80th-octave temperaments: Difference between revisions

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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. In addition, it is the first equal division to be consistent in the 19-limit.

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]].

Temperaments discussed elsewhere include

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

Optimal tuning (CTE): ~? = 471.~~7...~~

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]]~~.

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

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

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

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

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

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

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

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

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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

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

[[Category:80edo]]

~~[[Category:Temperament collections~~]]

[[Category:80edo]]]

@@ Line 1: / Line 1: @@
-{{Fractional-octave navigation|80}}
+{{Technical data page}}
+{{Infobox fractional-octave|80}}
-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. In addition, it is the first equal division to be consistent in the 19-limit.
+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]].
 Temperaments discussed elsewhere include
@@ Line 68: / Line 69: @@
 Mapping: [{{val| 80 1 343 -27 434 13 327 -69}}, {{val| 0 4 -5 8 -5 9 0 13}}]
-Optimal tuning (CTE): ~? = 471.7...
+Optimal tuning (CTE): ~67473/51376 = 471.732
 {{Optimal ET sequence|legend=1| 720, 1600, 2320, 3920 }}
 == 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
@@ Line 81: / Line 82: @@
 {{mapping|legend=1| 80 3 161 -23 252 197 | 0 5 1 10 1 4 }}
-Mapping generators: ~9295/9216, ~2304/1859
+: mapping generators: ~9295/9216, ~2304/1859
 Optimal tuning (CTE): ~2304/1859 = 371.385
@@ Line 93: / Line 94: @@
 {{mapping|legend=1| 80 3 161 -23 252 197 327 | 0 5 1 10 1 4 0 }}
-Mapping generators: ~459/455, ~1859/1500
+: mapping generators: ~459/455, ~1859/1500
 Optimal tuning (CTE): ~2275/1836 = 371.386
@@ Line 105: / Line 106: @@
 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
@@ Line 149: / Line 150: @@
 Optimal tuning (CTE): ~920/741 = 374.387
-[[Category:80edo]]
+{{Navbox fractional-octave}}
-[[Category:Temperament collections]]
+[[Category:80edo]]]