Semaphoresmic family: Difference between revisions

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Tunings and badnesses
Line 17: Line 17:
: Angle (7/6, 5/4) = 90 degrees
: Angle (7/6, 5/4) = 90 degrees


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/4 = 949.6154, ~5/4 = 379.7035
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.0000, ~7/4 = 952.2948, ~5/4 = 386.3137
: [[error map]]: {{val| 0.000 +2.635 0.000 -16.531 }}
* [[CWE]]: ~2 = 1200.0000, ~7/4 = 950.6890, ~5/4 = 382.3522
: error map: {{val| 0.000 -0.577 -3.962 -18.137 }}
<!-- * [[POTE]]: ~2 = 1200.0000, ~7/4 = 949.6154, ~5/4 = 379.7035
: error map: {{val| 0.000 -2.724 -6.610 -19.211 }} -->


{{Optimal ET sequence|legend=1| 4, 5, 9, 10, 14c, 15, 19 }}
{{Optimal ET sequence|legend=1| 4, 5, 9, 10, 14c, 15, 19 }}


[[Badness]]: 0.116 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.510


== Selenium ==
== Selenium ==
[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


Line 30: Line 36:
{{Mapping|legend=1| 1 0 2 2 5 | 0 2 0 1 1 | 0 0 1 0 -1 }}
{{Mapping|legend=1| 1 0 2 2 5 | 0 2 0 1 1 | 0 0 1 0 -1 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/4 = 951.7290, ~5/4 = 390.6487
[[Optimal tuning]]s:
* [[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 }}
* [[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 }}
<!-- * [[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 }} -->


{{Optimal ET sequence|legend=1| 4, 5, 9, 10, 15, 19, 24, 34 }}
{{Optimal ET sequence|legend=1| 4, 5, 9, 10, 15, 19, 24, 34 }}


[[Badness]]: 0.665 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.800


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 43: Line 55:
Mapping: {{mapping| 1 0 2 2 5 -1 | 0 2 0 1 1 3 | 0 0 1 0 -1 1 }}
Mapping: {{mapping| 1 0 2 2 5 -1 | 0 2 0 1 1 3 | 0 0 1 0 -1 1 }}


Optimal tuning (POTE): ~2 = 1\1, ~7/4 = 951.2441, ~5/4 = 389.5957
Optimal tunings:
* CTE: ~2 = 1200.0000, ~7/4 = 951.3729, ~5/4 = 389.6907
* CWE: ~2 = 1200.0000, ~7/4 = 951.2662, ~5/4 = 389.6120
<!-- * POTE: ~2 = 1200.0000, ~7/4 = 951.2441, ~5/4 = 389.5957 -->


{{Optimal ET sequence|legend=1| 5, 9, 10, 15, 19, 24, 34 }}
{{Optimal ET sequence|legend=0| 5, 9, 10, 15, 19, 24, 34 }}


Badness: 0.787 × 10<sup>-3</sup>
Badness (Sintel): 0.736


== Negric ==
== Negric ==
[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


Line 56: Line 71:
{{Mapping|legend=1| 1 2 2 3 0 | 0 -4 3 -2 0 | 0 0 0 0 1 }}
{{Mapping|legend=1| 1 2 2 3 0 | 0 -4 3 -2 0 | 0 0 0 0 1 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~15/14 = 125.6080, ~11/8 = 539.2342
[[Optimal tuning]]s:
* [[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 }}
* [[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 }}
<!-- * [[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 }} -->


{{Optimal ET sequence|legend=1| 9, 10, 19, 29, 38d, 67cde }}
{{Optimal ET sequence|legend=1| 9, 10, 19, 29, 38d, 67cdde, 105cdddee }}


[[Badness]]: 1.087 × 10<sup>-3</sup>
[[Badness]] (Sintel): 1.31


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 69: Line 90:
Mapping: {{mapping| 1 2 2 3 0 4 | 0 -4 3 -2 0 -3 | 0 0 0 0 1 0 }}
Mapping: {{mapping| 1 2 2 3 0 4 | 0 -4 3 -2 0 -3 | 0 0 0 0 1 0 }}


Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 125.5675, ~11/8 = 538.4845
Optimal tunings:
* CTE: ~2 = 1200.0000, ~14/13 = 124.4571, ~11/8 = 551.3179
* CWE: ~2 = 1200.0000, ~14/13 = 125.3543, ~11/8 = 540.9490
<!-- * POTE: ~2 = 1200.0000, ~14/13 = 125.5675, ~11/8 = 538.4845 -->


{{Optimal ET sequence|legend=1| 9, 10, 19, 29, 38df, 67cdef }}
{{Optimal ET sequence|legend=0| 9, 10, 19, 29, 38df, 67cddef, 105cdddeefff }}


Badness: 0.8076 × 10<sup>-3</sup>
Badness (Sintel): 0.755


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Semaphoresmic family| ]] <!-- main article -->
[[Category:Semaphoresmic family| ]] <!-- main article -->
[[Category:Rank 3]]
[[Category:Rank 3]]

Revision as of 13:32, 17 June 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-limit 49/48 temperament semaphore, and the 2.3.5.7 49/48 and 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:

  • CTE: ~2 = 1200.0000, ~7/4 = 952.2948, ~5/4 = 386.3137
error map: ⟨0.000 +2.635 0.000 -16.531]
  • 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:

  • CTE: ~2 = 1200.0000, ~7/4 = 951.8194, ~5/4 = 390.7202
error map: ⟨0.000 +1.684 +4.407 -17.007 +9.781]
  • 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:

  • CTE: ~2 = 1200.0000, ~7/4 = 951.3729, ~5/4 = 389.6907
  • 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:

  • CTE: ~2 = 1200.0000, ~15/14 = 124.8134, ~11/8 = 551.3179
error map: ⟨0.000 -1.209 -11.874 -18.453 0.000]
  • 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:

  • CTE: ~2 = 1200.0000, ~14/13 = 124.4571, ~11/8 = 551.3179
  • 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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