Semaphore and godzilla: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
BudjarnLambeth (talk | contribs)
autopatrolled, emailconfirmed
21,262 edits
m →See also: add link to orphaned page
Xenllium (talk | contribs)
autopatrolled, emailconfirmed
5,012 edits
No edit summary
Tags: Mobile edit Mobile web edit
 
(24 intermediate revisions by 6 users not shown)
Line 1: Line 1:
{{interwiki
{{interwiki
| en = Semaphore and godzilla
| de = Semiphor, Semaphor, Godzilla
| de = Semiphor, Semaphor, Godzilla
| en = Semaphore and Godzilla
| es =  
| es =  
| ja =  
| ja =  
}}
}}
'''Semaphore''', of the [[semaphoresmic clan]], is characterized by [[49/48]] being [[tempering out|tempered out]], so the [[generator]] represents [[8/7]] and [[7/6]] equally. This results in a very low [[complexity]] 2.3.7-[[subgroup]] [[regular temperament|temperament]], with the drawback that most intervals of 7 must be out of tune by at least half of the comma 49/48, or about 18 [[cent]]s. Semaphore is a play on the words "semi-" and "fourth."
{{Infobox regtemp
| Title = {{nowrap|Semaphore; Godzilla}}
| Subgroups = 2.3.7, 2.3.5.7, 2.3.5.7.13
| Comma basis = [[49/48]] (2.3.7); <br> [[49/48]], [[81/80]] (2.3.5.7); <br> [[49/48]], [[81/80]], [[91/90]] (L7.13)
| Edo join 1 = 5 | Edo join 2 = 19
| Mapping = 1; 2 8 1 11
| Generators = 7/4
| Generators tuning = 947.8
| Optimization method = CWE
| Pergen = (P8, P4/2)
| Color name = Zozoti
| MOS scales = [[4L&nbsp;1s]], [[5L&nbsp;4s]], [[5L&nbsp;9s]], [[5L&nbsp;14s]]
| Odd limit 1 = 9 | Mistuning 1 = 20.5 | Complexity 1 = 9
| Odd limit 2 = 2.3.5.7.13 15 | Mistuning 2 = 20.5 | Complexity 2 = 14
}}
'''Semaphore''', of the [[semaphoresmic clan]], is characterized by [[49/48]] being [[tempering out|tempered out]], so the [[generator]] represents [[7/4]] and [[12/7]] (or [[8/7]] and [[7/6]]) equally. This results in a very low [[complexity]] 2.3.7-[[subgroup]] [[regular temperament|temperament]], with the drawback that most intervals of 7 must be out of tune by at least half of the comma 49/48, or about 18 [[cent]]s. ''Semaphore'' is a play on the words "semi-" and "fourth".


If the [[5/1|5th harmonic]]'s intervals are desired, [[5/4]] can be sensibly mapped to &minus;8 generators by tempering out [[81/80]], making it a [[Meantone family #Extensions|meantone temperament]]. This temperament is '''godzilla'''. A more accurate but complex mapping of 5 can be found in [[immunity]], or 5/4 itself can be made the period by tempering out [[128/125]], resulting in [[triforce]].
If the [[5/1|5th harmonic]]'s intervals are desired, [[5/4]] can be sensibly mapped to +8 generators by tempering out [[81/80]], making it a [[Meantone family #Extensions|meantone temperament]]. This temperament is '''godzilla'''. Moreover, the generator can be taken to be [[26/15]], which maps [[13/8]] to +11 generators by tempering out [[91/90]] and [[105/104]]. This extends the temperament to the 2.3.5.7.13 subgroup, with an abundance of harmonic resource and little additional damage.  


See [[Semaphoresmic clan #Semaphore]] and [[Semaphoresmic clan #Godzilla|#Godzilla]] for technical data.  
A more accurate but complex mapping of 5 can be found in [[immunity]], or 5/4 itself can be made the period by tempering out [[128/125]], resulting in [[triforce]].
 
For technical information, see [[Semaphoresmic clan #Semaphore]] and [[Semaphoresmic clan #Godzilla|#Godzilla]]. For a discussion on 11- and 13-limit extensions, see [[Godzilla extensions]].


== Interval chains ==
== Interval chains ==
In the following tables, odd harmonics 1–13 and their inverses are in '''bold'''.
=== Semaphore ===
=== Semaphore ===
{| class="wikitable"
{| class="wikitable center-1 right-2"
|-
! # !! Cents* !! Approximate ratios
|-
| 0 || 0.0 || '''1/1'''
|-
| 1 || 950.7 || '''7/4''', 12/7
|-
| 2 || 701.4 || '''3/2'''
|-
|-
! Cents
| 3 || 452.1 || 9/7, 21/16
| 198.46
| 448.85
| 699.23
| 949.62
| 0
| 250.38
| 500.77
| 751.15
| 1001.54
|-
|-
! JI ratios
| 4 || 202.8 || '''9/8'''
| [[9/8|9/8]]
|-
| [[9/7|9/7]]
| 5 || 1153.4 || 27/14, 63/32
| [[3/2|3/2]]
| 12/7~7/4
| [[1/1|1/1]]
| 8/7~7/6
| [[4/3|4/3]]
| [[14/9|14/9]]
| [[16/9|16/9]]
|}
|}
<nowiki/>* In 2.3.7-subgroup CWE tuning, octave reduced


=== Godzilla ===
=== Godzilla ===
{| class="wikitable"
{| class="wikitable center-1 right-2"
|-
! # !! Cents* !! Approximate ratios
|-
| 0 || 0.0 || '''1/1'''
|-
| 1 || 948.0 || '''7/4''', 12/7, 26/15
|-
| 2 || 696.0 || '''3/2'''
|-
| 3 || 444.0 || 9/7, 13/10, 21/16
|-
| 4 || 192.0 || '''9/8''', 10/9
|-
| 5 || 1140.0 || 27/14, 39/20, 40/21, 52/27, 63/32
|-
| 6 || 888.0 || 5/3
|-
| 7 || 636.0 || 10/7, 13/9
|-
| 8 || 384.0 || '''5/4'''
|-
| 9 || 132.0 || 13/12, 15/14
|-
| 10 || 1080.0 || 13/7, 15/8
|-
| 11 || 828.0 || '''13/8'''
|-
| 12 || 576.0 || 25/18, 39/28, 45/32
|-
|-
! Cents
| 13 || 324.0 || 39/32
| 378.92
| 631.56
| 884.19
| 1136.83
| 189.46
| 442.10
| 694.73
| 947.37
| 0
| 252.63
| 505.27
| 757.90
| 1010.54
| 63.17
| 315.81
| 568.44
| 821.08
|-
|-
! JI ratios
| 14 || 72.1 || 25/24, 50/49
| [[5/4|5/4]]~16/13
| [[10/7|10/7]]~13/9
| [[5/3|5/3]]
| 27/14
| 10/9~9/8
| 9/7~13/10
| 3/2
| 12/7~7/4~26/15
| 1/1
| 8/7~7/6~15/13
| 4/3
| 14/9~20/13
| 16/9~9/5
| 28/27~21/20
| [[6/5|6/5]]
| [[7/5|7/5]]~18/13
| [[8/5|8/5]]~13/8
|}
|}
<nowiki/>* In 2.3.5.7.13-subgroup CWE tuning, octave reduced


== Scales ==
== Scales ==
Line 87: Line 94:


=== 5-note (proper) ===
=== 5-note (proper) ===
{| class="wikitable"
{| class="wikitable center-all"
|-
|-
! Small ("minor") interval
! Small ("minor") interval
| 198.46
| 202.8
| 448.85
| 452.1
| 699.23
| 701.4
| 949.62
| 950.7
|-
|-
! [[JI|JI]] intervals represented
! [[JI]] intervals represented
| 9/8
| 9/8
| 9/7~13/10
| 9/7~13/10
| 3/2
| 3/2
| 12/7~7/4~26/15
| 7/4~12/7
|-
|-
! Large ("major") interval
! Large ("major") interval
| 250.38
| 249.3
| 500.77
| 498.6
| 751.15
| 747.9
| 1001.54
| 997.2
|-
|-
! JI intervals represented
! JI intervals represented
| 8/7~7/6~15/13
| 7/6~8/7
| 4/3
| 4/3
| 14/9~20/13
| 14/9~20/13
Line 115: Line 122:


=== 9-note (improper) ===
=== 9-note (improper) ===
{{Main|5L 4s}}
{{Main| 5L 4s }}
{| class="wikitable"
 
{| class="wikitable center-all"
|-
|-
! Small ("minor") interval
! Small ("minor") interval
| 63.17
| 60.0
| 252.63
| 252.0
| 315.81
| 312.0
| 505.27
| 504.0
| 568.44
| 564.0
| 757.90
| 756.0
| 821.08
| 816.0
| 1010.54
| 1008.0
|-
|-
! JI intervals represented
! JI intervals represented
|  
|  
| 8/7~7/6~15/13
| 7/6~8/7
| 6/5
| 6/5
| 4/3
| 4/3
Line 136: Line 144:
| 14/9~20/13
| 14/9~20/13
| 8/5~13/8
| 8/5~13/8
| 16/9~9/5
| 9/5~16/9
|-
|-
! Large ("major") interval
! Large ("major") interval
| 189.46
| 192.0
| 378.92
| 384.0
| 442.10
| 444.0
| 631.56
| 636.0
| 694.73
| 696.0
| 884.19
| 888.0
| 947.37
| 948.0
| 1136.83
| 1140.0
|-
|-
! JI intervals represented
! JI intervals represented
| 10/9~9/8
| 9/8~10/9
| 5/4
| 5/4
| 9/7~13/10
| 9/7~13/10
Line 155: Line 163:
| 3/2
| 3/2
| 5/3
| 5/3
| 12/7~7/4~26/15
| 7/4~12/7
|  
|  
|}
|}


In 19edo, godzilla[9] has steps 3 3 1 3 1 3 1 3 1, and contains the following useful scales as subsets:
In 19edo, Godzilla[9] has steps 3 3 1 3 1 3 1 3 1, and contains the following useful scales as subsets:
* Meantone pentatonic (5 3 5 3 3)
* Meantone pentic (5 3 5 3 3)
* Altered diatonic I (3 4 3 1 3 4 1)
* Altered diatonic I (3 4 3 1 3 4 1)
* Altered diatonic II (3 4 3 1 4 3 1)
* Altered diatonic II (3 4 3 1 4 3 1)
Line 166: Line 174:
* Altered diatonic IV (3 3 4 1 3 4 1)
* Altered diatonic IV (3 3 4 1 3 4 1)


It does not, however, contain the ordinary diatonic scale. Godzilla[9] thus expands on the pentatonic scale, but in a different way than diatonic scales do.
It does not, however, contain the ordinary diatonic scale. Godzilla[9] thus expands on the pentic scale, but in a different way than diatonic scales do.
 
The four heptatonic subsets can be regarded as chromatic alterations of the diatonic scale, or alternatively as variants of Archytas' septimal diatonic scale, but with a greatly exaggerated difference between the two different whole tone sizes. All five of these subsets are very expressive melodically. Godzilla[9] combines all of these and is expressive in its own right; it could even be thought of as 19edo's answer to the well-loved Supra[7] diatonic scale of [[17edo]], as both are improper and made up of whole-tones and third-tones.


The four heptatonic subsets can be regarded as chromatic alterations of the diatonic scale, or alternatively as variants of Archytas' septimal diatonic scale, but with a greatly exaggerated difference between the two different whole tone sizes. All five of these subsets are very expressive melodically. Godzilla[9] combines all of these and is expressive in its own right; it could even be thought of as 19edo's answer to the well-loved supra[7] diatonic scale of [[17edo|17edo]], as both are improper and made up of whole-tones and third-tones.
Like Supra[7], Godzilla[9] is well stocked with subminor and supermajor triads; in this case they can be viewed as 6:7:9 and 10:13:15 since 19edo is a [[The Biosphere|biome]] temperament. Godzilla[9] has only ''one'' each of the more stable 5-limit major and minor triads, which might be considered a drawback, but could also be considered a strength for helping to establish a clearer tonal center (since all triads other than the tonic have tension in them).
 
== Tunings ==
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 2.3.7-subgroup norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~7/4 = 952.2948{{c}}
| CWE: ~7/4 = 950.6890{{c}}
| POTE: ~7/4 = 949.6154{{c}}
|}


Like supra[7], godzilla[9] is well stocked with subminor and supermajor triads; in this case they can be viewed as 6:7:9 and 10:13:15 since 19edo is a [[The_Biosphere|biome]] temperament. Godzilla[9] has only ''one'' each of the more stable 5-limit major and minor triads, which might be considered a drawback, but could also be considered a strength for helping to establish a clearer tonal center (since all triads other than the tonic have tension in them).
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~7/4 = 948.7959{{c}}
| CWE: ~7/4 = 947.8216{{c}}
| POTE: ~7/4 = 947.3650{{c}}
|}
 
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 2.3.5.7.13-subgroup norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~7/4 = 948.9311{{c}}
| CWE: ~7/4 = 948.0037{{c}}
| POTE: ~7/4 = 947.5708{{c}}
|}
 
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
|-
! Edo <br>generator
! [[Eigenmonzo|Unchanged interval <br>(eigenmonzo)]]*
! Generator (¢)
! Comments
|-
|
| 7/6
| 933.129
|
|-
| [[9edo|7\9]]
|
| 933.333
| 9cff val
|-
| [[14edo|11\14]]
|
| 942.857
| 14cf val, lower bound of 7- and 9-odd-limit diamond monotone
|-
|
| 9/7
| 945.028
|
|-
|
| 7/5
| 945.355
|
|-
|
| 13/7
| 947.170
|
|-
| [[19edo|15\19]]
|
| 947.368
| Lower bound of {{nowrap|no-11}} 13-odd-limit diamond monotone <br>{{nowrap|No-11}} 15-odd-limit diamond monotone (singleton)
|-
|
| 5/3
| 947.393
|
|-
|
| 13/9
| 948.088
|
|-
|
| 5/4
| 948.289
| 7-, 9-odd-limit, {{nowrap|no-11}} 13- and 15-odd-limit minimax
|-
|
| 13/12
| 948.730
|
|-
|
| 13/8
| 949.139
|
|-
| [[24edo|19\24]]
|
| 950.000
|
|-
|
| 3/2
| 950.978
|
|-
|
| 13/10
| 951.405
|
|-
| [[5edo|4\5]]
|
| 960.000
| Upper bound of 7-, 9-odd-limit, and {{nowrap|no-11}} 13-odd-limit diamond monotone
|-
|
| 7/4
| 968.826
|
|}
<nowiki/>* Besides the octave


== Music ==
== Music ==
Line 184: Line 336:


; [[Starshine]]
; [[Starshine]]
* [https://soundcloud.com/starshine99/rins-ufo-ride ''Rin's UFO Ride''] (2020) – Semaphore[9] in 19edo
* [https://soundcloud.com/starshine99/rins-ufo-ride ''Rin's UFO Ride''] (2020) – in Semaphore[9], 19edo tuning


== See also ==
== See also ==
* [[Diasem]], a [[maximum variety|max-variety-3]] JI [[detempering]] of semaphore
* [[Diasem]], a [[maximum variety|max-variety-3]] JI [[detempering]] of semaphore
* [[Semaphore-chromatic equivalence continuum]]
* [[Semaphore–chromatic equivalence continuum]]


[[Category:Temperaments]]
[[Category:Semaphore| ]] <!-- main article -->
[[Category:Semaphore| ]] <!-- main article -->
[[Category:Godzilla]] <!-- main article -->
[[Category:Godzilla]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Semaphoresmic clan]]
[[Category:Semaphoresmic clan]]
[[Category:Meantone family]]
[[Category:Meantone family]]
[[Category:Sensamagic clan]]
[[Category:Sensamagic clan]]

Latest revision as of 00:10, 23 May 2026

Semaphore; Godzilla
Subgroups 2.3.7, 2.3.5.7, 2.3.5.7.13
Comma basis 49/48 (2.3.7);
49/48, 81/80 (2.3.5.7);
49/48, 81/80, 91/90 (L7.13)
Reduced mapping ⟨1; 2 8 1 11]
ET join 5 & 19
Generators (CWE) ~7/4 = 947.8 ¢
MOS scales 4L 1s, 5L 4s, 5L 9s, 5L 14s
Ploidacot alpha-dicot
Pergen (P8, P4/2)
Color name Zozoti
Minimax error 9-odd-limit: 20.5 ¢;
2.3.5.7.13 15-odd-limit: 20.5 ¢
Target scale size 9-odd-limit: 9 notes;
2.3.5.7.13 15-odd-limit: 14 notes

Semaphore, of the semaphoresmic clan, is characterized by 49/48 being tempered out, so the generator represents 7/4 and 12/7 (or 8/7 and 7/6) equally. This results in a very low complexity 2.3.7-subgroup temperament, with the drawback that most intervals of 7 must be out of tune by at least half of the comma 49/48, or about 18 cents. Semaphore is a play on the words "semi-" and "fourth".

If the 5th harmonic's intervals are desired, 5/4 can be sensibly mapped to +8 generators by tempering out 81/80, making it a meantone temperament. This temperament is godzilla. Moreover, the generator can be taken to be 26/15, which maps 13/8 to +11 generators by tempering out 91/90 and 105/104. This extends the temperament to the 2.3.5.7.13 subgroup, with an abundance of harmonic resource and little additional damage.

A more accurate but complex mapping of 5 can be found in immunity, or 5/4 itself can be made the period by tempering out 128/125, resulting in triforce.

For technical information, see Semaphoresmic clan #Semaphore and #Godzilla. For a discussion on 11- and 13-limit extensions, see Godzilla extensions.

Interval chains

In the following tables, odd harmonics 1–13 and their inverses are in bold.

Semaphore

# Cents* Approximate ratios
0 0.0 1/1
1 950.7 7/4, 12/7
2 701.4 3/2
3 452.1 9/7, 21/16
4 202.8 9/8
5 1153.4 27/14, 63/32

* In 2.3.7-subgroup CWE tuning, octave reduced

Godzilla

# Cents* Approximate ratios
0 0.0 1/1
1 948.0 7/4, 12/7, 26/15
2 696.0 3/2
3 444.0 9/7, 13/10, 21/16
4 192.0 9/8, 10/9
5 1140.0 27/14, 39/20, 40/21, 52/27, 63/32
6 888.0 5/3
7 636.0 10/7, 13/9
8 384.0 5/4
9 132.0 13/12, 15/14
10 1080.0 13/7, 15/8
11 828.0 13/8
12 576.0 25/18, 39/28, 45/32
13 324.0 39/32
14 72.1 25/24, 50/49

* In 2.3.5.7.13-subgroup CWE tuning, octave reduced

Scales

Scala files:

5-note (proper)

Small ("minor") interval 202.8 452.1 701.4 950.7
JI intervals represented 9/8 9/7~13/10 3/2 7/4~12/7
Large ("major") interval 249.3 498.6 747.9 997.2
JI intervals represented 7/6~8/7 4/3 14/9~20/13 16/9

9-note (improper)

Main article: 5L 4s
Small ("minor") interval 60.0 252.0 312.0 504.0 564.0 756.0 816.0 1008.0
JI intervals represented 7/6~8/7 6/5 4/3 7/5~18/13 14/9~20/13 8/5~13/8 9/5~16/9
Large ("major") interval 192.0 384.0 444.0 636.0 696.0 888.0 948.0 1140.0
JI intervals represented 9/8~10/9 5/4 9/7~13/10 10/7~13/9 3/2 5/3 7/4~12/7

In 19edo, Godzilla[9] has steps 3 3 1 3 1 3 1 3 1, and contains the following useful scales as subsets:

  • Meantone pentic (5 3 5 3 3)
  • Altered diatonic I (3 4 3 1 3 4 1)
  • Altered diatonic II (3 4 3 1 4 3 1)
  • Altered diatonic III (4 3 3 1 4 3 1)
  • Altered diatonic IV (3 3 4 1 3 4 1)

It does not, however, contain the ordinary diatonic scale. Godzilla[9] thus expands on the pentic scale, but in a different way than diatonic scales do.

The four heptatonic subsets can be regarded as chromatic alterations of the diatonic scale, or alternatively as variants of Archytas' septimal diatonic scale, but with a greatly exaggerated difference between the two different whole tone sizes. All five of these subsets are very expressive melodically. Godzilla[9] combines all of these and is expressive in its own right; it could even be thought of as 19edo's answer to the well-loved Supra[7] diatonic scale of 17edo, as both are improper and made up of whole-tones and third-tones.

Like Supra[7], Godzilla[9] is well stocked with subminor and supermajor triads; in this case they can be viewed as 6:7:9 and 10:13:15 since 19edo is a biome temperament. Godzilla[9] has only one each of the more stable 5-limit major and minor triads, which might be considered a drawback, but could also be considered a strength for helping to establish a clearer tonal center (since all triads other than the tonic have tension in them).

Tunings

2.3.7-subgroup norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~7/4 = 952.2948 ¢ CWE: ~7/4 = 950.6890 ¢ POTE: ~7/4 = 949.6154 ¢
7-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~7/4 = 948.7959 ¢ CWE: ~7/4 = 947.8216 ¢ POTE: ~7/4 = 947.3650 ¢
2.3.5.7.13-subgroup norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~7/4 = 948.9311 ¢ CWE: ~7/4 = 948.0037 ¢ POTE: ~7/4 = 947.5708 ¢

Tuning spectrum

Edo
generator 		Unchanged interval
(eigenmonzo)* 		Generator (¢) Comments
7/6 933.129
7\9 933.333 9cff val
11\14 942.857 14cf val, lower bound of 7- and 9-odd-limit diamond monotone
9/7 945.028
7/5 945.355
13/7 947.170
15\19 947.368 Lower bound of no-11 13-odd-limit diamond monotone
No-11 15-odd-limit diamond monotone (singleton)
5/3 947.393
13/9 948.088
5/4 948.289 7-, 9-odd-limit, no-11 13- and 15-odd-limit minimax
13/12 948.730
13/8 949.139
19\24 950.000
3/2 950.978
13/10 951.405
4\5 960.000 Upper bound of 7-, 9-odd-limit, and no-11 13-odd-limit diamond monotone
7/4 968.826

* Besides the octave

Music

Cameron Bobro
Igliashon Jones
Roncevaux
Starshine

See also

Retrieved from "https://en.xen.wiki/index.php?title=Semaphore_and_godzilla&oldid=230775"