72nd-octave temperaments

Template:Fractional-octave navigation 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

Supporting ETs: 72, 1872f, 1944, 2016, 2088f, 2160fm, 3960