5L 4s: Difference between revisions

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'''5L 4s''', or '''semiquartal''', refers to the structure of [[MOS]] scales with generators ranging from 1\5 (one degree of [[5edo]] = 240¢) to 2\9 (two degrees of [[9edo]] = 266.7¢). In the case of 9edo, L and s are the same size; in the case of 5edo, s becomes so small it disappears (and all that remains are the five equal L's).

The familiar harmonic entropy minimum with this MOS pattern is [[Meantone_family#Godzilla|godzilla]], in which a generator is [[8/7|8/7]] or [[7/6|7/6]] (tempered to be the same interval, or even 37/32 if you like) so two of them make a [[4/3|4/3]]. However, in addition to godzilla (tempering out 81/80) and the 2.3.7 temperament [[Chromatic_pairs#semaphore|semaphore]], there is also a weird scale called "[[Pseudo-semaphore|pseudo-semaphore]]", in which two different flavors of [[3/2|3/2]] exist in the same scale: an octave minus two generators makes a sharp 3/2, and two octaves minus seven generators makes a flat 3/2.

Semiquartal tunings can be divided into two major ranges:

~~There are not really "good" temperament interpretations for 5L 4s scales except for semaphore or godzilla, but 5L 4s~~ tunings can be divided into two major ranges:

# [[Semaphore]] generated by semifourths flatter than 3\14 (257.14¢). This implies a diatonic fifth.

#: The generator could be viewed as a 15/13, and the resulting "ultramajor" chords and "inframinor" triads could be viewed as approximating 10:13:15 and 26:30:39. See [[Arto and Tendo Theory]].

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| style="text-align:center;" | ~~Pseudo-semaphore is around here~~

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| style="text-align:center;" | ~~Semaphore is around here~~

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| style="text-align:center;" | ~~Godzilla is around here~~

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L/s = 3

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== Tuning ranges ==

=== Semaphore ===

We view [[semaphore]] as any 5L 4s tuning where two [[semifourth]] generators make a ''diatonic'' ([[5L 2s]]) fourth, i.e. any tuning where the semifourth is between 1\5 (240¢) or 3\14 (257.14¢). One important sub-range of semaphore is given by stipulating that two semifourth generators must make a ''meantone'' fourth; i.e. that four fifths should approximate a [[5/4]] major third. This ~~results in~~ [[~~godzilla~~]] ~~temperament~~, ~~which is supported by~~ [[~~19edo~~]] and [[~~24edo~~]].

We view [[semaphore]] as any 5L 4s tuning where two [[semifourth]] generators make a ''diatonic'' ([[5L 2s]]) fourth, i.e. any tuning where the semifourth is between 1\5 (240¢) or 3\14 (257.14¢). One important sub-range of semaphore is given by stipulating that two semifourth generators must make a ''meantone'' fourth; i.e. that four fifths should approximate a [[5/4]] major third. This is supported by [[19edo]] (4\19), [[24edo]] (5\24), [[43edo]] (9\43), and [[62edo]] (13\62).

The sizes of the generator, large step and small step of 5L 4s are as follows in various semaphore tunings.

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 '''5L 4s''', or '''semiquartal''', refers to the structure of [[MOS]] scales with generators ranging from 1\5 (one degree of [[5edo]] = 240¢) to 2\9 (two degrees of [[9edo]] = 266.7¢). In the case of 9edo, L and s are the same size; in the case of 5edo, s becomes so small it disappears (and all that remains are the five equal L's).
-The familiar harmonic entropy minimum with this MOS pattern is [[Meantone_family#Godzilla|godzilla]], in which a generator is [[8/7|8/7]] or [[7/6|7/6]] (tempered to be the same interval, or even 37/32 if you like) so two of them make a [[4/3|4/3]]. However, in addition to godzilla (tempering out 81/80) and the 2.3.7 temperament [[Chromatic_pairs#semaphore|semaphore]], there is also a weird scale called "[[Pseudo-semaphore|pseudo-semaphore]]", in which two different flavors of [[3/2|3/2]] exist in the same scale: an octave minus two generators makes a sharp 3/2, and two octaves minus seven generators makes a flat 3/2.
+Semiquartal tunings can be divided into two major ranges:
-There are not really "good" temperament interpretations for 5L 4s scales except for semaphore or godzilla, but 5L 4s tunings can be divided into two major ranges:
 # [[Semaphore]] generated by semifourths flatter than 3\14 (257.14¢). This implies a diatonic fifth.
 #: The generator could be viewed as a 15/13, and the resulting "ultramajor" chords and "inframinor" triads could be viewed as approximating 10:13:15 and 26:30:39. See [[Arto and Tendo Theory]].
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 | | 12\59
 | | 244.068
-| style="text-align:center;" | Pseudo-semaphore is around here
+| style="text-align:center;" |
 |-
 | |
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 | |
 | | 249.057
-| style="text-align:center;" | Semaphore is around here
+| style="text-align:center;" |
 |-
 | |
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 | |
 | | 252.632
-| style="text-align:center;" | Godzilla is around here
+| style="text-align:center;" |
 L/s = 3
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 == Tuning ranges ==
 === Semaphore ===
-We view [[semaphore]] as any 5L 4s tuning where two [[semifourth]] generators make a ''diatonic'' ([[5L 2s]]) fourth, i.e. any tuning where the semifourth is between 1\5 (240¢) or 3\14 (257.14¢). One important sub-range of semaphore is given by stipulating that two semifourth generators must make a ''meantone'' fourth; i.e. that four fifths should approximate a [[5/4]] major third. This results in [[godzilla]] temperament, which is supported by [[19edo]] and [[24edo]].
+We view [[semaphore]] as any 5L 4s tuning where two [[semifourth]] generators make a ''diatonic'' ([[5L 2s]]) fourth, i.e. any tuning where the semifourth is between 1\5 (240¢) or 3\14 (257.14¢). One important sub-range of semaphore is given by stipulating that two semifourth generators must make a ''meantone'' fourth; i.e. that four fifths should approximate a [[5/4]] major third. This is supported by [[19edo]] (4\19), [[24edo]] (5\24), [[43edo]] (9\43), and [[62edo]] (13\62).
 The sizes of the generator, large step and small step of 5L 4s are as follows in various semaphore tunings.