# 80th-octave temperaments

(Redirected from Octodeca)

This page describes 80th-octave temperaments. 80edo is extremely accurate for the 17th harmonic, a relationship which is so far seen in all documented temperaments on this page. In addition, it is the first equal division to be consistent in the 19-limit.

Temperaments discussed elsewhere include

## Tetraicosic

Tetraicosic is described as 1600 & 2320, and named after the fact that 4 × 20 = 80.

Subgroup: 2.3.5.7

Comma list: [-52 17 12 -1, [25 42 -8 -26

Mapping: [80 1 343 -27], 0 4 -5 8]]

Mapping generators: ~31637227888/31381059609, ~7381125/5619712

Optimal tuning (CTE): ~7381125/5619712 = 471.7339

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 928760463360/928426965851, [49 -13 -14 -1 2

Mapping: [80 1 343 -27 434], 0 4 -5 8 -5]]

Optimal tuning (CTE): ~2278125/1734656 = 471.7342

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 9801/9800, 4100625/4100096, 14236560/14235529, 143327232/143286143

Mapping: [80 1 343 -27 434 13], 0 4 -5 8 -5 9]]

Optimal tuning (CTE): ~130/99 = 471.7322

### 17-limit

Subgroup 2.3.5.7.11.13.17

Comma list: 9801/9800, 14400/14399, 373527/373490, 1812608/1812525, 4685824/4685625

Mapping: [80 1 343 -27 434 13 327], 0 4 -5 8 -5 9 0]]

Optimal tuning (CTE): ~130/99 = 471.7...

### 19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 9801/9800, 10830/10839, 12636/12635, 23409/23408, 373527/373490, 32133332/32131125

Mapping: [80 1 343 -27 434 13 327 -69], 0 4 -5 8 -5 9 0 13]]

Optimal tuning (CTE): ~67473/51376 = 471.732

## Mercury

Named after the 80th element, defined as the 320 & 2000 temperament. 7 periods are mapped to ~17/16, 11 periods are mapped to ~11/10.

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 9801/9800, 67392/67375, 1399680/1399489, [14 -8 -8 5 2 -1

Mapping[80 3 161 -23 252 197], 0 5 1 10 1 4]]

Mapping generators: ~9295/9216, ~2304/1859

Optimal tuning (CTE): ~2304/1859 = 371.385

### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 9801/9800, 12376/12375, 67392/67375, 4230144/4229225, 494534656/494515125

Mapping[80 3 161 -23 252 197 327], 0 5 1 10 1 4 0]]

Mapping generators: ~459/455, ~1859/1500

Optimal tuning (CTE): ~2275/1836 = 371.386

### 19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 9801/9800, 12376/12375, 67392/67375, 392445/392392, 401408/401375, 1549184/1549125

Mapping: [80 3 161 -23 252 197 327 117], 0 5 1 10 1 4 0 9]]

Mapping generators: ~459/455, ~2275/1836

Optimal tuning (CTE): ~2275/1836 = 371.386

## Octodeca

Octodeca can be described as the 80 & 1920 temperament.

Subgroup: 2.3.5.7

Comma list: [21 60 -50, 184528125/184473632

Mapping[80 2 36 -25], 0 5 6 10]]

mapping generators: ~1071875/1062882 = 1\80, ~7476806640625/6025163444928 = 374.386

Optimal tuning (CTE): ~7476806640625/6025163444928 = 374.386

### 19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 5832/5831, 9801/9800, 89376/89375, 123201/123200, 392445/392392, 653184/653125

Mapping[80 2 36 -25 -127 -321 -327 -240], 0 5 6 10 6 -1 0 4]]

mapping generators: ~12000/12103 = 1\80, ~1625/1309 = 374.386

Optimal tuning (CTE): ~1625/1309 = 374.386

### 23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 5832/5831, 8625/8624, 9801/9800, 10626/10625, 89376/89375, 95013/95000, 123201/123200

Mapping[80 2 36 -25 -127 -321 -327 -240 -287], 0 5 6 10 6 -1 0 4 3]]

mapping generators: ~12000/12103 = 1\80, ~920/741 = 374.387

Optimal tuning (CTE): ~920/741 = 374.387