72nd-octave temperaments: Difference between revisions

Latest revision as of 13:47, 24 April 2026

Fractional-octave temperaments

← 71st-octave • 72nd-octave • 73rd-octave →

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 → Landscape microtemperaments

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

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-{{Fractional-octave navigation|72}}
+{{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.
 Temperaments discussed elsewhere include:
+* ''Omicronbeta'' → [[Compton family #Omicronbeta|Compton family]]
-* ''Omicronbeta'', → [[Compton family#Omicronbeta|Compton family]]
+* ''Hafnium'' → [[Landscape microtemperaments #Hafnium|Landscape microtemperaments]]
-== 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: [{{val|72 72 462 876 1302 1193}}, {{val|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|legend=1|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: [{{val|72 72 462 876 1302 1193 547}}, {{val|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|legend=1|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: [{{val|72 72 462 876 1302 1193 547 356}}, {{val|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: [{{val|72 72 462 876 1302 1193 547 356 378}}, {{val|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 ==

72nd-octave temperaments: Difference between revisions

Latest revision as of 13:47, 24 April 2026

Jamala

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