72nd-octave temperaments: Difference between revisions
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{{ | {{Infobox fractional-octave|72}} | ||
[[72edo]] is a notable tuning system due to its 17-limit consistency, being a multiple of 12, and usage in various ecclesiastical forms of chanting. | [[72edo]] is a notable tuning system due to its 17-limit consistency, being a multiple of 12, and usage in various ecclesiastical forms of chanting. | ||
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{{Optimal ET sequence|legend=1|1224, 1872def, 4320, 5544}} | {{Optimal ET sequence|legend=1|1224, 1872def, 4320, 5544}} | ||
===17-limit=== | |||
=== 17-limit === | |||
Subgroup: 2.3.5.7.11.13.17 | Subgroup: 2.3.5.7.11.13.17 | ||
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{{Optimal ET sequence|legend=1|1224, 1872def, 4320, 5544}} | {{Optimal ET sequence|legend=1|1224, 1872def, 4320, 5544}} | ||
===2.3.5.7.11.13.17.31 subgroup=== | === 2.3.5.7.11.13.17.31 subgroup === | ||
Since 4320edo and 5544edo have good 31st and 37th harmonics, addition of these subgroups is prescribed. | Since 4320edo and 5544edo have good 31st and 37th harmonics, addition of these subgroups is prescribed. | ||
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Optimal tuning (CTE): ~3/2 = 701.948 | Optimal tuning (CTE): ~3/2 = 701.948 | ||
===2.3.5.7.11.13.17.31.37 subgroup=== | === 2.3.5.7.11.13.17.31.37 subgroup === | ||
[[37/22]] is mapped to exact three-quarters of the octave. | [[37/22]] is mapped to exact three-quarters of the octave. | ||
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[[Optimal tuning]] ([[CTE]]): ~73205/53248 = 550.602 | [[Optimal tuning]] ([[CTE]]): ~73205/53248 = 550.602 | ||
{{Optimal ET sequence|legend=1|72, 1944, 2016, 3960}} | |||
{{Navbox fractional-octave}} | |||
Latest revision as of 03:55, 14 February 2025
72edo is a notable tuning system due to its 17-limit consistency, being a multiple of 12, and usage in various ecclesiastical forms of chanting.
Temperaments discussed elsewhere include:
- Omicronbeta, → Compton family
Hafnium
Hafnium is a period-72 temperament created by temperament-merging 4320edo and 5544edo, equal temperaments notable for their high divisibility. Named after the 72nd element.
Subgroup: 2.3.5.7.11.13
Comma list: 9801/9800, 250047/250000, 184549376/184528125, 463373664/463203125
Mapping: [⟨72 72 462 876 1302 1193], ⟨0 1 -7 -16 -25 -22]]
- mapping generators: ~105/104 = 1\72, ~3/2 = 701.948
Optimal tuning (CTE): ~3/2 = 701.948
Optimal ET sequence: 1224, 1872def, 4320, 5544
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 9801/9800, 12376/12375, 21879/21875, 194481/194480, 1713660/1713481, 97144749/97140736
Mapping: [⟨72 72 462 876 1302 1193 547], ⟨0 1 -7 -16 -25 -22 -6]]
- mapping generators: ~105/104 = 1\72, ~3/2 = 701.948
Optimal tuning (CTE): ~3/2 = 701.948
Optimal ET sequence: 1224, 1872def, 4320, 5544
2.3.5.7.11.13.17.31 subgroup
Since 4320edo and 5544edo have good 31st and 37th harmonics, addition of these subgroups is prescribed.
Subgroup: 2.3.5.7.11.13.17.31
Comma list: 9801/9800, 10881/10880, 12376/12375, 57629/57624, 179712/179707, 61456384/61448625
Sval mapping: [⟨72 72 462 876 1302 1193 547 356], ⟨0 1 -7 -16 -25 -22 -6 6]]
- mapping generators: ~105/104 = 1\72, ~3/2 = 701.948
Optimal tuning (CTE): ~3/2 = 701.948
2.3.5.7.11.13.17.31.37 subgroup
37/22 is mapped to exact three-quarters of the octave.
Subgroup: 2.3.5.7.11.13.17.31
Comma list: 10881/10880, 12376/12375, 16576/16575, 93093/93092, 954304/954261, 2737889/2737800, 126607131/126605120
Sval mapping: [⟨72 72 462 876 1302 1193 547 356 378], ⟨0 1 -7 -16 -25 -22 -6 6 -25]]
- mapping generators: ~105/104 = 1\72, ~3/2 = 701.948
Optimal tuning (CTE): ~3/2 = 701.948
Jamala
Jamala, described as the 1944 & 2016 temperament in the 2.5.11.13.19.41.47 subgroup was named by Eliora after a Ukrainian singer who won the 2016 Eurovision Song Contest with her song 1944.
Subgroup: 2.5.11.13.19.41.47
Comma basis: 47012251/47000000, 2502280/2501369, 410041489/410000000, 2680291328/2679296875, 52448351813/52428800000
Sval Mapping: [⟨72 2 183 -130 438 617 466], ⟨0 5 2 12 -4 -7 -2]]
- sval mapping generators: ~517/512 = 1\72, ~73205/53248 = 550.602
Optimal tuning (CTE): ~73205/53248 = 550.602
Optimal ET sequence: 72, 1944, 2016, 3960