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 [[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.

== Commatic relations ==

This comma identifies itself as the difference between the following superparticular pairs:

* [[85/84]] and [[120/119]]

* [[136/135]] and [[256/255]]

* [[273/272]] and [[4914/4913]]

It factors into the following superparticular pairs:

* [[561/560]] and [[595/594]]

* [[442/441]] and [[833/832]]

* [[385/384]] and [[1156/1155]]

* [[325/324]] and [[2601/2600]]

== Temperaments ==

Tempering it out 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 so is the whole tone, each representing 17/16~18/17. Odd edos cannot temper it out.

=== Semitonismic ===

[[Subgroup]]: 2.3.5.7.11.13.17

[[Mapping]]:

[{{val| 2 0 0 0 0 0 5 }}

{{val| 0 1 0 0 0 0 1 }}

{{val| 0 0 1 0 0 0 0 }}

{{val| 0 0 0 1 0 0 0 }}

{{val| 0 0 0 0 1 0 0 }}

{{val| 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

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

=== Semitonic ===

[[Subgroup]]: 2.3.17

[[Sval]] [[mapping]]: [{{val| 2 0 5 }} {{val| 0 1 1 }}]

Mapping generators: ~17/12, ~3

[[Optimal tuning]] ([[CTE]]): ~17/12 = 1\2, ~3/2 = 702.3472 (~17/16 = 102.3472)

== Etymology ==

@@ Line 6: / Line 6: @@
 '''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.
+== Commatic relations ==
+This comma identifies itself as the difference between the following superparticular pairs:
+* [[85/84]] and [[120/119]]
+* [[136/135]] and [[256/255]]
+* [[273/272]] and [[4914/4913]]
+It factors into the following superparticular pairs:
+* [[561/560]] and [[595/594]]
+* [[442/441]] and [[833/832]]
+* [[385/384]] and [[1156/1155]]
+* [[325/324]] and [[2601/2600]]
 == Temperaments ==
 Tempering it out 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 so is the whole tone, each representing 17/16~18/17. Odd edos cannot temper it out.
+=== Semitonismic ===
+[[Subgroup]]: 2.3.5.7.11.13.17
+[[Mapping]]: <br>
+[{{val| 2 0 0 0 0 0 5 }}<br>
+{{val| 0 1 0 0 0 0 1 }}<br>
+{{val| 0 0 1 0 0 0 0 }}<br>
+{{val| 0 0 0 1 0 0 0 }}<br>
+{{val| 0 0 0 0 1 0 0 }}<br>
+{{val| 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
+{{Val list|legend=1| 34d, 38df, 46, 58, 72, 130, 140, 190g, 212g, 270g, 342fg, 402defgg, 460gg, 532dfggg, 672defggg }}
+=== Semitonic ===
+[[Subgroup]]: 2.3.17
+[[Sval]] [[mapping]]: [{{val| 2 0 5 }} {{val| 0 1 1 }}]
+Mapping generators: ~17/12, ~3
+[[Optimal tuning]] ([[CTE]]): ~17/12 = 1\2, ~3/2 = 702.3472 (~17/16 = 102.3472)
+{{Val list|legend=1| 12, 58, 70, 82, 94, 106, 118, 224g }}
 == Etymology ==