4096/4095: Difference between revisions

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{{Infobox Interval
{{Infobox Interval
| Ratio = 4096/4095
| Name = minisma, minic comma
| Monzo = 12 -2 -1 -1 0 -1
| Color name = s3urg1, sathurugu 1sn, <br>Sathurugu comma
| Cents = 0.42272
| Comma = yes
| Name = schismina
| Color name = sathurugu comma
| FJS name = P1<sub>5,7,13</sub>
}}
}}
'''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 [[13-limit]] [[superparticular]] [[ratio]] 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 factors into ([[6656/6655]])⋅([[10648/10647]]). It is also the smallest superparticular ratio in the [[2.3.5.7.13 subgroup]].  


By tempering it out is defined the '''schisminic temperament''', which enables the '''schisminic chords''', the essentially tempered chords in the 21-odd-limit.  
== 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.  


The name lends itself to the general interval size measure knowns as the [[mina]]. In [[Sagittal notation]], 4096/4095 is the default comma represented by a mina or three [[tina]]s.  
=== 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 ==
== See also ==
* [[Unnoticeable comma]]
* [[List of superparticular intervals]]
* [[List of superparticular intervals]]
* [[Tridecimal schisma]] (disambiguation page)
== References ==
<references/>


[[Category:13-limit]]
[[Category:Minismic]]
[[Category:Unnoticeable commas]]
[[Category:Superparticular]]
[[Category:Octave-reduced subharmonics]]
[[Category:Schisminic]]
[[Category:Mina]]
[[Category:Mina]]
[[Category:Commas named after their interval size]]

Latest revision as of 18:41, 4 May 2026

Interval information
Ratio 4096/4095
Factorization 212 × 3-2 × 5-1 × 7-1 × 13-1
Monzo [12 -2 -1 -1 0 -1⟩
Size in cents 0.4227162¢
Names minisma,
minic comma
Color name s3urg1, sathurugu 1sn,
Sathurugu comma
FJS name [math]\displaystyle{ \text{P1}_{5,7,13} }[/math]
Special properties square superparticular,
reduced,
reduced subharmonic
Tenney norm (log2 nd) 23.9996
Weil norm (log2 max(n, d)) 24
Wilson norm (sopfr(nd)) 55
Comma size unnoticeable
S-expression S64
Open this interval in xen-calc

4096/4095, the minisma or minic comma, also described as the tridecimal schisma or tridecimal schismina[1], is an unnoticeable 13-limit superparticular comma of about 0.42 cents. It is the difference between the septimal comma (64/63) and the wilsorma (65/64), and between the septimal quartertone (36/35) and the tridecimal quartertone (1053/1024). It is also a Mersenne comma.

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 chords in the 21-odd-limit. You may find a list of good equal temperaments that support these temperaments below.

Minic

Subgroup: 2.3.5.7.13

Mapping:

[⟨ 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 tunings:

WE: ~2 = 1199.9720 ¢, ~3/2 = 701.9948 ¢, ~5/4 = 386.3824 ¢, ~7/4 = 968.9004 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.0114 ¢, ~5/4 = 386.3886 ¢, ~7/4 = 968.9218 ¢

Optimal ET sequence: 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:

[⟨ 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 tunings:

WE: ~2 = 1199.9720 ¢, ~3/2 = 701.9948 ¢, ~5/4 = 386.3824 ¢, ~7/4 = 968.9004 ¢, ~11/8 = 551.4020 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.0114 ¢, ~5/4 = 386.3886 ¢, ~7/4 = 968.9218 ¢, ~11/8 = 551.3984 ¢

Optimal ET sequence: 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 tinas) 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 nybbles (1.5 bytes).

See also

References

  1. ↑ Sagittal – A Microtonal Notation System by George D. Secor and David C. Keenan
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