Semaphoresmic family: Difference between revisions

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{{Technical data page}}
'''Semiphore family''' is a rank-3 temperament family of 2.3.5.7 that tempers out 49/48 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 called "[[Chromatic pairs #Semaphore|semaphore]]," and the 2.3.5.7 49/48 and 81/80 temperament called "[[Meantone family #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 =
== Semaphoresmic a.k.a. semiphore ==
Comma list: 49/48
[[Subgroup]]: 2.3.5.7


Mapping: [{{val| 1 0 2 2 }}, {{val| 0 2 0 1 }}, {{val| 0 0 1 0 }}]
[[Comma list]]: 49/48


Mapping: Generators: 2, 7/4, 5
{{Mapping|legend=1| 1 0 2 2 | 0 2 0 1 | 0 0 1 0 }}
 
: mapping generators: ~2, ~7/4, ~5


Lattice basis:  
Lattice basis:  
Line 13: Line 15:
: Angle (7/6, 5/4) = 90 degrees
: Angle (7/6, 5/4) = 90 degrees


{{Val list|legend=1| 4, 5, 9, 10, 15, 19 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1202.8324{{c}}, ~7/4 = 951.8567{{c}}, ~5/4 = 380.5997{{c}}
: [[error map]]: {{val| +2.832 +1.758 -0.049 -11.304 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/4 = 950.6890{{c}}, ~5/4 = 382.3522{{c}}
: error map: {{val| 0.000 -0.577 -3.962 -18.137 }}
 
{{Optimal ET sequence|legend=1| 4, 5, 9, 10, 14c, 15, 19, 48d, 53d, 67cdd, 86cdd }}
 
[[Badness]] (Sintel): 0.510
 
=== 2.3.5.7.13 subgroup ===
Subgroup: 2.3.5.7.13
 
Comma list: 49/48, 91/90
 
Sval mapping: {{mapping| 1 0 0 2 -1 | 0 2 0 1 3 | 0 0 1 0 1 }}
 
Optimal tunings:
* WE: ~2 = 1202.7771{{c}}, ~7/4 = 951.9629{{c}}, ~5/4 = 381.0131{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 950.6703{{c}}, ~5/4 = 382.2972{{c}}
 
{{Optimal ET sequence|legend=0| 5, 9, 10, 14cf, 15, 19, 48df, 53d, 67cddf, 72dd }}
 
Badness (Sintel): 0.380
 
== Selenium ==
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 49/48, 56/55


Badness: 0.000116
{{Mapping|legend=1| 1 0 2 2 5 | 0 2 0 1 1 | 0 0 1 0 -1 }}


= Selenium =
[[Optimal tuning]]s:
Comma list: 49/48, 56/55
* [[WE]]: ~2 = 1200.0412{{c}}, ~7/4 = 951.7616{{c}}, ~5/4 = 390.6621{{c}}
: [[error map]]: {{val| +0.041 +1.568 +4.431 -16.982 +9.905 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/4 = 951.7409{{c}}, ~5/4 = 390.6582{{c}}
: error map: {{val| 0.000 +1.527 +4.344 -17.085 +9.765 }}


Mapping: [{{val| 1 0 2 2 5 }}, {{val| 0 2 0 1 1 }}, {{val| 0 0 1 0 -1 }}]
{{Optimal ET sequence|legend=1| 4, 5, 9, 10, 15, 19, 24, 34 }}


{{Val list|legend=1| 4, 5, 9, 10, 15, 19, 24, 34 }}
[[Badness]] (Sintel): 0.800


Badness: 0.000665
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


== 13-limit ==
Comma list: 49/48, 56/55, 91/90
Comma list: 49/48, 56/55, 91/90


Mapping: [{{val| 1 0 2 2 5 -1 }}, {{val| 0 2 0 1 1 3 }}, {{val| 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 tunings:
* WE: ~2 = 1200.0790{{c}}, ~7/4 = 951.3067{{c}}, ~5/4 = 389.6213{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 951.2662{{c}}, ~5/4 = 389.6120{{c}}
 
{{Optimal ET sequence|legend=0| 5, 9, 10, 15, 19, 24, 34 }}
 
Badness (Sintel): 0.736
 
== Negric ==
[[Subgroup]]: 2.3.5.7.11


{{Val list|legend=1| 5, 9, 10, 15, 19, 24, 34 }}
[[Comma list]]: 49/48, 225/224


Badness: 0.000787
{{Mapping|legend=1| 1 2 2 3 0 | 0 -4 3 -2 0 | 0 0 0 0 1 }}


= Negric =
[[Optimal tuning]]s:
Comma list: 49/48, 225/224
* [[WE]]: ~2 = 1203.4810{{c}}, ~15/14 = 125.9724{{c}}, ~11/8 = 540.7985{{c}}
: [[error map]]: {{val| +3.481 +1.118 -1.435 -10.328 -0.076 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~15/14 = 125.4347{{c}}, ~11/8 = 541.8688{{c}}
: error map: {{val| 0.000 -3.694 -10.009 -19.695 -9.449 }}


Mapping: [{{val| 1 2 2 3 0 }}, {{val| 0 -4 3 -2 0 }}, {{val| 0 0 0 0 1 }}]
{{Optimal ET sequence|legend=1| 9, 10, 19, 29, 38d, 67cdde, 105cdddee }}


{{Val list|legend=1| 9, 10, 19, 29, 38d, 67cde }}
[[Badness]] (Sintel): 1.31


Badness: 0.001087
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


== 13-limit ==
Comma list: 49/48, 65/64, 91/90
Comma list: 49/48, 65/64, 91/90


Mapping: [{{val| 1 2 2 3 0 4 }}, {{val| 0 -4 3 -2 0 -3 }}, {{val| 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 tunings:
* WE: ~2 = 1203.6981{{c}}, ~14/13 = 125.9545{{c}}, ~11/8 = 540.1440{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 125.3543{{c}}, ~11/8 = 540.9490{{c}}


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


Badness: 80.756
Badness (Sintel): 0.755


[[Category:Theory]]
[[Category:Semaphoresmic family| ]] <!-- main article -->
[[Category:Temperament family]]
[[Category:Temperament families]]
[[Category:Semiphore]]
[[Category:Catalogs of rank-3 temperaments]]
[[Category:Rank 3]]

Latest revision as of 11:48, 16 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 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.

Semaphoresmic a.k.a. semiphore

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 = 380.5997 ¢
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, 48d, 53d, 67cdd, 86cdd

Badness (Sintel): 0.510

2.3.5.7.13 subgroup

Subgroup: 2.3.5.7.13

Comma list: 49/48, 91/90

Sval mapping: [⟨1 0 0 2 -1], ⟨0 2 0 1 3], ⟨0 0 1 0 1]]

Optimal tunings:

  • WE: ~2 = 1202.7771 ¢, ~7/4 = 951.9629 ¢, ~5/4 = 381.0131 ¢
  • CWE: ~2 = 1200.0000 ¢, ~7/4 = 950.6703 ¢, ~5/4 = 382.2972 ¢

Optimal ET sequence: 5, 9, 10, 14cf, 15, 19, 48df, 53d, 67cddf, 72dd

Badness (Sintel): 0.380

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 = 390.6621 ¢
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 = 389.6213 ¢
  • 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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