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5L 4s: Difference between revisions

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#: 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]].
#: 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]].
# [[Bug]], generated by semifourths sharper than 3\14 (257.14¢). This implies a "[[mavila]]" or superdiatonic fifth.
# [[Bug]], generated by semifourths sharper than 3\14 (257.14¢). This implies a "[[mavila]]" or superdiatonic fifth.
== Notation ==
This article uses [[diamond MOS notation]], with the convention JKLMNOPQR = LLSLSLSLS and pitch standard J = C4 = 261.6255653 Hz. The accidentals & and @ are used for raising and lowering by the chroma = L − S, respectively.
== 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 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.
{| class="wikitable right-2 right-3 right-4 right-5"
|-
!
! [[14edo]]
! [[19edo]]
! [[24edo]]
! [[29edo]]
|-
| generator (g)
| 3\14, 257.14
| 4\19, 252.63
| 5\24, 250.
| 6\29, 248.28
|-
| L (octave - 4g)
| 171.43
| 189.47
| 200.00
| 206.90
|-
| s (5g - octave)
| 85.71
| 63.16
| 50.00
| 41.38
|}
=== Bug ===
For convenience's sake, we view [[bug]] as any 5L 4s tuning where two [[semifourth]] generators make a ''superdiatonic'' ([[7L 2s]]) fourth, i.e. any tuning where the semifourth is between 3\14 (257.14¢) and 2\9 (266.67¢). [[23edo]]'s 5\23 (260.87¢) is an example of a bug generator.
The sizes of the generator, large step and small step of 5L 4s are as follows in various bug tunings.
{| class="wikitable right-2 right-3 right-4 right-5"
|-
!
! [[23edo]]
! [[32edo]]
! [[37edo]]
|-
| generator (g)
| 5\23, 252.63
| 7\32, 262.50
| 8\37, 259.46
|-
| L (octave - 4g)
| 156.52
| 150.00
| 162.16
|-
| s (5g - octave)
| 104.35
| 112.50
| 97.30
|}
== Intervals ==
== Modes == 
TODO: names
* LLsLsLsLs
* LsLLsLsLs
* LsLsLLsLs
* LsLsLsLLs
* LsLsLsLsL
* sLLsLsLsL
* sLsLLsLsL
* sLsLsLLsL
* sLsLsLsLL
One can think of 5L 4s modes as being built from two pentachords (division of the perfect fourth into four intervals) plus a whole tone. The possible pentachords are LsLs, sLLs, and sLsL.
== Chords ==
== Primodal theory ==
=== Nejis ===
==== 14nejis ====
# 95:100:105:110:116:122:128:135:141:148:156:164:172:180:190 (uses /19 prime family intervals while being pretty close to equal)
== Samples ==
[[File:Dream EP 14edo Sketch.mp3]]
''[[:File:Dream EP 14edo Sketch.mp3]]'' is a short swing ditty in [[14edo]] 5L 4s, in the 212121221 mode.
== Scale tree ==
== Scale tree ==
{| class="wikitable"
{| class="wikitable"
Line 657: Line 744:
| style="text-align:center;" |  
| style="text-align:center;" |  
|}
|}
== 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 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.
{| class="wikitable right-2 right-3 right-4 right-5"
|-
!
! [[14edo]]
! [[19edo]]
! [[24edo]]
! [[29edo]]
|-
| generator (g)
| 3\14, 257.14
| 4\19, 252.63
| 5\24, 250.
| 6\29, 248.28
|-
| L (octave - 4g)
| 171.43
| 189.47
| 200.00
| 206.90
|-
| s (5g - octave)
| 85.71
| 63.16
| 50.00
| 41.38
|}
=== Bug ===
For convenience's sake, we view [[bug]] as any 5L 4s tuning where two [[semifourth]] generators make a ''superdiatonic'' ([[7L 2s]]) fourth, i.e. any tuning where the semifourth is between 3\14 (257.14¢) and 2\9 (266.67¢). [[23edo]]'s 5\23 (260.87¢) is an example of a bug generator.
The sizes of the generator, large step and small step of 5L 4s are as follows in various bug tunings.
{| class="wikitable right-2 right-3 right-4 right-5"
|-
!
! [[23edo]]
! [[32edo]]
! [[37edo]]
|-
| generator (g)
| 5\23, 252.63
| 7\32, 262.50
| 8\37, 259.46
|-
| L (octave - 4g)
| 156.52
| 150.00
| 162.16
|-
| s (5g - octave)
| 104.35
| 112.50
| 97.30
|}
== Notation ==
This article uses [[diamond MOS notation]], with the convention JKLMNOPQR = LLSLSLSLS and pitch standard J = C4 = 261.6255653 Hz. The accidentals & and @ are used for raising and lowering by the chroma = L − S, respectively.
== Intervals ==
== Modes == 
TODO: names
* LLsLsLsLs
* LsLLsLsLs
* LsLsLLsLs
* LsLsLsLLs
* LsLsLsLsL
* sLLsLsLsL
* sLsLLsLsL
* sLsLsLLsL
* sLsLsLsLL
One can think of 5L 4s modes as being built from two pentachords (division of the perfect fourth into four intervals) plus a whole tone. The possible pentachords are LsLs, sLLs, and sLsL.
== Chords ==
== Primodal theory ==
=== Nejis ===
==== 14nejis ====
# 95:100:105:110:116:122:128:135:141:148:156:164:172:180:190 (uses /19 prime family intervals while being pretty close to equal)
== Samples ==
[[File:Dream EP 14edo Sketch.mp3]]
''[[:File:Dream EP 14edo Sketch.mp3]]'' is a short swing ditty in [[14edo]] 5L 4s, in the 212121221 mode.


[[Category:Scales]]
[[Category:Scales]]
[[Category:MOS scales]]
[[Category:MOS scales]]
[[Category:Abstract MOS patterns]]
[[Category:Abstract MOS patterns]]
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