4096/4095: Difference between revisions

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{{Infobox Interval

| Name = minisma, minic comma

| Color name = s3urg1, sathurugu 1sn, <br>Sathurugu comma

| Comma = yes

}}

'''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]].

'''~~4096/4095~~''', the '''~~schismina~~''', is a ~~ratio~~ of ~~about~~ 0.~~42 cents~~. It is the ~~difference between~~ [[~~1053/1024~~]] and [[36/35]].

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 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 ===

[[Subgroup]]: 2.3.5.7.13

[[Mapping]]: <br>

{| class="right-all"

|-

| [⟨ || 1 || 0 || 0 || 0 || 0 || 12 || ],

|-

| ⟨ || 0 || 1 || 0 || 0 || 0 || -2 || ],

|-

| ⟨ || 0 || 0 || 1 || 0 || 0 || -1 || ],

|-

| ⟨ || 0 || 0 || 0 || 1 || 0 || -1 || ],

|}

: mapping generators: ~2, ~3, ~5, ~7

[[Optimal tuning]]s:

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

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

{{Optimal ET sequence|legend=1| 31, 41, 46, 53, 77, 84, 87, 99, 118, 130, 140, 171, 224, 270, 441, 935, 1376, 1506, 1947, 2441, 2882, 5323f, 6699df, 8205cdf, 9140cdf, 9581cdff, 12022bcdff, 14904bcddfff, 19957bccdddffff }}

[[Badness]] (Sintel): 0.135

=== Minismic ===

[[Subgroup]]: 2.3.5.7.11.13

[[Mapping]]: <br>

{| class="right-all"

|-

| [⟨ || 1 || 0 || 0 || 0 || 0 || 12 || ],

|-

| ⟨ || 0 || 1 || 0 || 0 || 0 || -2 || ],

|-

| ⟨ || 0 || 0 || 1 || 0 || 0 || -1 || ],

|-

| ⟨ || 0 || 0 || 0 || 1 || 0 || -1 || ],

|-

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

|}

: mapping generators: ~2, ~3, ~5, ~7, ~11

[[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}}

: [[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}}

{{Optimal ET sequence|legend=1| 31, 41, 46, 53, 77, 84, 87, 118, 130, 183, 217, 224, 270, 494, 764, 935, 1012, 1236, 1506, 3236, 3323, 3817, 5323f, 9140cdef, 14463bcddeefff, 17786bccddeeefff }}

[[Badness]] (Sintel): 0.479

== Etymology ==

The names ''minisma'' and ''minic comma'' were given by [[Flora Canou]] in 2026 after the general interval size measure knowns as the ''[[mina]]'', because 4096/4095 is the default comma represented by a mina (or three [[tina]]s) in [[Sagittal notation]]. ''Mina'' is a clipping of ''schismina'', which describes a range of commas in Sagittal notation, and in turn comes from ''schisma'' (the interval region) + -in (diminutive suffix). [[Eufalesio]] proposed ''trinyblisma'', for the denominator is the largest integer that can be written in 3 {{w|nibble|nybbles}} (1.5 bytes).

== See also ==

* [[List of superparticular intervals]]

* [[Tridecimal schisma]] (disambiguation page)

== References ==

[[Category:Minismic]]

[[Category:Mina]]

[[Category:Commas named after their interval size]]

@@ Line 1: / Line 1: @@
-{{Infobox Interval}}
+{{Infobox Interval
+| Name = minisma, minic comma
+| Color name = s3urg1, sathurugu 1sn, <br>Sathurugu comma
+| Comma = yes
+}}
+'''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]].
-'''4096/4095''', the '''schismina''', is a ratio of about 0.42 cents. It is the difference between [[1053/1024]] and [[36/35]].
+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 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 ===
+[[Subgroup]]: 2.3.5.7.13
+[[Mapping]]: <br>
+{| class="right-all"
+|-
+| [⟨ || 1 || 0 || 0 || 0 || 0 || 12 || ],
+|-
+| ⟨ || 0 || 1 || 0 || 0 || 0 || -2 || ],
+|-
+| ⟨ || 0 || 0 || 1 || 0 || 0 || -1 || ],
+|-
+| ⟨ || 0 || 0 || 0 || 1 || 0 || -1 || ],
+|}
+: mapping generators: ~2, ~3, ~5, ~7
+[[Optimal tuning]]s:
+: [[WE]]: ~2 = 1199.9720{{c}}, ~3/2 = 701.9948{{c}}, ~5/4 = 386.3824{{c}}, ~7/4 = 968.9004{{c}}
+: [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.0114{{c}}, ~5/4 = 386.3886{{c}}, ~7/4 = 968.9218{{c}}
+{{Optimal ET sequence|legend=1| 31, 41, 46, 53, 77, 84, 87, 99, 118, 130, 140, 171, 224, 270, 441, 935, 1376, 1506, 1947, 2441, 2882, 5323f, 6699df, 8205cdf, 9140cdf, 9581cdff, 12022bcdff, 14904bcddfff, 19957bccdddffff }}
+[[Badness]] (Sintel): 0.135
+=== Minismic ===
+[[Subgroup]]: 2.3.5.7.11.13
+[[Mapping]]: <br>
+{| class="right-all"
+|-
+| [⟨ || 1 || 0 || 0 || 0 || 0 || 12 || ],
+|-
+| ⟨ || 0 || 1 || 0 || 0 || 0 || -2 || ],
+|-
+| ⟨ || 0 || 0 || 1 || 0 || 0 || -1 || ],
+|-
+| ⟨ || 0 || 0 || 0 || 1 || 0 || -1 || ],
+|-
+| ⟨ || 0 || 0 || 0 || 0 || 1 || 0 || ]]
+|}
+: mapping generators: ~2, ~3, ~5, ~7, ~11
+[[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}}
+: [[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}}
+{{Optimal ET sequence|legend=1| 31, 41, 46, 53, 77, 84, 87, 118, 130, 183, 217, 224, 270, 494, 764, 935, 1012, 1236, 1506, 3236, 3323, 3817, 5323f, 9140cdef, 14463bcddeefff, 17786bccddeeefff }}
+[[Badness]] (Sintel): 0.479
+== Etymology ==
+The names ''minisma'' and ''minic comma'' were given by [[Flora Canou]] in 2026 after the general interval size measure knowns as the ''[[mina]]'', because 4096/4095 is the default comma represented by a mina (or three [[tina]]s) in [[Sagittal notation]]. ''Mina'' is a clipping of ''schismina'', which describes a range of commas in Sagittal notation, and in turn comes from ''schisma'' (the interval region) + -in (diminutive suffix). [[Eufalesio]] proposed ''trinyblisma'', for the denominator is the largest integer that can be written in 3 {{w|nibble|nybbles}} (1.5 bytes).
+== See also ==
+* [[List of superparticular intervals]]
+* [[Tridecimal schisma]] (disambiguation page)
+== References ==
+<references/>
+[[Category:Minismic]]
+[[Category:Mina]]
+[[Category:Commas named after their interval size]]