# Quintosec family

(Redirected from Quintosec)

The quintosec family tempers out the quintosec comma, 140737488355328/140126044921875 = [47 -15 -10.

## Quintosec

Quintosec is naturally a 2.3.5.11-subgroup temperament. It was documented as qintosec on the temperament finder, which was probably due to a typo[1]. The original, intended spelling is now restored, and the distinctive no-u form is relegated to the lower-accuracy 7-limit extension considered below.

Subgroup: 2.3.5

Comma list: [47 -15 -10

Mapping[5 1 22], 0 2 -3]]

mapping generators: ~524288/455625, ~16384/10125
• CTE: ~524288/455625 = 1\5, ~16384/10125 = 831.1061 (~16/15 = 111.1061)
• POTE: ~524288/455625 = 1\5, ~16384/10125 = 831.105 (~16/15 = 111.105)

### 2.3.5.11 subgroup

Subgroup: 2.3.5.11

Comma list: 5632/5625, 26214400/26198073

Mapping: [5 1 22 45], 0 2 -3 -8]]

Optimal tunings:

• CTE: ~1024/891 = 1\5, ~160/99 = 831.0932 (~16/15 = 111.0932)
• CWE: ~1024/891 = 1\5, ~160/99 = 831.0743 (~16/15 = 111.0743)

Optimal ET sequence: 10e, 55e, 65, 205, 270, 335, 605, 940, 1545c

## Qintosec

Qintosec can be described as the 10 & 65d temperament, tempering out the marvel comma, 225/224 in the 7-limit.

Subgroup: 2.3.5.7

Comma list: 225/224, 2560000/2470629

Mapping[5 1 22 21], 0 2 -3 -2]]

Wedgie⟨⟨10 -15 -10 -47 -44 19]]

Optimal tunings:

• CTE: ~400/343 = 1\5, ~80/49 = 831.7095 (~15/14 = 111.7095)
• POTE: ~400/343 = 1\5, ~80/49 = 831.553 (~15/14 = 111.553)

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 243/242, 3840/3773

Mapping: [5 1 22 21 0], 0 2 -3 -2 5]]

Optimal tunings:

• CTE: ~55/48 = 1\5, ~80/49 = 831.1939 (~15/14 = 111.1939)
• POTE: ~55/48 = 1\5, ~80/49 = 831.311 (~15/14 = 111.311)

Optimal ET sequence: 10, 55d, 65d, 140de

### Qinto

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 332750/321489

Mapping: [5 1 22 21 -7], 0 2 -3 -2 7]]

Optimal tunings:

• CTE: ~220/189 = 1\5, ~44/27 = 832.4051 (~15/14 = 112.4051)
• POTE: ~220/189 = 1\5, ~44/27 = 832.448 (~15/14 = 112.448)

Optimal ET sequence: 10, 75

## Sengasec

Sengasec can be described as the 10 & 55 temperament, tempering out the cloudy comma, 16807/16384 and the sengic comma, 686/675 in the 7-limit.

Subgroup: 2.3.5.7

Comma list: 686/675, 16807/16384

Mapping[5 1 22 14], 0 2 -3 0]]

Wedgie⟨⟨10 -15 0 -47 -28 42]]

• CTE: ~8/7 = 1\5, ~45/28 = 831.1061 (~16/15 = 111.1061)
• POTE: ~8/7 = 1\5, ~45/28 = 831.113 (~16/15 = 111.113)

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 243/242, 385/384, 686/675

Mapping: [5 1 22 14 0], 0 2 -3 0 5]]

Optimal tunings:

• CTE: ~8/7 = 1\5, ~45/28 = 830.7772 (~16/15 = 110.7772)
• POTE: ~8/7 = 1\5, ~45/28 = 830.659 (~16/15 = 110.659)

Optimal ET sequence: 10, 55, 65

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 144/143, 243/242, 686/675

Mapping: [5 1 22 14 0 22], 0 2 -3 0 5 -1]]

Optimal tunings:

• CTE: ~8/7 = 1\5, ~45/28 = 830.8942 (~16/15 = 110.8942)
• POTE: ~8/7 = 1\5, ~45/28 = 830.910 (~16/15 = 110.910)

Optimal ET sequence: 10, 55, 65

## Decoid

Decoid tempers out 2401/2400 and 67108864/66976875, as well as the linus comma, [11 -10 -10 10. Either 8/7 or 16/15 can be used as its generator. It may be described as the 130 & 270 temperament, and as one might expect, 181\940 or 233\1210 makes for an excellent tuning choice.

Subgroup: 2.3.5.7

Comma list: 2401/2400, 67108864/66976875

Mapping[10 0 47 36], 0 2 -3 -1]]

mapping generators: ~15/14, ~8192/4725

Wedgie⟨⟨20 -30 -10 -94 -72 61]]

• CTE: ~15/14 = 1\10, ~8192/4725 = 951.1086 (~16/15 = 111.1086, or ~225/224 = 8.8914)
• POTE: ~15/14 = 1\10, ~8192/4725 = 951.099 (~16/15 = 111.099, or ~225/224 = 8.901)

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 5632/5625, 9801/9800

Mapping: [10 0 47 36 98], 0 2 -3 -1 -8]]

Optimal tunings:

• CTE: ~15/14 = 1\10, ~400/231 = 951.0943 (~16/15 = 111.0943, or ~225/224 = 8.9057)
• POTE: ~15/14 = 1\10, ~400/231 = 951.070 (~16/15 = 111.070, or ~225/224 = 8.930)

Optimal ET sequence: 10e, …, 130, 270, 670, 940, 1210, 2150c

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1001/1000, 1716/1715, 4096/4095

Mapping: [10 0 47 36 98 37], 0 2 -3 -1 -8 0]]

Optimal tunings:

• CTE: ~15/14 = 1\10, ~26/15 = 951.0943 (~16/15 = 111.0943, or ~196/195 = 8.9057)
• POTE: ~15/14 = 1\10, ~26/15 = 951.083 (~16/15 = 111.083, or ~196/195 = 8.917)

Optimal ET sequence: 10e, …, 130, 270, 940, 1210f, 1480cf