# 118th-octave temperaments

118th-octave temperaments

118edo is the 17th zeta peak edo, and it is accurate for harmonics 3 and 5, so various 118th-octave temperaments naturally occur through temperament merging of its supersets. Furthermore, one step of 118edo is in direct proximity to essential tempering commas 169/168 and 170/169.

## Parakleischis

118edo's is an excellent 5-limit system and its comma basis constitutes the parakleismic and schismic temperaments together. Parakleischis retains the 5-limit mapping from 118edo and leaves other harmonics as independent generators.

Subgroup: 2.3.5.7

Comma list: 32805/32768, 1224440064/1220703125

Mapping: [118 187 274 0], 0 0 0 1]]

mapping generators: ~15625/15552, ~7

Optimal tuning (CTE): ~7/4 = 968.7235

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 32805/32768, 137781/137500

Mapping: [118 187 274 0 77], 0 0 0 1 1]]

mapping generators: ~176/176, ~7

Optimal tuning (CTE): ~7/4 = 968.5117

### Peithoian

Peithoian is an extension of parakleischis which retains the 5-limit mapping of 118edo and provides the correction for 13th harmonic. 13-limit is the first prime limit that 118edo does not tune consistently, and the goal of peithoian temperament is to expand on that. Named after the minor planet 118 Peitho.

Subgroup: 2.3.5.7.11

Comma list: 32805/32768, 151263/151250, 1224440064/1220703125

Mapping: [118 187 274 0 -420], 0 0 0 2 5]]

Mapping generators: ~15625/15552, ~405504/153125

Optimal tuning (CTE): ~202752/153125 = 484.4837

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 32805/32768, 34398/34375, 384912/384475

Mapping: [118 187 274 0 -420 271], 0 0 0 2 5 1]]

Optimal tuning (CTE): ~8125/6144 = 484.4867

## Oganesson

Named after the 118th element. In the 13-limit, the period corresponds to 169/168, and in the 17-limit, it corresponds also to 170/169, meaning that 28561/28560 is tempered out. As opposed to being an extension of parakleischis, this has the generator that splits the third harmonic into three equal parts.

In the 7-limit and 11-limit, the period corresponds to bronzisma.

Subgroup: 2.3.5.7

Comma list: [30 10 -27 6, [77 -20 -5 -12

Mapping: [118 0 274 643], 0 3 0 -5]]

Mapping generators: ~2097152/2083725, ~1953125/1354752

Optimal tuning (CTE): ~1953125/1354752 = 634.0068

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 9801/9800, [13 -1 4 -16 7, [55 -7 -15 -2 -1

Mapping: [118 0 274 643 1094], 0 3 0 -5 -11]]

Optimal tuning (CTE): ~1953125/1354752 = 634.0085

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 4096/4095, 9801/9800, 537403776/537109375, 453874312332/453857421875

Mapping: [118 0 274 643 1094 499], 0 3 0 -5 -11 -1]]

Mapping generators: ~169/168, ~1124864/779625

Optimal tuning (CTE): ~1124864/779625 = 634.0087

### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 4096/4095, 9801/9800, 34391/34375, 361250/361179, 562432/562275

Mapping: [118 0 274 643 1094 499 607], 0 3 0 -5 -11 -1 2]]

Mapping generators: ~170/169, ~238/165

Optimal tuning (CTE): ~238/165 = 634.0080

### 19-limit

The closest superparticular to one step of 118edo is 171/170, so 19-limit extension for oganesson is prescribed.

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 4096/4095, 6175/6174, 9801/9800, 14365/14364, 28900/28899, 3438981/3437500

Mapping: [118 0 274 643 1094 499 607 1000], 0 3 0 -5 -11 -1 2 -8]]

mapping generators: ~171/170, ~238/165

Optimal tuning (CTE): ~238/165 = 634.006

Optimal ET sequence354, ..., 3422, 3776