4L 5s (3/1-equivalent): Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
No edit summary
Line 21: Line 21:


== Notation ==
== Notation ==
Bohlen-Pierce theory possesses a well-established<ref>[https://en.wikipedia.org/wiki/Bohlen%E2%80%93Pierce_scale#Intervals_and_scale_diagrams]</ref> [[nonoctave]] notation system for [[EDT]]s and no-twos music, which is based on this MOS scale as generated by approximately [[7/3]], relating it to [[Bohlen-Pierce-Stearns|BPS]] temperament, where two 7/3 generators are equated to 27/5. The preferred generator for any edt is its patent val approximation of 7/3.
Bohlen-Pierce theory possesses a well-established [[nonoctave]] notation system for [[EDT]]s and no-twos music, which is based on this MOS scale as generated by approximately [[7/3]], relating it to [[Bohlen-Pierce-Stearns|BPS]] temperament, where two 7/3 generators are equated to 27/5. The preferred generator for any edt is its patent val approximation of 7/3.


This notation uses 9 nominals: E F G H J A B C D E as presented in the E Cassiopeian (sLsLsLsLs) mode, and represented by a circle of generators going as follows: ...C# - A# - H# - F# - D - B - J - G - E - C - A - H - F - Db - Bb - Jb - Gb...
This notation uses 9 nominals: for compatibility with [[diamond-MOS notation]], the current recommendation is to use the notes J K L M N O P Q R J as presented in the J Cassiopeian (symmetric, sLsLsLsLs) mode, and represented by a circle of generators going as follows: ...Q# - O# - M# - K# - R - P - N - L - J - Q - O - M - K - Rb - Pb - Nb - Lb... However, an alternative convention (as used on Wikipedia <ref>[https://en.wikipedia.org/wiki/Bohlen%E2%80%93Pierce_scale#Intervals_and_scale_diagrams]</ref> and certain articles of this wiki) labels them C D E F G H J A B C in the C Andromedan (LssLsLsLs) mode, which rotates to the E Cassiopeian mode.
An extension of [[ups and downs notation]], in the obvious way, can be found at [[Lambda ups and downs notation]].


=== Examples ===
=== Examples ===
[[9edt]] (equalized):
[[9edt]] (equalized):
{|class="wikitable article-table"
{|Qlass="wikitable article-table"
|-
|-
!0
!0
Line 40: Line 41:
!9
!9
|-
|-
|E
|F
|G
|H
|J
|J
|A
|K
|B
|L
|C
|M
|D
|N
|E
|O
|P
|Q
|R
|J
|-
|-
|P0
|P0
Line 65: Line 66:
[[13edt]] (Bohlen-Pierce):
[[13edt]] (Bohlen-Pierce):


{|class="wikitable article-table"
{|Qlass="wikitable article-table"
|-
|-
!0
!0
Line 82: Line 83:
!13
!13
|-
|-
|E
|F
|F#, Gb
|G
|H
|H#, Jb
|J
|J
|A
|K
|A#, Bb
|K#, Gb
|B
|L
|C
|M
|Db, C#
|M#, Jb
|D
|N
|E
|O
|O#, Bb
|P
|Q
|Rb, C#
|R
|J
|-
|-
|P0
|P0
Line 115: Line 116:
[[30edt]]:
[[30edt]]:


{|class="wikitable article-table"
{|Qlass="wikitable article-table"
|-
|-
!0
!0
Line 149: Line 150:
!30
!30
|-
|-
|E
|J
|D#, Gbb
|R#, Gbb
|F
|K
|E#
|J#
|Gb
|Lb
|F#
|K#
|Hb
|Mb
|G
|L
|Fx, Jbb
|Kx, Jbb
|H
|M
|G#
|L#
|Nb
|M#
|Ob
|N
|Mx, Bbb
|O
|N#
|Pb
|O#
|Qb
|P
|Ox, Dbb
|Q
|P#
|Rb
|Q#
|Jb
|Jb
|H#
|R
|Ab
|Kb, Cx
|J
|J
|Hx, Bbb
|A
|J#
|Bb
|A#
|Cb
|B
|Ax, Dbb
|C
|B#
|Db
|C#
|Eb
|D
|Fb, Cx
|E
|-
|-
|P0
|P0

Revision as of 09:28, 23 August 2024

↖ 3L 4s⟨3/1⟩ ↑ 4L 4s⟨3/1⟩ 5L 4s⟨3/1⟩ ↗
← 3L 5s⟨3/1⟩ 4L 5s (3/1-equivalent) 5L 5s⟨3/1⟩ →
↙ 3L 6s⟨3/1⟩ ↓ 4L 6s⟨3/1⟩ 5L 6s⟨3/1⟩ ↘ 
┌╥┬╥┬╥┬╥┬┬┐
│║│║│║│║│││
│││││││││││
└┴┴┴┴┴┴┴┴┴┘
Scale structure
Step pattern LsLsLsLss
ssLsLsLsL
Equave 3/1 (1902.0 ¢)
Period 3/1 (1902.0 ¢)
Generator size(edt)
Bright 2\9 to 1\4 (422.7 ¢ to 475.5 ¢)
Dark 3\4 to 7\9 (1426.5 ¢ to 1479.3 ¢)
Related MOS scales
Parent 4L 1s⟨3/1⟩
Sister 5L 4s⟨3/1⟩
Daughters 9L 4s⟨3/1⟩, 4L 9s⟨3/1⟩
Neutralized 8L 1s⟨3/1⟩
2-Flought 13L 5s⟨3/1⟩, 4L 14s⟨3/1⟩
Equal tunings(edt)
Equalized (L:s = 1:1) 2\9 (422.7 ¢)
Supersoft (L:s = 4:3) 7\31 (429.5 ¢)
Soft (L:s = 3:2) 5\22 (432.3 ¢)
Semisoft (L:s = 5:3) 8\35 (434.7 ¢)
Basic (L:s = 2:1) 3\13 (438.9 ¢)
Semihard (L:s = 5:2) 7\30 (443.8 ¢)
Hard (L:s = 3:1) 4\17 (447.5 ¢)
Superhard (L:s = 4:1) 5\21 (452.8 ¢)
Collapsed (L:s = 1:0) 1\4 (475.5 ¢)

4L 5s⟨3/1⟩, also called Lambda, is a 3/1-equivalent (tritave-equivalent) moment of symmetry scale containing 4 large steps and 5 small steps, repeating every interval of 3/1 (1902.0 ¢). Generators that produce this scale range from 422.7 ¢ to 475.5 ¢, or from 1426.5 ¢ to 1479.3 ¢. Suggested for use as the analog of the diatonic scale when playing Bohlen-Pierce is this 9-note Lambda scale, which is the 4L 5s mos with equave 3/1. This can be thought of as a mos generated by a 3.5.7-subgroup rank-2 temperament called BPS (Bohlen-Pierce-Stearns) that eliminates only the comma 245/243, so that (9/7)2 is equated with 5/3. This is a very good temperament on the 3.5.7 subgroup, and additionally is supported by many edt's (and even edos!) besides 13edt.

Some low-numbered edos that support BPS are 19, 22, 27, 41, and 46, and some low-numbered edts that support it are 9, 13, 17, and 30, all of which make it possible to play BP music to some reasonable extent. These equal temperaments contain not only the Lambda "BP diatonic" scale, but, with the exception 9edt, also the 13-note "BP chromatic" mos scale, or BPS[13], which can be thought of as a "detempered" version of the 13edt Bohlen-Pierce scale. This scale may be a suitable melodic substitute for the "BP chromatic" scale, and is basically the same as how 19edo and 31edo do not contain 12edo as a subset, but they do contain the meantone[12] chromatic scale.

When playing this temperament in some edo, it may be desired to stretch/compress the tuning so that the tritave is pure, rather than the octave being pure - or in general, to minimize the error on the 3.5.7 subgroup while ignoring the error on 2/1.

One can add the octave to BPS temperament by simply creating a new mapping for 2/1. A simple way to do so is to map the 2/1 to +7 of the ~9/7 generators, minus a single tritave. This is sensi temperament, in essence treating it as a "3.5.7.2-subgroup extension" of the original 3.5.7-subgroup BPS temperament.

Modes

Scale degrees of the modes of 4L 5s⟨3/1⟩
UDP Cyclic
order 		Step
pattern 		Scale degree (mosdegree)
0 1 2 3 4 5 6 7 8 9
8|0 1 LsLsLsLss Perf. Maj. Perf. Maj. Maj. Maj. Maj. Aug. Maj. Perf.
7|1 3 LsLsLssLs Perf. Maj. Perf. Maj. Maj. Maj. Maj. Perf. Maj. Perf.
6|2 5 LsLssLsLs Perf. Maj. Perf. Maj. Maj. Min. Maj. Perf. Maj. Perf.
5|3 7 LssLsLsLs Perf. Maj. Perf. Min. Maj. Min. Maj. Perf. Maj. Perf.
4|4 9 sLsLsLsLs Perf. Min. Perf. Min. Maj. Min. Maj. Perf. Maj. Perf.
3|5 2 sLsLsLssL Perf. Min. Perf. Min. Maj. Min. Maj. Perf. Min. Perf.
2|6 4 sLsLssLsL Perf. Min. Perf. Min. Maj. Min. Min. Perf. Min. Perf.
1|7 6 sLssLsLsL Perf. Min. Perf. Min. Min. Min. Min. Perf. Min. Perf.
0|8 8 ssLsLsLsL Perf. Min. Dim. Min. Min. Min. Min. Perf. Min. Perf.

Proposed names

Lériendil proposes mode names derived from the constellations of the northern sky.

Modes of 4L 5s
UDP Cyclic
order		 Step
pattern
8|0 1 LsLsLsLss
7|1 3 LsLsLssLs
6|2 5 LsLssLsLs
5|3 7 LssLsLsLs
4|4 9 sLsLsLsLs
3|5 2 sLsLsLssL
2|6 4 sLsLssLsL
1|7 6 sLssLsLsL
0|8 8 ssLsLsLsL

Notation

Bohlen-Pierce theory possesses a well-established nonoctave notation system for EDTs and no-twos music, which is based on this MOS scale as generated by approximately 7/3, relating it to BPS temperament, where two 7/3 generators are equated to 27/5. The preferred generator for any edt is its patent val approximation of 7/3.

This notation uses 9 nominals: for compatibility with diamond-MOS notation, the current recommendation is to use the notes J K L M N O P Q R J as presented in the J Cassiopeian (symmetric, sLsLsLsLs) mode, and represented by a circle of generators going as follows: ...Q# - O# - M# - K# - R - P - N - L - J - Q - O - M - K - Rb - Pb - Nb - Lb... However, an alternative convention (as used on Wikipedia [1] and certain articles of this wiki) labels them C D E F G H J A B C in the C Andromedan (LssLsLsLs) mode, which rotates to the E Cassiopeian mode. An extension of ups and downs notation, in the obvious way, can be found at Lambda ups and downs notation.

Examples

9edt (equalized):

0 1 2 3 4 5 6 7 8 9
J K L M N O P Q R J
P0 P1 P2 P3 P4 P5 P6 P7 P8 P9

13edt (Bohlen-Pierce):

0 1 2 3 4 5 6 7 8 9 10 11 12 13
J K K#, Gb L M M#, Jb N O O#, Bb P Q Rb, C# R J
P0 m1 M1, d2 P2 m3 M3, m4 M4 m5 M5, m6 M6 P7 m8, A7 M8 P9

30edt:

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
J R#, Gbb K J# Lb K# Mb L Kx, Jbb M L# Nb M# Ob N Mx, Bbb O N# Pb O# Qb P Ox, Dbb Q P# Rb Q# Jb R Kb, Cx J
P0 MM8, dd2 m1 A0 d2 M1 mm3 P2 MM1, mm4 m3 A2 m4 M3 mm5 M4 MM3, mm6 m5 MM4 m6 M5 d7 M6 MM5, mm8 P7 MM6 m8 A7 d9 M8 mm1, AA7 P9

List of edts supporting the Lambda scale

Below is a list of equal temperaments which contain a 4L 5s scale using generators between 422.7 cents and 475.5 cents.

Template:Scale tree

Analogously to how the diatonic scale equalizes approaching 7edo and its small steps collapse to 0 in 5edo, this scale equalizes approaching 9edt and its small steps collapse in 4edt; therefore, temperaments setting the 7/3 generator to precisely 7\9edt and to precisely 3\4edt are analogs of whitewood and blackwood respectively; however, unlike for the diatonic scale, the just point is not close to the center of the tuning range, but approximately 1/4 of the way between 9edt and 4edt, being closely approximated by 37\48edt and extremely closely approximated by 118\153edt.

  1. ↑ [1]
Retrieved from "https://en.xen.wiki/index.php?title=4L_5s_(3/1-equivalent)&oldid=151739"