118th-octave temperaments: Difference between revisions

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[[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.
{{Infobox fractional-octave|118}}
[[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 ===
== Parakleischis ==
118edo and its multiples are members of both [[parakleismic]] and [[Schismatic family|schismic]], and from this it derives its name.
118edo's is an excellent 5-limit system and its comma basis constitutes the [[parakleismic]] and [[Schismatic family|schismic]] temperaments together. Parakleischis retains the 5-limit mapping from 118edo and leaves other harmonics as independent generators.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 10: Line 11:
[[Mapping]]: [{{val| 118 187 274 0 }}, {{val| 0 0 0 1 }}]
[[Mapping]]: [{{val| 118 187 274 0 }}, {{val| 0 0 0 1 }}]


Mapping generators: ~15625/15552, ~7
: mapping generators: ~15625/15552, ~7


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


{{Optimal ET sequence|legend=1| 118, 236, 354, 472, 2242, 2714b, 3186b, 3658b }}
{{Optimal ET sequence|legend=1| 118, 236, 354, 472, 2242, 2714b, 3186b, 3658b }}
Line 18: Line 19:
[[Badness]]: 0.145166
[[Badness]]: 0.145166


==== 11-limit ====
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 25: Line 26:
Mapping: [{{val| 118 187 274 0 77 }}, {{val| 0 0 0 1 1 }}]
Mapping: [{{val| 118 187 274 0 77 }}, {{val| 0 0 0 1 1 }}]


Optimal tuning (POTE): ~7/4 = 968.5117
: mapping generators: ~176/176, ~7
 
Optimal tuning (CTE): ~7/4 = 968.5117


{{Optimal ET sequence|legend=1| 118, 354, 472 }}
{{Optimal ET sequence|legend=1| 118, 354, 472 }}
Line 31: Line 34:
Badness: 0.049316
Badness: 0.049316


==== Centenniamajor ====
=== Peithoian ===
Named after the fact that 18 is the age of majority in most countries, and 100 (centennial) + 18 (major) = 118.
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 [[wikipedia:118 Peitho|118 Peitho]].


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 48: Line 51:
Badness: 0.357
Badness: 0.357


===== 13-limit =====
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 61: Line 64:
Badness: 0.122
Badness: 0.122


=== Oganesson ===
== 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.  
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.  


Line 76: Line 79:
[[Optimal tuning]] ([[CTE]]): ~1953125/1354752 = 634.0068
[[Optimal tuning]] ([[CTE]]): ~1953125/1354752 = 634.0068


{{Optimal ET sequence|legend=1| 354, 2360, 2714, 3068, 3442, 3776, 7198cd, 10974bccdd }}
{{Optimal ET sequence|legend=1| 354, 2360, 2714, 3068, 3422, 3776, 7198cd, 10974bccdd }}


[[Badness]]: 2.66
[[Badness]]: 2.66


==== 11-limit ====
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 89: Line 92:
Optimal tuning (CTE): ~1953125/1354752 = 634.0085
Optimal tuning (CTE): ~1953125/1354752 = 634.0085


{{Optimal ET sequence|legend=1| 354, 3068e, 3442, 3776, 11682ccdde }}
{{Optimal ET sequence|legend=1| 354, 3068e, 3422, 3776, 11682ccdde }}


Badness: 0.568
Badness: 0.568


==== 13-limit ====
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 108: Line 111:
Badness: 0.172
Badness: 0.172


==== 17-limit ====
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Line 121: Line 124:
{{Optimal ET sequence|legend=1| 354, 3068e, 3422, 3776 }}
{{Optimal ET sequence|legend=1| 354, 3068e, 3422, 3776 }}


Badness: 0.105
=== 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: [{{val| 118 0 274 643 1094 499 607 1000}}, {{val| 0 3 0 -5 -11 -1 2 -8}}]
 
: mapping generators: ~171/170, ~238/165
 
Optimal tuning (CTE): ~238/165 = 634.006
 
{{Optimal ET sequence|legend=1| 354, ..., 3422, 3776 }}
 
{{Navbox fractional-octave}}
 
[[Category:118edo]]

Latest revision as of 03:43, 14 February 2025

← 117th-octave • 118th-octave • 119th-octave →

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

Optimal ET sequence: 118, 236, 354, 472, 2242, 2714b, 3186b, 3658b

Badness: 0.145166

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

Optimal ET sequence: 118, 354, 472

Badness: 0.049316

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

Optimal ET sequence: 354, 944e, 1298

Badness: 0.357

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

Optimal ET sequence: 354, 944e, 1298

Badness: 0.122

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

Optimal ET sequence: 354, 2360, 2714, 3068, 3422, 3776, 7198cd, 10974bccdd

Badness: 2.66

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

Optimal ET sequence: 354, 3068e, 3422, 3776, 11682ccdde

Badness: 0.568

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

Optimal ET sequence: 354, 3068e, 3422, 3776

Badness: 0.172

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

Optimal ET sequence: 354, 3068e, 3422, 3776

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 sequence: 354, ..., 3422, 3776


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