289/288: Difference between revisions

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'''289/288''', the '''semitonisma''', also described as the ''septendecimal semitones comma'' and ''septendecimal 6-cent comma'', is a [[small comma|small]] [[17-limit]] (also 2.3.17 [[subgroup]]) [[superparticular]] [[comma]] of about 6 [[cent]]s. It is the difference between [[17/16]] and [[18/17]], the two septendecimal semitones, as well as between [[17/12]] and [[24/17]], the two septendecimal tritones. It is also the difference between two stacked [[17/16]] and [[9/8]].

'''289/288''', the '''semitonisma''', also described as the ''septendecimal semitones comma'' and ''septendecimal 6-cent comma'', is a [[17-limit]] (also 2.3.17 [[subgroup]]) [[superparticular]] [[comma]] of about 6 [[cent]]s. It is the difference between [[17/16]] and [[18/17]], the two septendecimal semitones, as well as between [[17/12]] and [[24/17]], the two septendecimal tritones. It is also the difference between two stacked [[17/16]] and [[9/8]].

== Commatic relations ==

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== Temperaments ==

Tempering out this comma in the full 17-limit results in the rank-6 '''semitonismic ~~temperament~~''', or in the 2.3.17 subgroup, the rank-2 '''semitonic ~~temperament~~'''. In either case the [[octave]] is split into two equal parts, each representing 17/12~24/17, and thus so is the whole tone, each representing 17/16~18/17. Odd edos cannot temper it out.

[[Tempering out]] this comma in the full 17-limit results in the rank-6 '''semitonismic''' temperament, or in the 2.3.17 subgroup, the rank-2 '''semitonic''' temperament. In either case the [[2/1|octave]] is split into two equal parts, each representing 17/12~24/17, and thus so is the whole tone, each representing 17/16~18/17. Odd edos cannot temper it out.

=== Semitonic ===

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| ⟨ || 0 || 0 || 0 || 0 || 0 || 1 || 0 || ]]

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: mapping generators: ~17/12, ~3, ~5, ~7, ~11, ~13

[[Optimal tuning]] ([[~~CTE~~]]): ~17/12 = 1\2, ~3/2 = 702.~~3472~~, 5/4, 7/4, 11/8, 13/8

[[Optimal tuning]]s:

* [[WE]]: ~17/12 = 600.1471{{c}}, ~3/2 = 701.9563{{c}}, ~5/4 = 385.7246{{c}}, ~7/4 = 968.2366{{c}}, ~11/8 = 550.4343{{c}}, ~13/8 = 839.6439{{c}}

* [[CWE]]: ~17/12 = 600.0000{{c}}, ~3/2 = 702.0260{{c}}, ~5/4 = 385.9237{{c}}, ~7/4 = 968.3544{{c}}, ~11/8 = 550.7369{{c}}, ~13/8 = 839.9061{{c}}

{{Optimal ET sequence|legend=1| 34d, 38df, 46, 58, 72, 130, 140, 190g, 212g, 270g, 342fg, 402defgg, 460gg, 532dfggg, 672defggg }}*

{{Optimal ET sequence|legend=1| 34d, 38df, 46, 58, 72, 130, 140, 190g, 212g, 270g, 342fg, 402defgg, 460gg, 532dfggg, 672defggg }} *

<nowiki>*</nowiki> [[optimal patent val]]: [[280edo|280]]

[[Badness]] (Sintel): 3.70

== Etymology ==

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== See also ==

* [[Small comma]]

* [[List of superparticular intervals]]

[[Category:Semitonismic]]

[[Category:Commas named for the intervals they stack]]

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 | Comma = yes
 }}
+'''289/288''', the '''semitonisma''', also described as the ''septendecimal semitones comma'' and ''septendecimal 6-cent comma'', is a [[small comma|small]] [[17-limit]] (also 2.3.17 [[subgroup]]) [[superparticular]] [[comma]] of about 6 [[cent]]s. It is the difference between [[17/16]] and [[18/17]], the two septendecimal semitones, as well as between [[17/12]] and [[24/17]], the two septendecimal tritones. It is also the difference between two stacked [[17/16]] and [[9/8]].
-'''289/288''', the '''semitonisma''', also described as the ''septendecimal semitones comma'' and ''septendecimal 6-cent comma'', is a [[17-limit]] (also 2.3.17 [[subgroup]]) [[superparticular]] [[comma]] of about 6 [[cent]]s. It is the difference between [[17/16]] and [[18/17]], the two septendecimal semitones, as well as between [[17/12]] and [[24/17]], the two septendecimal tritones. It is also the difference between two stacked [[17/16]] and [[9/8]].
 == Commatic relations ==
@@ Line 20: / Line 19: @@
 == Temperaments ==
-Tempering out this comma in the full 17-limit results in the rank-6 '''semitonismic temperament''', or in the 2.3.17 subgroup, the rank-2 '''semitonic temperament'''. In either case the [[octave]] is split into two equal parts, each representing 17/12~24/17, and thus so is the whole tone, each representing 17/16~18/17. Odd edos cannot temper it out.
+[[Tempering out]] this comma in the full 17-limit results in the rank-6 '''semitonismic''' temperament, or in the 2.3.17 subgroup, the rank-2 '''semitonic''' temperament. In either case the [[2/1|octave]] is split into two equal parts, each representing 17/12~24/17, and thus so is the whole tone, each representing 17/16~18/17. Odd edos cannot temper it out.
 === Semitonic ===
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 | ⟨ || 0 || 0 || 0 || 0 || 0 || 1 || 0 || ]]
 |}
 : mapping generators: ~17/12, ~3, ~5, ~7, ~11, ~13
-[[Optimal tuning]] ([[CTE]]): ~17/12 = 1\2, ~3/2 = 702.3472, 5/4, 7/4, 11/8, 13/8
+[[Optimal tuning]]s:
+* [[WE]]: ~17/12 = 600.1471{{c}}, ~3/2 = 701.9563{{c}}, ~5/4 = 385.7246{{c}}, ~7/4 = 968.2366{{c}}, ~11/8 = 550.4343{{c}}, ~13/8 = 839.6439{{c}}
+* [[CWE]]: ~17/12 = 600.0000{{c}}, ~3/2 = 702.0260{{c}}, ~5/4 = 385.9237{{c}}, ~7/4 = 968.3544{{c}}, ~11/8 = 550.7369{{c}}, ~13/8 = 839.9061{{c}}
-{{Optimal ET sequence|legend=1| 34d, 38df, 46, 58, 72, 130, 140, 190g, 212g, 270g, 342fg, 402defgg, 460gg, 532dfggg, 672defggg }}*
+{{Optimal ET sequence|legend=1| 34d, 38df, 46, 58, 72, 130, 140, 190g, 212g, 270g, 342fg, 402defgg, 460gg, 532dfggg, 672defggg }} *
 <nowiki>*</nowiki> [[optimal patent val]]: [[280edo|280]]
+[[Badness]] (Sintel): 3.70
 == Etymology ==
@@ Line 56: / Line 58: @@
 == See also ==
-* [[Small comma]]
 * [[List of superparticular intervals]]
 [[Category:Semitonismic]]
 [[Category:Commas named for the intervals they stack]]