80th-octave temperaments: Difference between revisions

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This page describes 80th-octave temperaments.
{{Technical data page}}
{{Infobox fractional-octave|80}}


80edo is extremely accurate for the 17th harmonic, a relationship which is seen in some of these temperaments.
This page describes 80th-octave temperaments. [[80edo]] is extremely accurate for the [[17/1|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]].


== Octogintic ==
Temperaments discussed elsewhere include
''Main article: [[Parkleiness temperaments#Octogintic]]''
 
* ''Octogintic'', → [[Parkleiness temperaments#Octogintic|Parkleiness temperaments]]


== Tetraicosic ==
== Tetraicosic ==
Tetraicosic is described as 720 & 1600, and named after the fact that 4 × 20 = 80.
Tetraicosic is described as 1600 & 2320, and named after the fact that 4 × 20 = 80.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 48: Line 50:


Badness: 0.0741
Badness: 0.0741
=== 17-limit ===
Subgroup 2.3.5.7.11.13.17
Comma list: 9801/9800, 14400/14399, 373527/373490, 1812608/1812525, 4685824/4685625
Mapping: [{{val| 80 1 343 -27 434 13 327}}, {{val| 0 4 -5 8 -5 9 0}}]
Optimal tuning (CTE): ~130/99 = 471.7...
{{Optimal ET sequence|legend=1| 720, 1600, 2320, 3920 }}
=== 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: [{{val| 80 1 343 -27 434 13 327 -69}}, {{val| 0 4 -5 8 -5 9 0 13}}]
Optimal tuning (CTE): ~67473/51376 = 471.732
{{Optimal ET sequence|legend=1| 720, 1600, 2320, 3920 }}


== Mercury ==
== 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]].
Named after the 80th element, defined as the 320 & 2000 temperament.
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 9801/9800, 67392/67375, 1399680/1399489, {{monzo|14 -8 -8  5  2 -1}}
 
{{mapping|legend=1| 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
 
{{Optimal ET sequence|legend=1|320, 2000, 4320}}
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 9801/9800, 12376/12375, 67392/67375, 4230144/4229225, 494534656/494515125
 
{{mapping|legend=1| 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


{{Optimal ET sequence|legend=1|320, 2000, 4320}}
=== 19-limit ===
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19
Line 59: Line 106:
Mapping: [{{val|80 3 161 -23 252 197 327 117}}, {{val|0 5 1 10 1 4 0 9}}]
Mapping: [{{val|80 3 161 -23 252 197 327 117}}, {{val|0 5 1 10 1 4 0 9}}]


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


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


Optimal GPV sequence: {{EDOs|320, 2000, 4320}}
{{Optimal ET sequence|legend=1|320, 2000, 4320}}
 
== Octodeca ==
Octodeca can be described as the 80 & 1920 temperament.
 
Subgroup: 2.3.5.7
 
Comma list: {{monzo|21 60 -50}}, 184528125/184473632
 
{{mapping|legend=1|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
 
[[Support]]ing [[ET]]s: {{EDOs|80, 1920, 2000}}
=== 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|legend=1|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|legend=1|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
 
{{Navbox fractional-octave}}


[[Category:80edo]]
[[Category:80edo]]]
[[Category:Temperament collections]]
[[Category:Rank 2]]

Latest revision as of 13:43, 13 April 2025

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

← 79th-octave • 80th-octave • 81st-octave →

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

Optimal ET sequence: 720, 1600, 2320, 3920

Badness: 1.49

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

Optimal ET sequence: 720, 1600, 2320, 3920

Badness: 0.178

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

Optimal ET sequence: 720, 1600, 2320, 3920

Badness: 0.0741

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...

Optimal ET sequence: 720, 1600, 2320, 3920

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

Optimal ET sequence: 720, 1600, 2320, 3920

Mercury

Named after the 80th element, defined as the 320 & 2000 temperament.

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

Optimal ET sequence: 320, 2000, 4320

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

Optimal ET sequence: 320, 2000, 4320

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

Optimal ET sequence: 320, 2000, 4320

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

Supporting ETs: 80, 1920, 2000

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


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