7L 3s: Difference between revisions

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{{Infobox MOS
{{Infobox MOS}}
| Name = Neapolitan-dicotonic
{{MOS intro|Scale Signature=7L 3s}}
| Periods = 1
7L 3s represents [[temperament]]s such as [[mohajira]]/[[mohaha]]/[[mohoho]], among others, whose generators are around a neutral third. The [[Mohaha7|seven]] and [[Mohaha10|ten-note]] forms of mohaha/mohoho form a [[Chromatic pairs#Mohaha|chromatic pair]].
| nLargeSteps = 7
| nSmallSteps = 3
| Equalized = 3
| Paucitonic = 2
| Pattern = LLLsLLsLLs
| Equave = 15/7}}
{{Infobox MOS
| Name = dicotonic
| Periods = 1
| nLargeSteps = 7
| nSmallSteps = 3
| Equalized = 3
| Paucitonic = 2
| Pattern = LLLsLLsLLs
}}


'''7L 3s(<15/7>)''' refers to the structure of [[MOSScales|moment of symmetry scales]] built from a 10-tone chain of (neutral) thirds (assuming a period of an octave or a minor ninth):
== Name ==
{{TAMNAMS name}}


L s L L L s L L s L
== Scale properties ==
{{TAMNAMS use}}


Graham Breed has a [http://x31eq.com/7plus3.htm page on his website] dedicated to 7+3 scales. He proposes calling the large step "t" for "tone", lowercase because the large step is a narrow neutral tone, and the small step "q" for "quartertone", because the small step is often close to a quartertone. (Note that the small step is not a quartertone in every instance of 7+3, so do not take that "q" literally.) Thus we have:
=== Intervals ===
{{MOS intervals}}


t q t t t q t t q t
=== Generator chain ===
==Names==
{{MOS genchain}}
This MOS is called '''dicotonic''' (named after the abstract temperaments [[dicot]] and more specifically 11-limit [[Dicot_family#Dichotic|dichotic]]) in [[TAMNAMS]].


==Intervals==
=== Modes ===
The generator (g) will fall between 343 cents (2\7 - two degrees of [[7edo]] and 360 cents (3\10 - three degrees of [[10edo]]), hence a neutral third.
{{MOS mode degrees}}


2g, then, will fall between 686 cents (4\7) and 720 cents (3\5), the range of [[5L 2s|diatonic]] fifths.
== Theory ==
=== Neutral intervals ===
7L 3s combines the familiar sound of perfect fifths and fourths with the unfamiliar sounds of neutral intervals, thus making it compatible with [[Arabic, Turkish, Persian|Arabic]] and [[Arabic, Turkish, Persian|Turkish]] scales, but not with traditional Western scales. Notable intervals include:
* The '''perfect 3-mosstep''', the scale's dark generator, whose range is around that of a neutral third.
* The '''perfect 7-mosstep''', the scale's bright generator, the inversion of the perfect 3-mosstep, whose range is around that of a neutral sixth.
* The '''minor mosstep''', or '''small step''', which ranges form a [[quartertone]] to a minor second.
* The '''major mosstep''', or '''large step''', which ranges from a submajor second to a [[sinaic]], or trienthird (around 128{{c}}).
* The '''major 4-mosstep''', whose range coincides with that of a perfect fourth.
* The '''minor 6-mosstep''', the inversion of the major 4-mosstep, whose range coincides with that of a perfect 5th.


The "large step" will fall between 171 cents (1\7) and 120 cents (1\10), ranging from a submajor second to a [[sinaic]].
=== Quartertone and tetrachordal analysis===
Due to the presence of [[quartertone]]-like intervals, [[Graham Breed]] has proposed the terms ''tone'' (abbreviated as ''t'') and ''quartertone'' (abbreviated as ''q'') as alternatives for large and small steps. This interpretation only makes sense for step ratios in which the small step approximates a quartertone. Additionally, Breed has also proposed a larger tone size, abbreviated using a capital ''T'', to refer to the combination of ''t'' and ''q''. Through this addition of a larger step, 7-note subsets of 7L&nbsp;3s can be constructed. Some of these subsets are identical to that of 3L 4s, such as {{dash|''T, t, T, t, T, t, t''}}, but Breed states that non-MOS patterns are possible, such as {{dash|''T, t, t, T, t, t, T''}}.


The "small step" will fall between 0 cents and 120 cents, sometimes sounding like a minor second, and sometimes sounding like a quartertone or smaller microtone.
Additionally, due to the presence of fourth and fifth-like intervals, 7L&nbsp;3s can be analyzed as a [[tetrachord|tetrachordal scale]]. Since the major 4-dicostep, the fourth-like interval, is reached using 4 steps rather than 3 (3 tones and 1 quartertone), Andrew Heathwaite offers an additional step ''A'', for ''augmented second'', to refer to the combination of two tones (''t''). Thus, the possible tetrachords can be built as a combination of a (large) tone and two (regular) tones ({{dash|''T'', ''t'', ''t''}}), or an augmented step, small tone, and quartertone ({{dash|''A'', ''t'', ''q''}}).


The most frequent interval, then is the neutral third (and its inversion, the neutral sixth), followed by the perfect fourth and fifth. Thus, 7+3 combines the familiar sound of perfect fifths and fourths with the unfamiliar sounds of neutral intervals. They are compatible with Arabic and Turkish scales, but not with traditional Western ones.
==Scale tree==
 
{{MOS tuning spectrum
Note: In TAMNAMS, a k-step interval class in dicotonic may be called a "k-step", "k-mosstep", or "k-dicostep". 1-indexed terms such as "mos(k+1)th" are discouraged for non-diatonic mosses.
| 6/5 = [[Restles]]&nbsp;↑
{| class="wikitable"
| 7/5 = [[Beatles]]
!# generators up
| 3/2 = [[Suhajira]] / ringo
!Notation (1/1 = 0)
| 12/5 = [[Hemif]] / [[hemififths]]
!name
| 5/2 = [[Mohaha]] / [[neutrominant]] / [[mohamaq]]
!In L's and s's
| 13/5 = [[Hemif]] / [[salsa]] / [[karadeniz]]
!# generators up
| 4/1 = [[Mohaha]] / [[migration]] / [[mohajira]]
!Notation of 2/1 inverse
| 6/1 = [[Mohaha]] / [[ptolemy]]
!name
| 13/8 = Golden [[suhajira]]
!In L's and s's
}}
|-
| colspan="8" style="text-align:center" |The 10-note MOS has the following intervals (from some root):
|-
|0
|0
|perfect unison
|0
|0
|0
|perfect 10-step
|7L+3s
|-
|1
|7
|perfect 7-step
|5L+2s
| -1
|3
|perfect 3-step
|2L+1s
|-
|2
|4
|major 4-step
|3L+1s
| -2
|6
|minor 6-step
|4L+2s
|-
|3
|1
|major (1-)step
|1L
| -3
|9v
|minor 9-step
|6L+3s
|-
|4
|8
|major 8-step
|6L+2s
| -4
|2v
|minor 2-step
|1L+1s
|-
|5
|5
|major 5-step
|4L+1s
| -5
|5v
|minor 5-step
|3L+2s
|-
|6
|2
|major 2-step
|2L
| -6
|8v
|minor 8-step
|5L+3s
|-
|7
|9
|major 9-step
|7L+2s
| -7
|1v
|minor (1-)step
|1s
|-
|8
|6^
|major 6-step
|5L+1s
| -8
|4v
|minor 4-step
|2L+2s
|-
|9
|3^
|augmented 3-step
|3L
| -9
|7v
|diminished 7-step
| 4L+3s
|-
|10
|0^
|augmented unison
|1L-1s
| -10
|0v
|diminished 10-step
|6L+4s
|-
|11
|7^
|augmented 7-step
|6L+1s
| -11
|3v
|diminished 3-step
| 1L+2s
|-
| colspan="8" style="text-align:center" |The chromatic 17-note MOS (either [[7L 10s]], [[10L 7s]], or [[17edo]]) also has the following intervals (from some root):
|-
|12
|4^
|augmented 4-step
|4L
| -12
|6v
|diminished 6-step
|3L+3s
|-
|13
|1^
|augmented (1-)step
|2L-1s
| -13
|9w
|diminished 9-step
|5L+4s
|-
|14
|8^
|augmented 8-step
|8L+1s
| -14
|2w
|diminished 2-step
|2s
|-
|15
|5^
| augmented 5-step
|5L
| -15
|5w
|diminished 5-step
|2L+3s
|-
|16
|2^
|augmented 2-step
| 3L-1s
| -16
|8w
|diminished 8-step
|4L+4s
|}
==Scale tree ==
The generator range reflects two extremes: one where L = s (3\10), and another where s = 0 (2\7). Between these extremes, there is an infinite continuum of possible generator sizes. By taking freshman sums of the two edges (adding the numerators, then adding the denominators), we can fill in this continuum with compatible edos, increasing in number of tones as we continue filling in the in-betweens. Thus, the smallest in-between edo would be (3+2)\(10+7) = 5\17 – five degrees of [[17edo]]:
 
{| class="wikitable center-all"
! colspan="4" rowspan="2" |Generator
! colspan="2" | Cents
! rowspan="2" |''ed17\16''
! rowspan="2" |L
! rowspan="2" |s
! rowspan="2" |L/s
! rowspan="2" |Comments
|-
!edo
!Neapolitan (normalized)
|-
|7\10|| || || ||840.000||933.333 (1171.242 śata)
|''1190.000''||1||1||1.000||
|-
| 40\57||  || || ||842.105||923.077 (1158.371 śata)
|''1192.9825''|| 6||5||1.200||(Neapolitan-)Restles↑
|-
|
|73\104
|
|
|842.308
|922.105 (1157.152 śata)
|''1193.269''
|11
|9
|1.222
|
|-
|
|
|106\151
|
|842.384
|921.739 (1156.692 śata)
|''1193.3775''
|16
|13
|1.231
|
|-
| 33\47||  || ||  || 842.553||920.930 (1155.677 śata)
|''1193.617''||5|| 4||1.250 ||
|-
|
|92\131
|
|
|842.748
|920.000 (1154.510 śata)
|''1193.893''
|14
|11
|1.273
|
|-
| 59\84|| || || ||842.857 ||919.4805 (1153.858 śata)
|''1194.048''||9||7||1.286||
|-
|
|85\121
|
|
|842.975
|918.919 (1153.153 śata)
|''1194.215''
|13
|10
|1.300
|
|-
|
|
|111\158
|
|843.038
|918.621 (1152.779 śata)
|''1194.304''
|17
|13
|1.308
|
|-
| 26\37|| || || ||843.243||917.647 (1151.557 śata)
|''1194.595''||4 || 3||1.333 ||
|-
|
|97\138
|
|
|843.478
|916.535 (1150.162 śata)
|''1194.9275''
|15
|11
|1.364
|
|-
|71\101|| || || ||843.564||916.129 (1149.652 śata)
|''1195.0495''||11||8||1.375||
|-
|45\64||  ||  || || 843.750||915.254 (1148.554 śata)
|''1195.3125''||7|| 5||1.400||(Neapolitan-)Beatles
|-
|64\91|| || || ||843.956||914.286 (1147.339 śata)
|''1195.604''||10||7 || 1.428 ||
|-
|
|83\118
|
|
|844.068
|913.7615 (1146.681 śata)
|''1195.763''
|13
|9
|1.444
|
|-
|
|
|102\145
|
|844.138
|913.433 (1146.269 śata)
|''1195.862''
|16
|11
|1.4545
|
|-
|
|
|
|121\172
|844.186
|913.2075 (1145.986 śata)
|''1195.930''
|19
|13
|1.4615
|
|-
|19\27|| || || || 844.444|| 912.000 (1144.471 śata)
|''1196.296''||3||2 || 1.500||L/s = 3/2, (Neapolitan-)suhajira/ringo
|-
|
|
|
|107\152
|844.736
|910.638 (1142.762 śata)
|''1196.7105''
|17
|11
|1.5455
|
|-
|
|
|88\125
|
|844.8
|910.345 (1142.3935 śata)
|''1196.800''
|14
|9
|1.556
|
|-
| || 69\98||  || ||844.898||909.890 (1141.823 śata)
|''1196.939''||11||7 || 1.571||
|-
|50\71|| || || ||845.070|| 909.091 (1140.820 śata)
|''1197.183''||8||5||1.600||
|-
|  ||81\115|| || ||845.217||908.411 (1139.967 śata)
|''1197.391''||13||8 || 1.625||Golden (Neapolitan-)suhajira
|-
| 31\44|| || || ||845.455||907.317 (1138.594 śata)
|''1197.727''||5|| 3||1.667 ||
|-
| ||74\105|| || ||845.714 || 906.122 (1137.095 śata)
|''1198.095''||12 || 7||1.714 ||
|-
|43\61|| || || ||845.902||905.263 (1136.0165 śata)
|''1198.361''||7||4||1.750||
|-
|
|
|98\139
|
|846.043
|904.615 (1135.204 śata)
|''1198.561''
|16
|9
|1.778
|
|-
| ||55\78||  ||  ||846.154||904.110 (1134.569 śata)
|''1198.718''||9||5||1.800 ||
|-
|
|
|67\95
|
|846.316
|903.371 (1133.642 śata)
|''1198.947''
|11
|6
|1.833
|
|-
|
|
|
|79\112
|846.429
|902.857 (1132.997 śata)
|''1199.107''
|13
|7
|1.857
|
|-
|12\17|| ||  ||  || 847.059||900.000 (1129.412 śata)
|''1200.000''||2||1||2.000 ||Basic (Neapolitan-)dicotonic<br>(Generators smaller than this are proper)
|-
|  ||  ||53\75||  ||848.000||352.000
| ||9||4||2.250||
|-
| ||41\58|| || ||848.273||351.724
| ||7||3||2.333||
|-
| || ||70\99|| ||848.485||351.515
| ||12||5||2.400||(Neapolitan-)Hemif/hemififths
|-
|29\41|| || || || 848.780 || 351.220
| ||5|| 2||2.500||(Neapolitan-)Mohaha/neutrominant
|-
| || ||75\106|| ||849.057 || 350.943
| || 13||5 || 2.600|| (Neapolitan-)Hemif/salsa/karadeniz
|-
| ||46\65|| ||  ||849.231 ||350.769
| ||8||3 || 2.667 || (Neapolitan-)Mohaha/mohamaq
|-
| || ||63\89|| ||849.438||350.562
| ||11||4||2.750||
|-
|17\24|| || || ||850.000||350.000
| ||3 || 1||3.000 ||L/s = 3/1
|-
| || ||56\79|| ||850.633||349.367
| ||10||3 ||3.333||
|-
| ||39\55|| || ||850.909||349.091
| ||7||2 || 3.500||
|-
| || ||61\86|| || 851.613|| 358.837
| ||11||3||3.667||
|-
|22\31|| ||  || || 851.613||348.387
| ||4 || 1||4.000||(Neapolitan-)Mohaha/migration/mohajira
|-
| || ||49\69|| ||852.174|| 347.826
| ||9||2 ||4.500||
|-
| ||27\38||  ||  ||852.632||347.368
| ||5||1||5.000 ||
|-
| ||  ||32\45|| ||853.333||346.667
| || 6||1 ||6.000||(Neapolitan-)Mohaha/ptolemy
|-
|5\7|| || || ||857.143|| 342.867
| || 1||0 ||→ inf||
|}
 
The scale produced by stacks of 5\17 is the [[17edo neutral scale]]. Between 11/38 and 16/55, with 9/31 in between, is the mohajira/mohaha/mohoho range, where mohaha and mohoho use the MOS as the chromatic scale of a [[Chromatic pairs|chromatic pair]].
 
Other compatible edos include: [[37edo]], [[27edo]], [[44edo]], [[41edo]], [[24edo]], [[31edo]].
 
You can also build this scale by stacking neutral thirds that are not members of edos – for instance, frequency ratios 11:9, 49:40, 27:22, 16:13 – or the square root of 3:2 (a bisected just perfect fifth).
 
==Rank-2 temperaments==
==7-note subsets ==
If you stop the chain at 7 tones, you have a heptatonic scale of the form [[3L 4s]]:
 
L s s L s L s
 
The large steps here consist of t+s of the 10-tone system, and the small step is the same as t. Graham proposes calling the large step here T for "tone," uppercase because it is a wider tone than t. Thus, we have:
 
T t t T t T t
 
This scale (and its rotations) is not the only possible heptatonic scale. Graham also gives us:
 
T t t T t t T
 
which is not a complete moment of symmetry scale in itself, but a subset of one.
 
== Tetrachordal structure==
Due to the frequency of perfect fourths and fifths in this scale, it can also be analyzed as a [[tetrachord|tetrachordal scale]]. The perfect fourth can be traversed by 3 t's and a q, or 2 t's and a T.
 
I ([[File:external-6855b5f4f272812f2538853afd1c4157-withext.jpg|16px]] - [http://www.wikispaces.com/user/view/Andrew_Heathwaite Andrew Heathwaite]) offer "a" to refer to a step of 2t (for "augmented second")
 
Thus, the possible tetrachords are:
 
T t t
 
t T t
 
t t T
 
a q t
 
a t q
 
t a q
 
t q a
 
q a t


q t a
== External links==
* [http://x31eq.com/7plus3.htm Graham Breed's page on 7L&nbsp;3s] (which covers 3L&nbsp;7s to an extent)


[[Category:Abstract MOS patterns]]
[[Category:10-tone scales]]

Latest revision as of 14:02, 5 May 2025

↖ 6L 2s ↑ 7L 2s 8L 2s ↗
← 6L 3s 7L 3s 8L 3s →
↙ 6L 4s ↓ 7L 4s 8L 4s ↘ 
┌╥╥╥┬╥╥┬╥╥┬┐
│║║║│║║│║║││
││││││││││││
└┴┴┴┴┴┴┴┴┴┴┘
Scale structure
Step pattern LLLsLLsLLs
sLLsLLsLLL
Equave 2/1 (1200.0 ¢)
Period 2/1 (1200.0 ¢)
Generator size
Bright 7\10 to 5\7 (840.0 ¢ to 857.1 ¢)
Dark 2\7 to 3\10 (342.9 ¢ to 360.0 ¢)
TAMNAMS information
Name dicoid
Prefix dico-
Abbrev. di
Related MOS scales
Parent 3L 4s
Sister 3L 7s
Daughters 10L 7s, 7L 10s
Neutralized 4L 6s
2-Flought 17L 3s, 7L 13s
Equal tunings
Equalized (L:s = 1:1) 7\10 (840.0 ¢)
Supersoft (L:s = 4:3) 26\37 (843.2 ¢)
Soft (L:s = 3:2) 19\27 (844.4 ¢)
Semisoft (L:s = 5:3) 31\44 (845.5 ¢)
Basic (L:s = 2:1) 12\17 (847.1 ¢)
Semihard (L:s = 5:2) 29\41 (848.8 ¢)
Hard (L:s = 3:1) 17\24 (850.0 ¢)
Superhard (L:s = 4:1) 22\31 (851.6 ¢)
Collapsed (L:s = 1:0) 5\7 (857.1 ¢)

7L 3s, named dicoid in TAMNAMS, is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 7 large steps and 3 small steps, repeating every octave. Generators that produce this scale range from 840 ¢ to 857.1 ¢, or from 342.9 ¢ to 360 ¢. 7L 3s represents temperaments such as mohajira/mohaha/mohoho, among others, whose generators are around a neutral third. The seven and ten-note forms of mohaha/mohoho form a chromatic pair.

Name

TAMNAMS suggests the temperament-agnostic name dicoid as the name of 7L 3s. The name derives from dichotic and dicot temperament. Although this name is directly based off of a temperament, tunings of dichotic and dicot cover the entire tuning range of 7L 3s; see TAMNAMS/Appendix #Dicoid (7L 3s) for more information.

Scale properties

This article uses TAMNAMS conventions for the names of this scale's intervals and scale degrees. The use of 1-indexed ordinal names is reserved for interval regions.

Intervals

Intervals of 7L 3s
Intervals Steps
subtended 		Range in cents
Generic Specific Abbrev.
0-dicostep Perfect 0-dicostep P0dis 0 0.0 ¢
1-dicostep Minor 1-dicostep m1dis s 0.0 ¢ to 120.0 ¢
Major 1-dicostep M1dis L 120.0 ¢ to 171.4 ¢
2-dicostep Minor 2-dicostep m2dis L + s 171.4 ¢ to 240.0 ¢
Major 2-dicostep M2dis 2L 240.0 ¢ to 342.9 ¢
3-dicostep Perfect 3-dicostep P3dis 2L + s 342.9 ¢ to 360.0 ¢
Augmented 3-dicostep A3dis 3L 360.0 ¢ to 514.3 ¢
4-dicostep Minor 4-dicostep m4dis 2L + 2s 342.9 ¢ to 480.0 ¢
Major 4-dicostep M4dis 3L + s 480.0 ¢ to 514.3 ¢
5-dicostep Minor 5-dicostep m5dis 3L + 2s 514.3 ¢ to 600.0 ¢
Major 5-dicostep M5dis 4L + s 600.0 ¢ to 685.7 ¢
6-dicostep Minor 6-dicostep m6dis 4L + 2s 685.7 ¢ to 720.0 ¢
Major 6-dicostep M6dis 5L + s 720.0 ¢ to 857.1 ¢
7-dicostep Diminished 7-dicostep d7dis 4L + 3s 685.7 ¢ to 840.0 ¢
Perfect 7-dicostep P7dis 5L + 2s 840.0 ¢ to 857.1 ¢
8-dicostep Minor 8-dicostep m8dis 5L + 3s 857.1 ¢ to 960.0 ¢
Major 8-dicostep M8dis 6L + 2s 960.0 ¢ to 1028.6 ¢
9-dicostep Minor 9-dicostep m9dis 6L + 3s 1028.6 ¢ to 1080.0 ¢
Major 9-dicostep M9dis 7L + 2s 1080.0 ¢ to 1200.0 ¢
10-dicostep Perfect 10-dicostep P10dis 7L + 3s 1200.0 ¢

Generator chain

Generator chain of 7L 3s
Bright gens Scale degree Abbrev.
16 Augmented 2-dicodegree A2did
15 Augmented 5-dicodegree A5did
14 Augmented 8-dicodegree A8did
13 Augmented 1-dicodegree A1did
12 Augmented 4-dicodegree A4did
11 Augmented 7-dicodegree A7did
10 Augmented 0-dicodegree A0did
9 Augmented 3-dicodegree A3did
8 Major 6-dicodegree M6did
7 Major 9-dicodegree M9did
6 Major 2-dicodegree M2did
5 Major 5-dicodegree M5did
4 Major 8-dicodegree M8did
3 Major 1-dicodegree M1did
2 Major 4-dicodegree M4did
1 Perfect 7-dicodegree P7did
0 Perfect 0-dicodegree
Perfect 10-dicodegree		 P0did
P10did
−1 Perfect 3-dicodegree P3did
−2 Minor 6-dicodegree m6did
−3 Minor 9-dicodegree m9did
−4 Minor 2-dicodegree m2did
−5 Minor 5-dicodegree m5did
−6 Minor 8-dicodegree m8did
−7 Minor 1-dicodegree m1did
−8 Minor 4-dicodegree m4did
−9 Diminished 7-dicodegree d7did
−10 Diminished 10-dicodegree d10did
−11 Diminished 3-dicodegree d3did
−12 Diminished 6-dicodegree d6did
−13 Diminished 9-dicodegree d9did
−14 Diminished 2-dicodegree d2did
−15 Diminished 5-dicodegree d5did
−16 Diminished 8-dicodegree d8did

Modes

Scale degrees of the modes of 7L 3s
UDP Cyclic
order 		Step
pattern 		Scale degree (dicodegree)
0 1 2 3 4 5 6 7 8 9 10
9|0 1 LLLsLLsLLs Perf. Maj. Maj. Aug. Maj. Maj. Maj. Perf. Maj. Maj. Perf.
8|1 8 LLsLLLsLLs Perf. Maj. Maj. Perf. Maj. Maj. Maj. Perf. Maj. Maj. Perf.
7|2 5 LLsLLsLLLs Perf. Maj. Maj. Perf. Maj. Maj. Min. Perf. Maj. Maj. Perf.
6|3 2 LLsLLsLLsL Perf. Maj. Maj. Perf. Maj. Maj. Min. Perf. Maj. Min. Perf.
5|4 9 LsLLLsLLsL Perf. Maj. Min. Perf. Maj. Maj. Min. Perf. Maj. Min. Perf.
4|5 6 LsLLsLLLsL Perf. Maj. Min. Perf. Maj. Min. Min. Perf. Maj. Min. Perf.
3|6 3 LsLLsLLsLL Perf. Maj. Min. Perf. Maj. Min. Min. Perf. Min. Min. Perf.
2|7 10 sLLLsLLsLL Perf. Min. Min. Perf. Maj. Min. Min. Perf. Min. Min. Perf.
1|8 7 sLLsLLLsLL Perf. Min. Min. Perf. Min. Min. Min. Perf. Min. Min. Perf.
0|9 4 sLLsLLsLLL Perf. Min. Min. Perf. Min. Min. Min. Dim. Min. Min. Perf.

Theory

Neutral intervals

7L 3s combines the familiar sound of perfect fifths and fourths with the unfamiliar sounds of neutral intervals, thus making it compatible with Arabic and Turkish scales, but not with traditional Western scales. Notable intervals include:

  • The perfect 3-mosstep, the scale's dark generator, whose range is around that of a neutral third.
  • The perfect 7-mosstep, the scale's bright generator, the inversion of the perfect 3-mosstep, whose range is around that of a neutral sixth.
  • The minor mosstep, or small step, which ranges form a quartertone to a minor second.
  • The major mosstep, or large step, which ranges from a submajor second to a sinaic, or trienthird (around 128 ¢).
  • The major 4-mosstep, whose range coincides with that of a perfect fourth.
  • The minor 6-mosstep, the inversion of the major 4-mosstep, whose range coincides with that of a perfect 5th.

Quartertone and tetrachordal analysis

Due to the presence of quartertone-like intervals, Graham Breed has proposed the terms tone (abbreviated as t) and quartertone (abbreviated as q) as alternatives for large and small steps. This interpretation only makes sense for step ratios in which the small step approximates a quartertone. Additionally, Breed has also proposed a larger tone size, abbreviated using a capital T, to refer to the combination of t and q. Through this addition of a larger step, 7-note subsets of 7L 3s can be constructed. Some of these subsets are identical to that of 3L 4s, such as T – t – T – t – T – t – t, but Breed states that non-MOS patterns are possible, such as T – t – t – T – t – t – T.

Additionally, due to the presence of fourth and fifth-like intervals, 7L 3s can be analyzed as a tetrachordal scale. Since the major 4-dicostep, the fourth-like interval, is reached using 4 steps rather than 3 (3 tones and 1 quartertone), Andrew Heathwaite offers an additional step A, for augmented second, to refer to the combination of two tones (t). Thus, the possible tetrachords can be built as a combination of a (large) tone and two (regular) tones (T – t – t), or an augmented step, small tone, and quartertone (A – t – q).

Scale tree

Scale tree and tuning spectrum of 7L 3s
Generator(edo) Cents Step ratio Comments
Bright Dark L:s Hardness
7\10 840.000 360.000 1:1 1.000 Equalized 7L 3s
40\57 842.105 357.895 6:5 1.200 Restles ↑
33\47 842.553 357.447 5:4 1.250
59\84 842.857 357.143 9:7 1.286
26\37 843.243 356.757 4:3 1.333 Supersoft 7L 3s
71\101 843.564 356.436 11:8 1.375
45\64 843.750 356.250 7:5 1.400 Beatles
64\91 843.956 356.044 10:7 1.429
19\27 844.444 355.556 3:2 1.500 Soft 7L 3s
Suhajira / ringo
69\98 844.898 355.102 11:7 1.571
50\71 845.070 354.930 8:5 1.600
81\115 845.217 354.783 13:8 1.625 Golden suhajira
31\44 845.455 354.545 5:3 1.667 Semisoft 7L 3s
74\105 845.714 354.286 12:7 1.714
43\61 845.902 354.098 7:4 1.750
55\78 846.154 353.846 9:5 1.800
12\17 847.059 352.941 2:1 2.000 Basic 7L 3s
Scales with tunings softer than this are proper
53\75 848.000 352.000 9:4 2.250
41\58 848.276 351.724 7:3 2.333
70\99 848.485 351.515 12:5 2.400 Hemif / hemififths
29\41 848.780 351.220 5:2 2.500 Semihard 7L 3s
Mohaha / neutrominant / mohamaq
75\106 849.057 350.943 13:5 2.600 Hemif / salsa / karadeniz
46\65 849.231 350.769 8:3 2.667
63\89 849.438 350.562 11:4 2.750
17\24 850.000 350.000 3:1 3.000 Hard 7L 3s
56\79 850.633 349.367 10:3 3.333
39\55 850.909 349.091 7:2 3.500
61\86 851.163 348.837 11:3 3.667
22\31 851.613 348.387 4:1 4.000 Superhard 7L 3s
Mohaha / migration / mohajira
49\69 852.174 347.826 9:2 4.500
27\38 852.632 347.368 5:1 5.000
32\45 853.333 346.667 6:1 6.000 Mohaha / ptolemy
5\7 857.143 342.857 1:0 → ∞ Collapsed 7L 3s

External links

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