4096/4095: Difference between revisions

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'''4096/4095''', the '''minisma''' or '''minic comma''', also described as the ''tridecimal schisma'' or ''tridecimal schismina''<ref>[https://sagittal.org/sagittal.pdf Sagittal – A Microtonal Notation System] by [[George Secor|George D. Secor]] and [[Dave Keenan|David C. Keenan]]</ref>, is an [[unnoticeable comma|unnoticeable]] [[13-limit]] [[superparticular]] [[comma]] of about 0.42 cents. It is the difference between the [[64/63|septimal comma (64/63)]] and the [[65/64|wilsorma (65/64)]], and between the [[36/35|septimal quartertone (36/35)]] and the [[1053/1024|tridecimal quartertone (1053/1024)]]. It is also a [[Mersenne comma]].

It is also the smallest superparticular ratio in the [[2.3.5.7.13 subgroup]].

It factors into ([[6656/6655]])⋅([[10648/10647]]). It is also the smallest superparticular ratio in the [[2.3.5.7.13 subgroup]].

== Temperaments ==

[[Tempering out]] this comma in the full 13-limit defines the '''minismic''' temperament, or in the 2.3.5.7.13 subgroup, the '''minic''' temperament. In either case, it equates the tridecimal quartertone with the septimal one, and enables the [[minismic chords]], the [[essentially tempered chord]]s in the [[21-odd-limit]]. You may find a list of good [[equal temperament]]s that [[support]] these temperaments below.

[[Tempering out]] this comma in the full 13-limit defines the rank-5 '''minismic''' temperament, or in the 2.3.5.7.13 subgroup, the rank-4 '''minic''' temperament. In either case, it equates the tridecimal quartertone with the septimal one, and enables the [[minismic chords]], the [[essentially tempered chord]]s in the [[21-odd-limit]]. You may find a list of good [[equal temperament]]s that [[support]] these temperaments below.

=== Minic ===

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[[Optimal tuning]]s:

: [[WE]]: ~2 = 1199.9720{{c}}, ~3/2 = 701.9948{{c}}, ~5/4 = 386.3824{{c}}, ~7/4 = 968.9004{{c}}, ~11/8 = ~551.4020{{c}}

: [[WE]]: ~2 = 1199.9720{{c}}, ~3/2 = 701.9948{{c}}, ~5/4 = 386.3824{{c}}, ~7/4 = 968.9004{{c}}, ~11/8 = 551.4020{{c}}

: [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.0114{{c}}, ~5/4 = 386.3886{{c}}, ~7/4 = 968.9218{{c}}, ~11/8 = 551.3984{{c}}

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== See also ==

* [[List of superparticular intervals]]

* [[Tridecimal schisma]] (disambiguation page)

== References ==

@@ Line 6: / Line 6: @@
 '''4096/4095''', the '''minisma''' or '''minic comma''', also described as the ''tridecimal schisma'' or ''tridecimal schismina''<ref>[https://sagittal.org/sagittal.pdf Sagittal – A Microtonal Notation System] by [[George Secor|George D. Secor]] and [[Dave Keenan|David C. Keenan]]</ref>, is an [[unnoticeable comma|unnoticeable]] [[13-limit]] [[superparticular]] [[comma]] of about 0.42 cents. It is the difference between the [[64/63|septimal comma (64/63)]] and the [[65/64|wilsorma (65/64)]], and between the [[36/35|septimal quartertone (36/35)]] and the [[1053/1024|tridecimal quartertone (1053/1024)]]. It is also a [[Mersenne comma]].
-It is also the smallest superparticular ratio in the [[2.3.5.7.13 subgroup]].
+It factors into ([[6656/6655]])⋅([[10648/10647]]). It is also the smallest superparticular ratio in the [[2.3.5.7.13 subgroup]].
 == Temperaments ==
-[[Tempering out]] this comma in the full 13-limit defines the '''minismic''' temperament, or in the 2.3.5.7.13 subgroup, the '''minic''' temperament. In either case, it equates the tridecimal quartertone with the septimal one, and enables the [[minismic chords]], the [[essentially tempered chord]]s in the [[21-odd-limit]]. You may find a list of good [[equal temperament]]s that [[support]] these temperaments below.
+[[Tempering out]] this comma in the full 13-limit defines the rank-5 '''minismic''' temperament, or in the 2.3.5.7.13 subgroup, the rank-4 '''minic''' temperament. In either case, it equates the tridecimal quartertone with the septimal one, and enables the [[minismic chords]], the [[essentially tempered chord]]s in the [[21-odd-limit]]. You may find a list of good [[equal temperament]]s that [[support]] these temperaments below.
 === Minic ===
@@ Line 54: / Line 54: @@
 [[Optimal tuning]]s:
-: [[WE]]: ~2 = 1199.9720{{c}}, ~3/2 = 701.9948{{c}}, ~5/4 = 386.3824{{c}}, ~7/4 = 968.9004{{c}}, ~11/8 = ~551.4020{{c}}
+: [[WE]]: ~2 = 1199.9720{{c}}, ~3/2 = 701.9948{{c}}, ~5/4 = 386.3824{{c}}, ~7/4 = 968.9004{{c}}, ~11/8 = 551.4020{{c}}
 : [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.0114{{c}}, ~5/4 = 386.3886{{c}}, ~7/4 = 968.9218{{c}}, ~11/8 = 551.3984{{c}}
@@ Line 66: / Line 66: @@
 == See also ==
 * [[List of superparticular intervals]]
+* [[Tridecimal schisma]] (disambiguation page)
 == References ==