5L 5s: Difference between revisions

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{{Infobox MOS
{{Infobox MOS
| Name = blackwood, pentasymmetric
| Name = pentawood
| Periods = 5
| Periods = 5
| nLargeSteps = 5
| nLargeSteps = 5
| nSmallSteps = 5
| nSmallSteps = 5
| Equalized = 1
| Equalized = 1
| Paucitonic = 0
| Collapsed = 0
| Pattern = LsLsLsLsLs
| Pattern = LsLsLsLsLs
}}
}}
'''5L 5s''', '''blackwood''' or '''pentasymmetric''' refers to the structure of octave-equivalent [[MOS]] scales with period 1\5 (one degree of [[5edo]] = 240¢) and generators ranging from 1\10 (one degree of [[10edo]] = 120¢) to 1\5 (240¢). In the case of 10edo, 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).
{{MOS intro}}


The true MOS, LsLsLsLsLs, is always proper because there is only one small step per period, but because there are 5 periods in an octave, there are a wealth of near-MOSes in which multiples of the period (that is, intervals of an even number of steps) are the only generic intervals that come in more than two different flavors. Specifically, there are 6 others: LLssLsLsLs, LLssLLssLs, LLsLssLsLs, LLsLssLLss, LLsLsLssLs, LLsLsLsLss. [[15edo]] is right on the boundary of being [[Rothenberg_propriety|proper]].
There is only one significant [[harmonic entropy]] minimum with this MOS pattern: [[limmic temperaments#5-limit_.28blackwood.29|blackwood]], in which intervals of the prime numbers 3 and 7 are all represented using steps of [[5edo|5edo]], and the generator reaches intervals of 5 like 6/5, 5/4, or 7/5.
 
In addition to the true MOS form (LsLsLsLsLs and sLsLsLsLsL), there are 6 near-MOS forms – LLssLsLsLs, LLssLLssLs, LLsLssLsLs, LLsLssLLss, LLsLsLssLs, LLsLsLsLss – in which the period and its multiples (intervals of 2, 4, 6, and 8 mossteps) have more than two varieties. These forms are proper if the bright generator is less than 160¢.


Some important properties of Blackwood are:
* Blackwood[10] has the most 5-odd-limit consonant triads it is possible to have in a 10-note 5-limit scale.
* Because it is a 10-note scale with a period of 1/5 of an octave, any arbitrary harmony will occur either 5 or 10 times within the 10-note scale, and for otonal harmonies consisting of three or more notes, the utonal counterpart of the harmony will also occur either 5 or 10 times within the scale; this is a property that is only held by other scales with 5 periods per octave.
* Blackwood[10] is also a "mode of limited transposition" like the Diminished and Augmented scales in 12edo: since the scale is built by applying the generator only a single time within each period, the scale has only two modes.
== Intervals ==
== Intervals ==
{{MOS intervals}}


{| class="wikitable"
==Modes==
|-
{{MOS mode degrees}}
| | Step of 15edo
| | Cent Value
| | Interval Class
| | Guitar Notation
| | Decimal Notation
| | Approximated Ratios
| | Pseudo-Diatonic Category
|-
| | 0
| | 0
| | Unison
| | E
| | 1
| | 1/1
| | Unison
|-
| | 1
| | 80
| | Minor 2nd, Augmented Unison*
| | E#, Gbb
| | 2b, 1#
| | 16/15, 21/20, 22/21, 25/24
| | Minor 2nd
|-
| | 2
| | 160
| | Major 2nd, Diminished 3rd
| | Gb, Ex
| | 2, 3b
| | 10/9, 11/10, 12/11, 15/14
| | Flat Major 2nd
|-
| | 3
| | 240
| | Perfect 3rd, Augmented 2nd, Diminished 4th
| | G
| | 3, 2#, 4bb
| | 7/6, 8/7, 9/8
| | Major 2nd/Subminor 3rd
|-
| | 4
| | 320
| | Minor 4th, Augmented 3rd
| | G#, Abb
| | 4b, 3#
| | 6/5, 11/9
| | Minor 3rd
|-
| | 5
| | 400
| | Major 4th, Diminished 5th
| | Ab, Gx
| | 4, 5b
| | 5/4, 14/11,
| | Major 3rd
|-
| | 6
| | 480
| | Perfect 5th, Augmented 4th, Diminished 6th
| | A
| | 5, 4#, 6bb
| | 4/3, 21/16, 9/7
| | Perfect Fourth
|-
| | 7
| | 560
| | Minor 6th, Augmented 5th
| | A#, Bbb
| | 6b, 5#
| | 7/5, 11/8,
| | Augmented Fourth
|-
| | 8
| | 640
| | Major 6th, Diminished 7th
| | Bb, Ax
| | 6, 7b
| | 10/7, 16/11
| | Diminished 5th
|-
| | 9
| | 720
| | Perfect 7th, Augmented 6th, Diminished 8th
| | B
| | 7, 6#, 8bb
| | 3/2, 32/21, 14/9
| | Perfect Fifth
|-
| | 10
| | 800
| | Minor 8th, Augmented 7th
| | B#, Dbb
| | 8b, 7#
| | 8/5, 11/7,
| | Minor 6th
|-
| | 11
| | 880
| | Major 8th, Diminished 9th
| | Db, Bx
| | 8, 9b
| | 5/3, 18/11
| | Major 6th
|-
| | 12
| | 960
| | Perfect 9th, Augmented 8th, Diminished 10th
| | D
| | 9, 8#, 0bb
| | 12/7, 7/4, 16/9
| | Minor 7th/Supermajor 6th
|-
| | 13
| | 1040
| | Minor 10th, Augmented 9th
| | D#, Ebb
| | 0b, 9#
| | 9/5, 20/11, 11/6, 28/15
| | Sharp Minor 7th
|-
| | 14
| | 1120
| | Major 10th, Diminished Undecave
| | Eb, Dx
| | 0, 1b
| | 15/8, 40/21, 48/25
| | Major 7th
|-
| | 15
| | 1200
| | Undecave ("Octave")
| | E
| | 1
| | 2/1
| | Octave
|}
*Augmented and diminished intervals do not occur in the 10-note MOS scale, but can occur in chromatically-altered MODMOSs.
 
== Chords ==
 
=== Basic Functional Chords ===
All of the familiar triads and tetrads of the diatonic scale are found plentifully in Blackwood[10], which is pretty obvious when you just look at the notes available in the major and minor modes:
 
{| class="wikitable"
|-
| |
| | 1st
| | 2nd
| | 3rd
| | 4th
| | 5th
| | 6th
| | 7th
| | 8th
| | 9th
| | 10th
| | 11-ave
|-
| | Major Mode (cents)
| | 0
| | 160
| | 240
| | 400
| | 480
| | 640
| | 720
| | 880
| | 960
| | 1120
| | 1200
|-
| | Minor Mode (cents)
| | 0
| | 80
| | 240
| | 320
| | 480
| | 560
| | 720
| | 800
| | 960
| | 1040
| | 1200
|}
Looking at this table, one can see approximations to all sorts of functional chords; if it's not immediately obvious, I'll spell it out in the following tables:
 
{| class="wikitable"
|-
| | Diatonic Chord Name
| | Decatonic Name
 
(if different)
| | Tuning (cents)
| | Spelling 1
| | Spelling 2
| | Degrees of Major Mode Found On:
| | Degrees of Minor Mode Found On:
|-
| | Major Triad
| | Same
| | 0-400-720
| | E-Ab-B
| | 1-4-7
| | Odd
| | Even
|-
| | Minor Triad
| | Same
| | 0-320-720
| | E-G#-B
| | 1-4b-7
| | Even
| | Odd
|-
| | Diminished
| | Same
| | 0-320-560
| | E-G#-A#
| | 1-4b-6b
| | Even
| | Odd
|-
| | Sus2
| | Sus3
| | 0-240-720
| | E-G-B
| | 1-3-7
| | All
| | All
|-
| | Sus4
| | Sus5
| | 0-480-720
| | E-A-B
| | 1-5-7
| | All
| | All
|-
| | Major 7th (maj7)
| | Major 10th
| | 0-400-720-1120
| | E-Ab-B-Eb
| | 1-4-7-0
| | Odd
| | Even
|-
| | Minor 7th (min7)
| | Minor 10th
| | 0-320-720-1040
| | E-G#-B-D#
| | 1-4b-7-0b
| | Even
| | Odd
|-
| | Dominant 7th (7)
| | Major 9th
| | 0-400-720-960
| | E-Ab-B-D
| | 1-4-7-9
| | Odd
| | Even
|-
| | Half-Diminished 7th (m7b5)
| | Diminished 10th
| | 0-320-560-1040
| | E-G#-A#-D#
| | 1-4b-6b-0b
| | Even
| | Odd
|-
| | Diminished 7th
| | Diminished 9th
| | 0-320-560-960
| | E-G#-A#-D
| | 1-4b-6b-9
| | Even
| | Odd
|}
 
=== Additional Functional Chords ===
 
{| class="wikitable"
|-
| | Diatonic Chord Name
| | Decatonic Name
 
(if different)
| | Tuning (cents)
| | Spelling 1
| | Spelling 2
| | Degrees of Major:
| | Degrees of Minor Mode:
|-
| | Major 6th (M6)
| | Major 8th
| | 0-400-720-880
| | E-Ab-B-Db
| | 1-4-7-8
| | Odd
| | Even
|-
| | Minor-Major 6th (m6)
| | Minor 9th
| | 0-320-720-960
| | E-G#-B-D
| | 1-4b-7-9
| | Even
| | Odd
|-
| | Diminished(bb3) (Dim(bb3))
| | Sus3-Maj6
| | 0-240-640
| | E-G-Bb
| | 1-3-6
| | Odd
| | Even
|-
| | Double-Diminished (Dim(bb3)(bb5))
| | Sus3-Min6
| | 0-240-560
| | E-G-A#
| | 1-3-6b
| | Even
| | Odd
|-
| | Major-Diminished (Maj(b5))
| | Major-Sus6
| | 0-400-640
| | E-Ab-Bb
| | 1-4-6
| | Odd
| | Even
|}
 
=== Approximate JI Chords ===
The dominant 7th and minor-major 6th are both 7-limit chords (4:5:6:7 and 1/(4:5:6:7), respectively). The diminished triad also approximates 5:6:7. There are no full otonal 11-limit hexads in the 10-note scale, but there are lots of smaller 11-limit chords (otonal and utonal) approximated:
 
{| class="wikitable"
|-
| | Otonal Harmonics
| | Utonal Harmonics
| | Tuning (cents)
| | Spelling 1
| | Spelling 2
| | Degrees of Major:
| | Degrees of Minor:
|-
| | 5:7:9
| | 1/(6:8:11)
| | 0-560-1040
| |
| |
| |
| |
|-
| | 6:7:11
| | 1/(5:8:9)
| | 0-240-1040
| |
| |
| |
| |
|-
| | 6:7:9:11
| |
| | 0-240-720-1040
| |
| |
| |
| |
|-
| | 6:8:11
| | 1/(5:7:9)
| | 0-480-1040
| |
| |
| |
| |
|-
| | 7:8:9, 6:7:8
| | 1/(7:8:9), 1/(6:7:8)
| | 0-240-480
| |
| |
| |
| |
|-
| | 7:10:12
| | 1/(9:11:16)
| | 0-640-960
| |
| |
| |
| |
|-
| | 8:9:11
| | 1/(5:6:7)
| | 0-240-560
| |
| |
| |
| |
|-
| | 8:9:11:14
| |
| | 0-240-560-960
| |
| |
| |
| |
|-
| | 8:11:14
| |
| | 0-560-960
| |
| |
| |
| |
|-
| | 9:10:12:14
| |
| | 0-160-480-720
| |
| |
| |
| |
|-
| | 9:11:16
| |
| | 0-320-960
| |
| |
| |
| |
|-
| | 11:14:16
| |
| | 0-400-640
| |
| |
| |
| |
|-
| | 11:14:16:18
| |
| | 0-400-640-880
| |
| |
| |
| |
|}
 
== Diatonic Modal Harmony in Blackwood[10] ==
Because 15edo is not a meantone temperament, and thus does not temper out the syntonic comma of 81/80, the usual 5L2s diatonic scale is not available. In fact, in 15edo the syntonic comma, which is normally only 21.51 cents, is tuned quite wide: it is mapped to one step of 15edo, and is thus 80 cents! However, one can approximate the diatonic scale (or rather, approximate the various untempered 5-limit JI versions of it) using 3 step-sizes—a large whole tone of 240 cents representing 9/8, a small whole tone of 160 cents representing 10/9, and a semitone of 80 cents representing 16/15. Since these non-MOS diatonic scales do not temper out the syntonic comma, they will only have at most five consonant 5-limit triads (unless an 8th note is included in the right place). They may have even fewer, depending on how the steps are permuted (for instance, if the step-pattern 240-240-80-240-160-160-80 is used, only two consonant 5-limit triads are available).
 
However, if one insists on using only the versions of the diatonic scale that have the maximum number of consonant triads available, then it turns out all of these scales will be 7-note subsets of Blackwood[10]. They will also be the most compact arrangement of those five consonant triads possible on the 5-limit triangular lattice, which is just a fancy way of saying those five chords will be maximally connected to each other by common tones. This suggests that one can approach melody in Blackwood[10] by treating it not as one 10-note scale, but as several related 7-note scales, each of which functions like a 5-limit untempered version of the diatonic scale.
 
This approach allows one to apply the usual principles of diatonic tonality and modality, with the caveat that each familiar mode of the diatonic scale will come in two flavors, depending which of the 7 notes one wants to build consonant triads on. The two flavors will share six notes in common, but one of the seven will differ by 81/80 (i.e. one step of 15edo). Interesting relationships do arise if one maintains the tonic but switches through its different modes (i.e. 1 mixolydian to 1 ionian to 1 lydian), and an "extra" mode appears, because five fifths close (hence V/V/V/V = IV, unlike in meantone where V/V/V/V = iii). All together, there are 14 modes of the 7-note diatonic to be found in Blackwood[10] if we keep the same tonic, and 20 if we allow alterations of the tonic.
 
== MODMOSes ==
 
=== Single-alteration MODMOSes ===
<span style="line-height: 1.5;">0-160-240-400-480-</span><span style="color: #0000ff; line-height: 1.5;">560</span><span style="line-height: 1.5;">-720-880-960-1120-1200</span>
 
0-80-240-320-480-560-720-800-960-<span style="color: #ff0000;">1120</span>-1200
 
=== Double-alteration MODMOSes ===
<span style="line-height: 1.5;">0-160-240-400-480-</span><span style="color: #0000ff; line-height: 1.5;">560</span><span style="line-height: 1.5;">-720-</span><span style="color: #0000ff; line-height: 1.5;">800</span><span style="line-height: 1.5;">-960-1120-1200</span>
 
0-<span style="color: #ff0000;">160</span>-240-320-480-560-720-800-960-<span style="color: #ff0000; line-height: 1.5;">1120</span><span style="line-height: 1.5;">-1200</span>
 
0-160-240-<span style="color: #0000ff;">320</span>-480-640-720-<span style="color: #0000ff;">800</span>-960-1120-1200
 
0-<span style="color: #ff0000;">160</span>-240-320-480-<span style="color: #ff0000;">640</span>-720-800-960-1040-1200


== Scale tree ==
== Scale tree ==
{{MOS tuning spectrum
| 6/5 = Qintosec&nbsp;↑
| 7/5 = Warlock
| 13/8 = Unnamed golden tuning
| 7/4 = Quinkee
| 2/1 = Blacksmith is optimal around here
| 9/4 = Trisedodge
| 13/5 = Unnamed golden tuning
| 6/1 = Cloudtone&nbsp;↓
}}


{| class="wikitable"
[[Category:Pentawood| ]]
|-
[[Category:10-tone scales]]
! colspan="5" | Generator
<!-- main article -->
! | Cents
! | Comments
|-
| | 0\5
| |
| |
| |
| |
| | 0
| style="text-align:center;" |
|-
| |
| |
| |
| |
| | 1\30
| | 40
| |
|-
| |
| |
| |
| | 1\25
| |
| | 48
| |
|-
| |
| |
| | 1\20
| |
| |
| | 60
| style="text-align:center;" |
|-
| |
| |
| |
| | 2\35
| |
| | 68.57
| |
|-
| |
| |
| |
| |
| | 3\50
| | 72
| |
|-
| |
| | 1\15
| |
| |
| |
| | 80
| style="text-align:center;" | 5-limit Blackwood is around here
 
Optimum rank range (L/s=2/1) for MOS
|-
| |
| |
| |
| | 3\40
| |
| | 90
| style="text-align:center;" |
|-
| |
| |
| |
| |
| | 5\65
| | 92.31
| style="text-align:center;" | Golden blackwood
|-
| |
| |
| | 2\25
| |
| |
| | 96
| style="text-align:center;" |
|-
|
|
|
|3\35
|
|102.86
|
|-
|
|
|
|
|4\45
|103.33
|
|-
| | 1\10
| |
| |
| |
| |
| | 120
| style="text-align:center;" |
|}
 
== Music ==
[http://micro.soonlabel.com/gene_ward_smith/Others/Roncevaux/Pocahontas_by_Roncevaux_on_SoundCloud___Hear_the_world_s_sounds.mp3 Pocahontas] by [https://soundcloud.com/lois-lancaster/pocahontas Roncevaux (Löis Lancaster)]; Blackwood[10] in 15edo
 
[[Category:Scales]]
[[Category:MOS scales]]
[[Category:Abstract MOS patterns]]

Latest revision as of 13:19, 13 March 2025

↖ 4L 4s ↑ 5L 4s 6L 4s ↗
← 4L 5s 5L 5s 6L 5s →
↙ 4L 6s ↓ 5L 6s 6L 6s ↘ 
┌╥┬╥┬╥┬╥┬╥┬┐
│║│║│║│║│║││
││││││││││││
└┴┴┴┴┴┴┴┴┴┴┘
Scale structure
Step pattern LsLsLsLsLs
sLsLsLsLsL
Equave 2/1 (1200.0 ¢)
Period 1\5 (240.0 ¢)
Generator size
Bright 1\10 to 1\5 (120.0 ¢ to 240.0 ¢)
Dark 0\5 to 1\10 (0.0 ¢ to 120.0 ¢)
TAMNAMS information
Name pentawood
Prefix pentawd-
Abbrev. pw
Related MOS scales
Parent none
Sister 5L 5s (self)
Daughters 10L 5s, 5L 10s
Neutralized 10edo
2-Flought 15L 5s, 5L 15s
Equal tunings
Equalized (L:s = 1:1) 1\10 (120.0 ¢)
Supersoft (L:s = 4:3) 4\35 (137.1 ¢)
Soft (L:s = 3:2) 3\25 (144.0 ¢)
Semisoft (L:s = 5:3) 5\40 (150.0 ¢)
Basic (L:s = 2:1) 2\15 (160.0 ¢)
Semihard (L:s = 5:2) 5\35 (171.4 ¢)
Hard (L:s = 3:1) 3\20 (180.0 ¢)
Superhard (L:s = 4:1) 4\25 (192.0 ¢)
Collapsed (L:s = 1:0) 1\5 (240.0 ¢)

5L 5s, named pentawood in TAMNAMS, is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 5 large steps and 5 small steps, with a period of 1 large step and 1 small step that repeats every 240.0 ¢, or 5 times every octave. Generators that produce this scale range from 120 ¢ to 240 ¢, or from 0 ¢ to 120 ¢. Scales of the true MOS form, where every period is the same, are proper because there is only one small step per period.

There is only one significant harmonic entropy minimum with this MOS pattern: blackwood, in which intervals of the prime numbers 3 and 7 are all represented using steps of 5edo, and the generator reaches intervals of 5 like 6/5, 5/4, or 7/5.

In addition to the true MOS form (LsLsLsLsLs and sLsLsLsLsL), there are 6 near-MOS forms – LLssLsLsLs, LLssLLssLs, LLsLssLsLs, LLsLssLLss, LLsLsLssLs, LLsLsLsLss – in which the period and its multiples (intervals of 2, 4, 6, and 8 mossteps) have more than two varieties. These forms are proper if the bright generator is less than 160¢.

Intervals

Intervals of 5L 5s
Intervals Steps
subtended 		Range in cents
Generic Specific Abbrev.
0-pentawdstep Perfect 0-pentawdstep P0pws 0 0.0 ¢
1-pentawdstep Minor 1-pentawdstep m1pws s 0.0 ¢ to 120.0 ¢
Major 1-pentawdstep M1pws L 120.0 ¢ to 240.0 ¢
2-pentawdstep Perfect 2-pentawdstep P2pws L + s 240.0 ¢
3-pentawdstep Minor 3-pentawdstep m3pws L + 2s 240.0 ¢ to 360.0 ¢
Major 3-pentawdstep M3pws 2L + s 360.0 ¢ to 480.0 ¢
4-pentawdstep Perfect 4-pentawdstep P4pws 2L + 2s 480.0 ¢
5-pentawdstep Minor 5-pentawdstep m5pws 2L + 3s 480.0 ¢ to 600.0 ¢
Major 5-pentawdstep M5pws 3L + 2s 600.0 ¢ to 720.0 ¢
6-pentawdstep Perfect 6-pentawdstep P6pws 3L + 3s 720.0 ¢
7-pentawdstep Minor 7-pentawdstep m7pws 3L + 4s 720.0 ¢ to 840.0 ¢
Major 7-pentawdstep M7pws 4L + 3s 840.0 ¢ to 960.0 ¢
8-pentawdstep Perfect 8-pentawdstep P8pws 4L + 4s 960.0 ¢
9-pentawdstep Minor 9-pentawdstep m9pws 4L + 5s 960.0 ¢ to 1080.0 ¢
Major 9-pentawdstep M9pws 5L + 4s 1080.0 ¢ to 1200.0 ¢
10-pentawdstep Perfect 10-pentawdstep P10pws 5L + 5s 1200.0 ¢

Modes

Scale degrees of the modes of 5L 5s
UDP Cyclic
order 		Step
pattern 		Scale degree (pentawddegree)
0 1 2 3 4 5 6 7 8 9 10
5|0(5) 1 LsLsLsLsLs Perf. Maj. Perf. Maj. Perf. Maj. Perf. Maj. Perf. Maj. Perf.
0|5(5) 2 sLsLsLsLsL Perf. Min. Perf. Min. Perf. Min. Perf. Min. Perf. Min. Perf.

Scale tree

Scale tree and tuning spectrum of 5L 5s
Generator(edo) Cents Step ratio Comments(always proper)
Bright Dark L:s Hardness
1\10 120.000 120.000 1:1 1.000 Equalized 5L 5s
6\55 130.909 109.091 6:5 1.200 Qintosec ↑
5\45 133.333 106.667 5:4 1.250
9\80 135.000 105.000 9:7 1.286
4\35 137.143 102.857 4:3 1.333 Supersoft 5L 5s
11\95 138.947 101.053 11:8 1.375
7\60 140.000 100.000 7:5 1.400 Warlock
10\85 141.176 98.824 10:7 1.429
3\25 144.000 96.000 3:2 1.500 Soft 5L 5s
11\90 146.667 93.333 11:7 1.571
8\65 147.692 92.308 8:5 1.600
13\105 148.571 91.429 13:8 1.625 Unnamed golden tuning
5\40 150.000 90.000 5:3 1.667 Semisoft 5L 5s
12\95 151.579 88.421 12:7 1.714
7\55 152.727 87.273 7:4 1.750 Quinkee
9\70 154.286 85.714 9:5 1.800
2\15 160.000 80.000 2:1 2.000 Basic 5L 5s
Blacksmith is optimal around here
9\65 166.154 73.846 9:4 2.250 Trisedodge
7\50 168.000 72.000 7:3 2.333
12\85 169.412 70.588 12:5 2.400
5\35 171.429 68.571 5:2 2.500 Semihard 5L 5s
13\90 173.333 66.667 13:5 2.600 Unnamed golden tuning
8\55 174.545 65.455 8:3 2.667
11\75 176.000 64.000 11:4 2.750
3\20 180.000 60.000 3:1 3.000 Hard 5L 5s
10\65 184.615 55.385 10:3 3.333
7\45 186.667 53.333 7:2 3.500
11\70 188.571 51.429 11:3 3.667
4\25 192.000 48.000 4:1 4.000 Superhard 5L 5s
9\55 196.364 43.636 9:2 4.500
5\30 200.000 40.000 5:1 5.000
6\35 205.714 34.286 6:1 6.000 Cloudtone ↓
1\5 240.000 0.000 1:0 → ∞ Collapsed 5L 5s
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