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===Vital statistics===
{{Technical data page}}
[[Comma|Comma]]: hemimage 10976/10935
The '''hemimage family''' of [[rank-3 temperament]]s [[tempering out|tempers out]] the [[hemimage comma]] ({{monzo|legend=1| 5 -7 -1 3 }}, [[ratio]]: 10976/10935).


7-limit minimax
== Hemimage ==
Hemimage is generated by a [[3/2|perfect fifth]] and a [[~]][[28/27]] subminor second. Two of the subminor seconds make ~[[15/14]], and three make ~[[10/9]]. The ubiquitous presence of this little interval gives rise to the highly intense [[hemimage bleeding chord]]s.


[|1 0 0 0>, |5/8 1/8 -1/8 3/8>,
[[Subgroup]]: 2.3.5.7
|5/8 -7/8 7/8 3/8>, |0 0 0 1>]


[[Eigenmonzo|Eigenmonzos]]: 2, 8/7, 6/5
[[Comma list]]: 10976/10935


9-limit minimax
{{Mapping|legend=1| 1 0 -1 -2 | 0 1 2 3 | 0 0 3 1 }}
: mapping generators: ~2, ~3, ~28/27


[|1 0 0 0>, |5/9 2/9 -1/9 1/3>,
[[Optimal tuning]]s:
|10/9 -14/9 7/9 2/3>, |0 0 0 1>]
* [[WE]]: ~2 = 1199.8555{{c}}, ~3/2 = 702.6069{{c}}, ~28/27 = 60.4665{{c}}
: [[error map]]: {{val| -0.145 +0.507 +0.155 -0.683 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.6137{{c}}, ~28/27 = 60.4761{{c}}
: error map: {{val| 0.000 +0.659 +0.342 -0.509 }}


[[Eigenmonzo|Eigenmonzos]]: 2, 10/9, 8/7
[[Minimax tuning]]:
* [[7-odd-limit]]
: {{monzo list| 1 0 0 0 | 5/8 1/8 -1/8 3/8 | 5/8 -7/8 7/8 3/8 | 0 0 0 1 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5/3.7
* [[9-odd-limit]]
: {{monzo list| 1 0 0 0 | 5/9 2/9 -1/9 1/3 | 10/9 -14/9 7/9 2/3 | 0 0 0 1 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5.7


Map: [<1 0 2 -1|, <0 1 2 3|, <0 0 -3 -1|]
{{Optimal ET sequence|legend=1| 17c, 19, 39d, 41, 58, 77, 80, 99, 217, 239, 338, 357, 456, 695 }}


[[generator|Generators]]: 2, 3, 27/14
[[Badness]] (Sintel): 0.878


[[EDO|Edos]]: 14, 19, 41, 58, 77, 80, [[99edo|99]], [[118edo|118]], [[140edo|140]], [[217edo|217]], [[316edo|316]], [[338edo|338]], [[357edo|357]], [[415edo|415]], [[456edo|456]], [[654edo|654]], [[695edo|695]]
[[Projection pair]]: <code>5 10976/2187</code> to 2.3.7


Badness: 0.000199
== Uni ==
[[Subgroup]]: 2.3.5.7.11


[[Projection_pair|Projection pair]]: 5 10976/2187 to 2.3.7
[[Comma list]]: 540/539, 896/891


==Uni==
{{Mapping|legend=1| 1 0 -1 -2 5 | 0 1 2 3 -1 | 0 0 3 1 1 }}
[[Comma|Commas]]: 540/539, 896/891


[[Minimax_tuning|Minimax tuning]]
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.1497{{c}}, ~3/2 = 703.3014{{c}}, ~28/27 = 60.0617{{c}}
: [[error map]]: {{val| -0.859 +0.496 -0.376 +0.290 +2.041 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.6475{{c}}, ~28/27 = 59.9675{{c}}
: error map: {{val| 0.000 +1.693 +0.884 +2.084 +5.002 }}


[|1 0 0 0 0&gt;, |16/11 6/11 1/11 0 -3/11&gt;,  
[[Minimax tuning]]:
|0 0 1 0 0&gt;, |19/11 14/11 6/11 0 -7/11&gt;,  
* [[11-odd-limit]]
|32/11 -10/11 2/11 0 5/11&gt;]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 16/11 6/11 1/11 0 -3/11 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| 19/11 14/11 6/11 0 -7/11 }}, {{monzo| 32/11 -10/11 2/11 0 5/11 }}]
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.11/9


[[Eigenmonzo|Eigenmonzos]]: 2, 11/9, 5/4
{{Optimal ET sequence|legend=1| 17c, 19, 22, 39d, 41, 58, 80, 99e, 121, 179e, 220e, 300bdee, 399bcdeee }}


Map: [&lt;1 0 2 -1 6|, &lt;0 1 2 3 -1|, &lt;0 0 -3 -1 -1|]
[[Badness]] (Sintel): 0.844


[[generator|Generators]]: 2, 3, 27/14
=== 13-limit  ===
Subgroup: 2.3.5.7.11.13


[[EDO|Edos]]: [[19edo|19]], [[22edo|22]], [[41edo|41]], [[58edo|58]], [[80edo|80]], [[121edo|121]]
Comma list: 352/351, 364/363, 540/539


Badness: 0.000702
Mapping: {{mapping| 1 0 -1 -2 5 10 | 0 1 2 3 -1 -4 | 0 0 3 1 1 1 }}


===13-limit===
Optimal tunings:
Commas: 352/351, 364/363, 540/539
* WE: ~2 = 1199.0867{{c}}, ~3/2 = 703.4048{{c}}, ~28/27 = 60.0034{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.9457{{c}}, ~28/27 = 59.8167{{c}}


Map: [&lt;1 0 2 -1 6 11|, &lt;0 1 2 3 -1 -4|, &lt;0 0 -3 -1 -1 -1|]
{{Optimal ET sequence|legend=0| 17c, 22, 36f, 39d, 41, 58, 80, 99ef, 121, 179ef, 300bdeef }}


EDOs: 22, 41, 58, 80, 121
Badness (Sintel): 0.993


Badness: 0.00106
=== Tin ===
Subgroup: 2.3.5.7.11.13


===Tin===
Comma list: 144/143, 196/195, 729/728
Commas: 144/143, 196/195, 729/728


Map: [&lt;1 0 2 -1 6 -2|, &lt;0 1 2 3 -1 3|, &lt;0 0 -3 -1 -1 1|]
Mapping: {{mapping| 1 0 -1 -2 5 -1 | 0 1 2 3 -1 3 | 0 0 3 1 1 -1 }}


EDOs: 19, 36, 41, 58
Optimal tunings:  
* WE: ~2 = 1198.8801{{c}}, ~3/2 = 702.5101{{c}}, ~28/27 = 60.9480{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.8952{{c}}, ~28/27 = 60.9129{{c}}


Badness: 0.00107
{{Optimal ET sequence|legend=0| 17c, 19, 22f, 36, 39df, 41, 58, 99ef, 157eff }}


==Turan==
Badness (Sintel): 1.00
[[Comma|Commas]]: 10976/10935, 441/440


[[Minimax_tuning|Minimax tuning]]
== Turan ==
[[Subgroup]]: 2.3.5.7.11


[|1 0 0 0 0&gt;, |19/12 -2/3 1/12 0 1/4&gt;,
[[Comma list]]: 441/440, 10976/10935
|19/6 -10/3 7/6 0 1/2&gt;,
|37/12 -8/3 7/12 0 3/4&gt;,  
|19/6 -10/3 1/6 0 3/2&gt;]


[[Eigenmonzo|Eigenmonzos]]: 2, 10/9, 11/9
{{Mapping|legend=1| 1 0 -1 -2 -6 | 0 1 2 3 6 | 0 0 3 1 -1 }}


Map: [&lt;1 0 2 -1 -7|, &lt;0 1 2 3 6|, &lt;0 0 -3 -1 1|]
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.0289{{c}}, ~3/2 = 702.1366{{c}}, ~28/27 = 60.8231{{c}}
: [[error map]]: {{val| +0.029 +0.211 +0.458 -1.564 +0.679 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.1290{{c}}, ~28/27 = 60.8257{{c}}
: error map: {{val| 0.000 +0.174 +0.421 -1.613 +0.630 }}


[[generator|Generators]]: 2, 3, 27/14
[[Minimax tuning]]:
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 19/12 -2/3 1/12 0 1/4 }}, {{monzo| 19/6 -10/3 7/6 0 1/2 }}, {{monzo| 37/12 -8/3 7/12 0 3/4 }}, {{monzo| 19/6 -10/3 1/6 0 3/2 }}]
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5.11/9


[[EDO|Edos]]: [[41edo|41]], [[58edo|58]], [[118edo|118]], [[159edo|159]], [[217edo|217]], [[335edo|335]]
{{Optimal ET sequence|legend=1| 17c, 19e, 36e, 39dee, 41, 58, 99e, 118, 159, 217, 335 }}


Badness: 0.000936
[[Badness]] (Sintel): 1.13


[[Category:Theory]]
[[Category:Hemimage family| ]] <!-- main article -->
[[Category:Temperament family]]
[[Category:Temperament families]]
[[Category:Planar temperament]]
[[Category:Catalogs of rank-3 temperaments]]
[[Category:Hemimage]]

Latest revision as of 12:00, 14 August 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The hemimage family of rank-3 temperaments tempers out the hemimage comma (monzo: [5 -7 -1 3⟩, ratio: 10976/10935).

Hemimage

Hemimage is generated by a perfect fifth and a ~28/27 subminor second. Two of the subminor seconds make ~15/14, and three make ~10/9. The ubiquitous presence of this little interval gives rise to the highly intense hemimage bleeding chords.

Subgroup: 2.3.5.7

Comma list: 10976/10935

Mapping: [⟨1 0 -1 -2], ⟨0 1 2 3], ⟨0 0 3 1]]

mapping generators: ~2, ~3, ~28/27

Optimal tunings:

  • WE: ~2 = 1199.8555 ¢, ~3/2 = 702.6069 ¢, ~28/27 = 60.4665 ¢
error map: ⟨-0.145 +0.507 +0.155 -0.683]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.6137 ¢, ~28/27 = 60.4761 ¢
error map: ⟨0.000 +0.659 +0.342 -0.509]

Minimax tuning:

[[1 0 0 0⟩, [5/8 1/8 -1/8 3/8⟩, [5/8 -7/8 7/8 3/8⟩, [0 0 0 1⟩]
unchanged-interval (eigenmonzo) basis: 2.5/3.7
[[1 0 0 0⟩, [5/9 2/9 -1/9 1/3⟩, [10/9 -14/9 7/9 2/3⟩, [0 0 0 1⟩]
unchanged-interval (eigenmonzo) basis: 2.9/5.7

Optimal ET sequence: 17c, 19, 39d, 41, 58, 77, 80, 99, 217, 239, 338, 357, 456, 695

Badness (Sintel): 0.878

Projection pair: 5 10976/2187 to 2.3.7

Uni

Subgroup: 2.3.5.7.11

Comma list: 540/539, 896/891

Mapping: [⟨1 0 -1 -2 5], ⟨0 1 2 3 -1], ⟨0 0 3 1 1]]

Optimal tunings:

  • WE: ~2 = 1199.1497 ¢, ~3/2 = 703.3014 ¢, ~28/27 = 60.0617 ¢
error map: ⟨-0.859 +0.496 -0.376 +0.290 +2.041]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.6475 ¢, ~28/27 = 59.9675 ¢
error map: ⟨0.000 +1.693 +0.884 +2.084 +5.002]

Minimax tuning:

[[1 0 0 0 0⟩, [16/11 6/11 1/11 0 -3/11⟩, [0 0 1 0 0⟩, [19/11 14/11 6/11 0 -7/11⟩, [32/11 -10/11 2/11 0 5/11⟩]
unchanged-interval (eigenmonzo) basis: 2.5.11/9

Optimal ET sequence: 17c, 19, 22, 39d, 41, 58, 80, 99e, 121, 179e, 220e, 300bdee, 399bcdeee

Badness (Sintel): 0.844

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 364/363, 540/539

Mapping: [⟨1 0 -1 -2 5 10], ⟨0 1 2 3 -1 -4], ⟨0 0 3 1 1 1]]

Optimal tunings:

  • WE: ~2 = 1199.0867 ¢, ~3/2 = 703.4048 ¢, ~28/27 = 60.0034 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.9457 ¢, ~28/27 = 59.8167 ¢

Optimal ET sequence: 17c, 22, 36f, 39d, 41, 58, 80, 99ef, 121, 179ef, 300bdeef

Badness (Sintel): 0.993

Tin

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 196/195, 729/728

Mapping: [⟨1 0 -1 -2 5 -1], ⟨0 1 2 3 -1 3], ⟨0 0 3 1 1 -1]]

Optimal tunings:

  • WE: ~2 = 1198.8801 ¢, ~3/2 = 702.5101 ¢, ~28/27 = 60.9480 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.8952 ¢, ~28/27 = 60.9129 ¢

Optimal ET sequence: 17c, 19, 22f, 36, 39df, 41, 58, 99ef, 157eff

Badness (Sintel): 1.00

Turan

Subgroup: 2.3.5.7.11

Comma list: 441/440, 10976/10935

Mapping: [⟨1 0 -1 -2 -6], ⟨0 1 2 3 6], ⟨0 0 3 1 -1]]

Optimal tunings:

  • WE: ~2 = 1200.0289 ¢, ~3/2 = 702.1366 ¢, ~28/27 = 60.8231 ¢
error map: ⟨+0.029 +0.211 +0.458 -1.564 +0.679]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.1290 ¢, ~28/27 = 60.8257 ¢
error map: ⟨0.000 +0.174 +0.421 -1.613 +0.630]

Minimax tuning:

[[1 0 0 0 0⟩, [19/12 -2/3 1/12 0 1/4⟩, [19/6 -10/3 7/6 0 1/2⟩, [37/12 -8/3 7/12 0 3/4⟩, [19/6 -10/3 1/6 0 3/2⟩]
unchanged-interval (eigenmonzo) basis: 2.9/5.11/9

Optimal ET sequence: 17c, 19e, 36e, 39dee, 41, 58, 99e, 118, 159, 217, 335

Badness (Sintel): 1.13

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