# Hemimean family

(Redirected from Belobog)

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