Semaphoresmic family: Difference between revisions

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{{Technical data page}}
{{Technical data page}}
The '''semaphoresmic family''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out|tempers out]] [[49/48]] in the full [[7-limit]], and thereby identifies the septimal minor third, [[7/6]], and the septimal whole tone, [[8/7]]. It also splits the fourth into two of these intervals; hence the name, which sounds like "semi-fourth". Related to this is the 2.3.7-limit 49/48 temperament [[Semaphoresmic clan #Semaphore|semaphore]], and the 2.3.5.7 49/48 and 81/80 temperament [[Semaphoresmic clan #Godzilla|godzilla]].
The '''semaphoresmic family''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out|tempers out]] [[49/48]] in the full [[7-limit]], and thereby identifies the septimal minor third, [[7/6]], and the septimal whole tone, [[8/7]]. It also splits the fourth into two of these intervals; hence the name, which sounds like "semi-fourth". Related to this is the 2.3.7-subgroup {49/48} temperament [[Semaphoresmic clan #Semaphore|semaphore]], and the 7-limit {49/48, 81/80} temperament [[Semaphoresmic clan #Godzilla|godzilla]].


== Semiphore ==
== Semiphore ==
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.0000, ~7/4 = 952.2948, ~5/4 = 386.3137
* [[WE]]: ~2 = 1202.8324, ~7/4 = 951.8567, ~5/4 = 386.2644
: [[error map]]: {{val| 0.000 +2.635 0.000 -16.531 }}
: [[error map]]: {{val| +2.832 +1.758 -0.049 -11.304 }}
* [[CWE]]: ~2 = 1200.0000, ~7/4 = 950.6890, ~5/4 = 382.3522
* [[CWE]]: ~2 = 1200.0000, ~7/4 = 950.6890, ~5/4 = 382.3522
: error map: {{val| 0.000 -0.577 -3.962 -18.137 }}
: error map: {{val| 0.000 -0.577 -3.962 -18.137 }}
<!-- * [[CTE]]: ~2 = 1200.0000, ~7/4 = 952.2948, ~5/4 = 386.3137
: [[error map]]: {{val| 0.000 +2.635 0.000 -16.531 }} -->
<!-- * [[POTE]]: ~2 = 1200.0000, ~7/4 = 949.6154, ~5/4 = 379.7035
<!-- * [[POTE]]: ~2 = 1200.0000, ~7/4 = 949.6154, ~5/4 = 379.7035
: error map: {{val| 0.000 -2.724 -6.610 -19.211 }} -->
: error map: {{val| 0.000 -2.724 -6.610 -19.211 }} -->
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.0000, ~7/4 = 951.8194, ~5/4 = 390.7202
* [[WE]]: ~2 = 1200.0412, ~7/4 = 951.7616, ~5/4 = 390.7445
: [[error map]]: {{val| 0.000 +1.684 +4.407 -17.007 +9.781 }}
: [[error map]]: {{val| +0.041 +1.568 +4.431 -16.982 +9.905 }}
* [[CWE]]: ~2 = 1200.0000, ~7/4 = 951.7409, ~5/4 = 390.6582
* [[CWE]]: ~2 = 1200.0000, ~7/4 = 951.7409, ~5/4 = 390.6582
: error map: {{val| 0.000 +1.527 +4.344 -17.085 +9.765 }}
: error map: {{val| 0.000 +1.527 +4.344 -17.085 +9.765 }}
<!-- * [[CTE]]: ~2 = 1200.0000, ~7/4 = 951.8194, ~5/4 = 390.7202
: [[error map]]: {{val| 0.000 +1.684 +4.407 -17.007 +9.781 }} -->
<!-- * [[POTE]]: ~2 = 1200.0000, ~7/4 = 951.7290, ~5/4 = 390.6487
<!-- * [[POTE]]: ~2 = 1200.0000, ~7/4 = 951.7290, ~5/4 = 390.6487
: error map: {{val| 0.000 +1.503 +4.335 -17.097 +9.762 }} -->
: error map: {{val| 0.000 +1.503 +4.335 -17.097 +9.762 }} -->
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.0000, ~7/4 = 951.3729, ~5/4 = 389.6907
* WE: ~2 = 1200.0790, ~7/4 = 951.3067, ~5/4 = 389.7793
* CWE: ~2 = 1200.0000, ~7/4 = 951.2662, ~5/4 = 389.6120
* CWE: ~2 = 1200.0000, ~7/4 = 951.2662, ~5/4 = 389.6120
<!-- * CTE: ~2 = 1200.0000, ~7/4 = 951.3729, ~5/4 = 389.6907 -->
<!-- * POTE: ~2 = 1200.0000, ~7/4 = 951.2441, ~5/4 = 389.5957 -->
<!-- * POTE: ~2 = 1200.0000, ~7/4 = 951.2441, ~5/4 = 389.5957 -->


Line 72: Line 77:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.0000, ~15/14 = 124.8134, ~11/8 = 551.3179
* [[WE]]: ~2 = 1203.4810, ~15/14 = 125.9724, ~11/8 = 540.7985
: [[error map]]: {{val| 0.000 -1.209 -11.874 -18.453 0.000 }}
: [[error map]]: {{val| +3.481 +1.118 -1.435 -10.328 -0.076 }}
* [[CWE]]: ~2 = 1200.0000, ~15/14 = 125.4347, ~11/8 = 541.8688
* [[CWE]]: ~2 = 1200.0000, ~15/14 = 125.4347, ~11/8 = 541.8688
: error map: {{val| 0.000 -3.694 -10.009 -19.695 -9.449 }}
: error map: {{val| 0.000 -3.694 -10.009 -19.695 -9.449 }}
<!-- * [[CTE]]: ~2 = 1200.0000, ~15/14 = 124.8134, ~11/8 = 551.3179
: [[error map]]: {{val| 0.000 -1.209 -11.874 -18.453 0.000 }} -->
<!-- * [[POTE]]: ~2 = 1200.0000, ~15/14 = 125.6080, ~11/8 = 539.2342
<!-- * [[POTE]]: ~2 = 1200.0000, ~15/14 = 125.6080, ~11/8 = 539.2342
: error map: {{val| 0.000 -4.387 -9.490 -20.042 -12.084 }} -->
: error map: {{val| 0.000 -4.387 -9.490 -20.042 -12.084 }} -->
Line 91: Line 98:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.0000, ~14/13 = 124.4571, ~11/8 = 551.3179
* WE: ~2 = 1203.6981, ~14/13 = 125.9545, ~11/8 = 540.1440
* CWE: ~2 = 1200.0000, ~14/13 = 125.3543, ~11/8 = 540.9490
* CWE: ~2 = 1200.0000, ~14/13 = 125.3543, ~11/8 = 540.9490
<!-- * CTE: ~2 = 1200.0000, ~14/13 = 124.4571, ~11/8 = 551.3179 -->
<!-- * POTE: ~2 = 1200.0000, ~14/13 = 125.5675, ~11/8 = 538.4845 -->
<!-- * POTE: ~2 = 1200.0000, ~14/13 = 125.5675, ~11/8 = 538.4845 -->


Revision as of 09:35, 14 July 2025

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 semaphoresmic family of rank-3 temperaments tempers out 49/48 in the full 7-limit, and thereby identifies the septimal minor third, 7/6, and the septimal whole tone, 8/7. It also splits the fourth into two of these intervals; hence the name, which sounds like "semi-fourth". Related to this is the 2.3.7-subgroup {49/48} temperament semaphore, and the 7-limit {49/48, 81/80} temperament godzilla.

Semiphore

This temperament is also known as semaphoresmic.

Subgroup: 2.3.5.7

Comma list: 49/48

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

mapping generators: ~2, ~7/4, ~5

Lattice basis:

7/6 length = 0.7627, 5/4 length = 2.322
Angle (7/6, 5/4) = 90 degrees

Optimal tunings:

  • WE: ~2 = 1202.8324, ~7/4 = 951.8567, ~5/4 = 386.2644
error map: ⟨+2.832 +1.758 -0.049 -11.304]
  • CWE: ~2 = 1200.0000, ~7/4 = 950.6890, ~5/4 = 382.3522
error map: ⟨0.000 -0.577 -3.962 -18.137]

Optimal ET sequence: 4, 5, 9, 10, 14c, 15, 19

Badness (Sintel): 0.510

Selenium

Subgroup: 2.3.5.7.11

Comma list: 49/48, 56/55

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

Optimal tunings:

  • WE: ~2 = 1200.0412, ~7/4 = 951.7616, ~5/4 = 390.7445
error map: ⟨+0.041 +1.568 +4.431 -16.982 +9.905]
  • CWE: ~2 = 1200.0000, ~7/4 = 951.7409, ~5/4 = 390.6582
error map: ⟨0.000 +1.527 +4.344 -17.085 +9.765]

Optimal ET sequence: 4, 5, 9, 10, 15, 19, 24, 34

Badness (Sintel): 0.800

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 56/55, 91/90

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

Optimal tunings:

  • WE: ~2 = 1200.0790, ~7/4 = 951.3067, ~5/4 = 389.7793
  • CWE: ~2 = 1200.0000, ~7/4 = 951.2662, ~5/4 = 389.6120

Optimal ET sequence: 5, 9, 10, 15, 19, 24, 34

Badness (Sintel): 0.736

Negric

Subgroup: 2.3.5.7.11

Comma list: 49/48, 225/224

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

Optimal tunings:

  • WE: ~2 = 1203.4810, ~15/14 = 125.9724, ~11/8 = 540.7985
error map: ⟨+3.481 +1.118 -1.435 -10.328 -0.076]
  • CWE: ~2 = 1200.0000, ~15/14 = 125.4347, ~11/8 = 541.8688
error map: ⟨0.000 -3.694 -10.009 -19.695 -9.449]

Optimal ET sequence: 9, 10, 19, 29, 38d, 67cdde, 105cdddee

Badness (Sintel): 1.31

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 65/64, 91/90

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

Optimal tunings:

  • WE: ~2 = 1203.6981, ~14/13 = 125.9545, ~11/8 = 540.1440
  • CWE: ~2 = 1200.0000, ~14/13 = 125.3543, ~11/8 = 540.9490

Optimal ET sequence: 9, 10, 19, 29, 38df, 67cddef, 105cdddeefff

Badness (Sintel): 0.755

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