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{{Infobox MOS
{{Infobox MOS}}
| Name = 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
}}


'''7L 3s''' 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):
== 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 '''dicoid''' (DY-koid; named after the abstract temperament [[dicot]]) in [[TAMNAMS]].
==Interval ranges==
The generator (g) will fall between 343 cents (2\7 - two degrees of [[7edo|7edo]] and 360 cents (3\10 - three degrees of [[10edo|10edo]]), hence a neutral third.


2g, then, will fall between 686 cents (4\7) and 720 cents (3\5), the range of [[5L 2s|diatonic]] fifths.
=== Modes ===
{{MOS mode degrees}}


The "large step" will fall between 171 cents (1\7) and 120 cents (1\10), ranging from a submajor second to a [[sinaic]].
== 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 "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.
=== 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 {{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 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.
Additionally, due to the presence of fourth and fifth-like intervals, 7L 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''}}).


==Scale tree==
==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|17edo]]:
{{MOS tuning spectrum
 
| 6/5 = [[Restles]] ↑
{| class="wikitable"
| 7/5 = [[Beatles]]
|-
| 3/2 = [[Suhajira]] / ringo
! colspan="6" | Generator (fraction of octave)
| 12/5 = [[Hemif]] / [[hemififths]]
! | Generator (cents)
| 5/2 = [[Mohaha]] / [[neutrominant]] / [[mohamaq]]
! | Comments
| 13/5 = [[Hemif]] / [[salsa]] / [[karadeniz]]
|-
| 4/1 = [[Mohaha]] / [[migration]] / [[mohajira]]
| | 3\10
| 6/1 = [[Mohaha]] / [[ptolemy]]
| |
| 13/8 = Golden [[suhajira]]
| |
}}
| |
| |
| |
| | 360
| style="text-align:center;" |
|-
| |
| |
| |
| |
| |
| | 17\57
| | 357.895
| |
|-
| |
| |
| |
| |
| | 14\47
| |
| | 357.45
| |
|-
| |
| |
| |
| | 11\37
| |
| |
| | 356.76
| |
|-
| |
| |
| | 8\27
| |
| |
| |
| | 355.56
| style="text-align:center;" | Optimum rank range (L/s=3/2)
|-
| |
| |
| |
| |
| |
| |
| | 355.11
| |
|-
| |
| |
| |
| |
| | 21\71
| |
| | 354.93
| style="text-align:center;" |
|-
| |
| |
| |
| |
| |
| | 34\115
| | 354.78
| style="text-align:center;" | Golden neutral thirds
|-
| |
| |
| |
| | 13\44
| |
| |
| | 354.55
| style="text-align:center;" |
|-
| |
| |
| |
| |
| |
| |
| | 354.19
| |
|-
| |
| | 5\17
| |
| |
| |
| |
| | 352.94
| style="text-align:center;" | Boundary of propriety: generators larger than this are proper
|-
| |
| |
| |
| | 12\41
| |
| |
| | 351.22
| style="text-align:center;" |
|-
| |
| |
| |
| |
| |
| | 31\106
| | 350.94
| style="text-align:center;" | No-5's neutral thirds (observing 81/80) is around here
|-
| |
| |
| |
| |
| | 19\65
| |
| | 350.77
| style="text-align:center;" |
|-
| |
| |
| |
| |
| |
| |
| | 350.64
| style="text-align:center;" | <span style="display: block; text-align: center;">L/s = e</span>
|-
| |
| |
| | 7\24
| |
| |
| |
| | 350
| style="text-align:center;" |
|-
| |
| |
| |
| |
| |
| |
| | 349.72
| style="text-align:center;" | <span style="display: block; text-align: center;">L/s = pi</span>
|-
| |
| |
| |
| |
| | 16\55
| |
| | 349.09
| |
|-
| |
| |
| |
| | 9\31
| |
| |
| | 348.39
| style="text-align:center;" | Mohajira (tempering out 81/80) is around here
|-
| |
| |
| |
| |
| | 11\38
| |
| | 347.37
| |
|-
| |
| |
| |
| |
| |
| | 13\45
| | 346.67
| |
|-
| | 2\7
| |
| |
| |
| |
| |
| | 342.86
| style="text-align:center;" |
|}
 
The scale produced by stacks of 5\17 is the [[17edo_neutral_scale|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|37edo]], [[27edo|27edo]], [[44edo|44edo]], [[41edo|41edo]], [[24edo|24edo]], [[31edo|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|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:Scales]]
[[Category:10-tone scales]]
[[Category:MOS scales]]
[[Category:Abstract MOS patterns]]
Retrieved from "https://en.xen.wiki/w/7L_3s"