2601/2600: Difference between revisions
→Temperaments: note about tuning exact sixth-tones |
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== Commatic relations == | == Commatic relations == | ||
In terms of commas, it is the difference between the following pairs: | In terms of commas, it is the difference between the following pairs: | ||
* [[289/288]] and [[325/324]] | * [[289/288]] and [[325/324]] (both of these commas are also within the 2.3.5.13.17 subgroup, while most others listed here are not) | ||
* [[561/560]] and [[715/714]] | * [[561/560]] and [[715/714]] | ||
* [[833/832]] and [[1225/1224]] | * [[833/832]] and [[1225/1224]] | ||
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* [[1701/1700]] and [[4914/4913]] | * [[1701/1700]] and [[4914/4913]] | ||
* [[2401/2400]] and [[31213/31212]] | * [[2401/2400]] and [[31213/31212]] | ||
* [[2431/2430]] and [[37180/37179]] | |||
== Temperaments == | == Temperaments == | ||
Tempering out this comma in the 17-limit results in the rank-6 '''sextantonismic temperament''', or in the 2.3.5.13.17 subgroup, the rank-4 '''sextantonic temperament'''. In either case [[26/25]] is split into two equal parts, each representing 51/50~52/51, and enables the [[sextantonismic chords]]. If [[140625/140608]] is also added to the comma list, the sixth-tone above becomes literally a sixth of [[9/8]] and is tuned exactly middle of 51/50 and 52/51. This temperament, however, strongly suggests also tempering out [[9801/9800]] and/or [[12376/12375]] since 2601/2600 = (9801/9800)(12376/12375)<sup>2</sup>(140625/140608). | Tempering out this comma in the 17-limit results in the rank-6 '''sextantonismic temperament''', or in the 2.3.5.13.17 subgroup, the rank-4 '''sextantonic temperament'''. In either case [[26/25]] is split into two equal parts, each representing 51/50~52/51, and enables the [[sextantonismic chords]]. If [[140625/140608]] is also added to the comma list, the sixth-tone above becomes literally a sixth of [[9/8]] and is tuned exactly middle of 51/50 and 52/51. This temperament, however, strongly suggests also tempering out [[9801/9800]] and/or [[12376/12375]] since 2601/2600 = (9801/9800)(12376/12375)<sup>2</sup>(140625/140608). | ||
=== Sextantonic === | |||
[[Subgroup]]: 2.3.5.13.17 | |||
{{Mapping|legend=2| 1 0 0 2 1 | 0 1 0 0 -1 | 0 0 0 0 1 | 0 0 0 2 1 }} | |||
: sval mapping generators: ~2, ~3, ~5, ~51/20 | |||
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.9252, ~5/4 = 386.3777, ~51/40 = 420.3045 | |||
{{Optimal ET sequence|legend=1| 12, 22f, 26, 31, 34, 72, 106, 137, 171, 183, 217, 277, 354, 388, 460, 494, 677, 814, 848, 3609g, 4069g, 4457g, 4917gg }} | |||
=== Sextantonismic === | === Sextantonismic === | ||
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<nowiki>*</nowiki> [[optimal patent val]]: [[2932edo|2932]] | <nowiki>*</nowiki> [[optimal patent val]]: [[2932edo|2932]] | ||
== Etymology == | == Etymology == | ||
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[[Category:Sextantonismic]] | [[Category:Sextantonismic]] | ||
[[Category:Commas named for the intervals they stack]] | |||