7 |
| * [[CTE]]: ~22/21 = 80.000, ~13/8 = 840.528 (~40/39 = 39.472)
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| * [[POTE]]: ~22/21 = 80.000, ~13/8 = 852.924 (~40/39 = 27.076)
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| {{Optimal ET sequence|legend=1| 15, 30 }}
| | [[Optimal tuning]] ([[CTE]]): ~22/21 = 80.0000, ~8/7 = 231.0071 (~1029/1024 = 8.9929) |
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| [[Badness]] (Smith): 0.028944
| | {{Optimal ET sequence|legend=1| 120, 135, 660, 795, 930, 1065, 1200, 1335, 2535e }} |
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| == Pentadecal ==
| | [[Badness]] (Sintel): 0.951 |
| Subgroup: 2.3.5
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| [[Comma]]: 30517578125/29386561536
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| [[Mapping]]: [{{Val| 15 0 11 }}, {{Val| 0 1 1 }}]
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| [[POTE generator]]: ~3/2 = 702.652
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| {{Optimal ET sequence|legend=1| 15, 45, 60, 75, 135, 210c, 345cc }}
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| [[Badness]]: 0.692042 | |
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| === 7-limit ===
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| The ''pentadecal'' temperament tempers out the 15-5/3-comma (Quintriyo, pentadecatonic minor thirds comma), {{monzo|-11 -15 15}}. This temperament can be described as 15&60 temperament, tempering out the [[cloudy comma]], 16807/16384 and the [[875/864|keema]], 875/864 in the 7-limit.
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| [[Subgroup]]: 2.3.5.7
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| [[Comma list]]: 875/864, 16807/16384
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| {{Mapping|legend=1| 15 0 11 42 | 0 1 1 0 }}
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| : mapping generators: ~21/20, ~3
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| [[Optimal tuning]] ([[POTE]]): ~21/20 = 1\15, ~3/2 = 700.223 (~126/125 = 19.777)
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| {{Optimal ET sequence|legend=1| 15, 45, 60 }}
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| [[Badness]]: 0.114833
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| === 11-limit ===
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| Subgroup: 2.3.5.7.11
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| Comma list: 100/99, 385/384, 1372/1331
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| Mapping: {{mapping| 15 0 11 42 52 | 0 1 1 0 0 }}
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| : mapping generators: ~21/20, ~3
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| Optimal tuning (POTE): ~21/20 = 1\15, ~3/2 = 702.733 (~56/55 = 17.267)
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| {{Optimal ET sequence|legend=1| 15, 45, 60, 75de, 135de }}
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| Badness: 0.077420
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| ==== 13-limit ====
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| Subgroup: 2.3.5.7.11.13
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| Comma list: 100/99, 105/104, 144/143, 1372/1331
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| Mapping: {{mapping| 15 0 11 42 52 8 | 0 1 1 0 0 2 }}
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| : mapping generators: ~21/20, ~3
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| Optimal tuning (POTE): ~21/20 = 1\15, ~3/2 = 701.715 (~91/90 = 18.285)
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| {{Optimal ET sequence|legend=1| 15, 45f, 60, 135de, 195cddee }}
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| Badness: 0.051740
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| === Quindecal ===
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| Subgroup: 2.3.5.7.11
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| Comma list: 121/120, 441/440, 875/864
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| Mapping: {{mapping| 15 0 11 42 28 | 0 1 1 0 1 }}
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| : mapping generators: ~21/20, ~3
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| Optimal tuning (POTE): ~21/20 = 1\15, ~3/2 = 700.318 (~126/125 = 19.682)
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| {{Optimal ET sequence|legend=1| 15, 45e, 60e }}
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| Badness: 0.044405
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| ==== 13-limit ====
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| Subgroup: 2.3.5.7.11.13
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| Comma list: 121/120, 196/195, 352/351, 875/864
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| Mapping: {{mapping| 15 0 11 42 28 103 | 0 1 1 0 1 -2 }}
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| : mapping generators: ~21/20, ~3
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| Optimal tuning (POTE): ~21/20 = 1\15, ~3/2 = 701.647 (~65/64 = 18.353)
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| {{Optimal ET sequence|legend=1| 15f, 45e, 60e }}
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| Badness: 0.055361
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| {{Fractional-octave footer}} | | {{Fractional-octave footer}} |