Hemimean family

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The hemimean family of temperaments are rank-3 temperaments tempering out 3136/3125.

The hemimean comma, 3136/3125, is the ratio between the septimal semicomma (126/125) and the septimal kleisma (225/224). This fact alone makes hemimean a very notable rank-3 temperament, as any non-meantone tuning of hemimean will split the syntonic comma (81/80) into two equal parts, each representing 126/125~225/224.

Other equivalences characteristic to hemimean are 128/125~50/49 and 49/45~(25/24)2.

Hemimean

Subgroup: 2.3.5.7

Comma list: 3136/3125 (hemimean)

Mapping[1 0 0 -3], 0 1 0 0], 0 0 2 5]]

mapping generators: ~2, ~3, ~56/25

Mapping to lattice: [0 0 2 5], 0 1 0 0]]

Lattice basis:

28/25 length = 0.5055, 3/2 length = 1.5849
Angle (28/25, 3/2) = 90 degrees

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.9550, ~28/25 = 193.6499

[[1 0 0 0, [0 1 0 0, [6/5 0 0 2/5, [0 0 0 1]
Eigenmonzo (unchanged-interval) basis: 2.3.7

Optimal ET sequence12, 19, 31, 68, 80, 87, 99, 217, 229, 328, 347, 446, 675c

Complexity spectrum: 5/4, 7/5, 4/3, 6/5, 8/7, 7/6, 9/8, 10/9, 9/7

Projection pairs: 5 3136/625 7 68841472/9765625 to 2.3.25/7

Hemimean orion

As the second generator of hemimean, 28/25, is close to 19/17, and as the latter is the mediant of 10/9 and 9/8, it is natural to extend hemimean to the 2.3.5.7.17.19 subgroup by tempering out (28/25)/(19/17) = 476/475, or equivalently stated, the semiparticular (5/4)/(19/17)2 = 1445/1444. Notice 3136/3125 = (476/475)(2128/2125) and that 2128/2125 = (1216/1215)(1701/1700), so it makes sense to temper out 1216/1215 and/or 1701/1700 as well. An interesting tuning not in the optimal ET sequence is 111edo. This temperament finds the harmonic 17 and 19 at (+5, +1) and (+5, +2), respectively, with virtually no additional error.

The S-expression-based comma list for the 2.3.5.7.17.19 subgroup extension is { S16/S18, S17/S19, S18/S20(, (S16*S17)/(S19*S20) = S16/S18 * S17/S19 * S18/S20) }.

Subgroup: 2.3.5.7.17

Comma list: 1701/1700, 3136/3125

Sval mapping: [1 0 0 -3 -5], 0 1 0 0 5], 0 0 2 5 1]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.1960, ~28/25 = 193.6548

Optimal ET sequence12, 19g, 31g, …, 87, 99, 217, 229, 316, 328h, 446, 545c, 873cg

2.3.5.7.17.19 subgroup

Subgroup: 2.3.5.7.17.19

Comma list: 476/475, 1216/1215, 1445/1444

Sval mapping: [1 0 0 -3 -5 -6], 0 1 0 0 5 5], 0 0 2 5 1 2]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.132, ~19/17 = 193.647

Optimal ET sequence12, 19gh, 31gh, …, 87, 99, 118, 210gh, 217, 229, 328h, 446

Semiorion

Semiorion is an alternative subgroup extension of lower complexity, which splits the octave into two. The S-expression-based comma list for the 2.3.5.7.17.19 subgroup extension is {S17, S19, S16/S18(, S18/S20, 476/475 = S16/S20 * S17/S19)}.

Subgroup: 2.3.5.7.17

Comma list: 289/288, 3136/3125

Sval mapping: [2 0 0 -6 5], 0 1 0 0 1], 0 0 2 5 0]]

sval mapping generators: ~17/12, ~3, ~56/25

Optimal tuning (CTE): ~17/12 = 1\2, ~3/2 = 702.3471, ~28/25 = 193.6499

2.3.5.7.17.19 subgroup

Subgroup: 2.3.5.7.17.19

Comma list: 289/288, 361/360, 476/475

Mapping: [2 0 0 -6 5 3], 0 1 0 0 1 1], 0 0 2 5 0 1]]

Optimal tuning (CTE): ~17/12 = 1\2, ~3/2 = 702.509, ~28/25 = 193.669

Optimal ET sequence12, …, 50, 68, 106d, 118, 248g, 316g

Belobog

Subgroup: 2.3.5.7.11

Comma list: 441/440, 3136/3125

Mapping[1 0 0 -3 -9], 0 1 0 0 2], 0 0 2 5 8]]

mapping generators: ~2, ~3, ~56/25

Mapping to lattice: [0 -2 2 5 4], 0 -1 0 0 -2]]

Lattice basis:

28/25 length = 0.3829, 16/15 length = 1.1705
Angle (28/25, 16/15) = 93.2696

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.7205, ~28/25 = 193.5545

[[1 0 0 0 0, [27/22 6/11 -5/22 -3/11 5/22, [24/11 -4/11 -2/11 2/11 2/11, [27/11 -10/11 -5/11 5/11 5/11, [24/11 -4/11 -13/11 2/11 13/11]
Eigenmonzo (unchanged-interval) basis: 2.9/7.11/5

Projection pairs: 5 3136/625 7 68841472/9765625 11 1700108992512/152587890625 to 2.3.25/7

Scales: belobog31

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 441/440, 1001/1000, 3136/3125

Mapping: [1 0 0 -3 -9 15], 0 1 0 0 2 -2], 0 0 2 5 8 -7]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.8219, ~28/25 = 193.5816

Bellowblog

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351, 625/624

Mapping: [1 0 0 -3 -9 -4], 0 1 0 0 2 -1], 0 0 2 5 8 8]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.5667, ~28/25 = 193.2493

Siebog

Subgroup: 2.3.5.7.11

Comma list: 540/539, 3136/3125

Mapping[1 0 0 -3 8], 0 1 0 0 3], 0 0 2 5 -8]]

mapping generators: ~2, ~3, ~56/25

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.1636, ~28/25 = 193.8645

[[1 0 0 0 0, [0 1 0 0 0, [8/5 3/5 1/5 0 -1/5, [1 3/2 1/2 0 -1/2, [8/5 3/5 -4/5 0 4/5]
Eigenmonzo (unchanged-interval) basis: 2.3.11/5

Triglav

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 3136/3125

Mapping[1 0 2 2 1], 0 1 2 5 2], 0 0 -4 -10 -1]]

mapping generators: ~2, ~3, ~18/11

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.2875, ~18/11 = 854.3132