User:Ganaram inukshuk/7L 3s - Revision history

ArrowHead294 at 18:23, 3 March 2025

2025-03-03T18:23:50Z

← Older revision		Revision as of 18:23, 3 March 2025
Line 29:		Line 29:
	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 (''T''-''t''-''t''), or an augmented step, small tone, and quartertone (''A''-''t''-''q'').		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 (''T''-''t''-''t''), or an augmented step, small tone, and quartertone (''A''-''t''-''q'').
	==Scale tree ==		==Scale tree ==
	{{Scale ~~tree\|~~7L 3s\|~~Comments=~~6/5: Restles ↑;		{{MOS tuning spectrum
	7/5: Beatles;		\| Scale Signature = 7L 3s
	3/2: Suhajira / ringo;		\| 6/5 = Restles ↑
	12/5: Hemif / hemififths;		\| 7/5 = Beatles
	5/2: Mohaha / neutrominant;		\| 3/2 = Suhajira / ringo
	13/5: Hemif / salsa / karadeniz;		\| 12/5 = Hemif / hemififths
	8/3: Mohaha / mohamaq;		\| 5/2 = Mohaha / neutrominant
	4/1: Mohaha / migration / mohajira;		\| 13/5 = Hemif / salsa / karadeniz
	6/1: Mohaha / ptolemy;		\| 8/3 = Mohaha / mohamaq
	13/8: Golden suhajira}}		\| 4/1 = Mohaha / migration / mohajira
			\| 6/1 = Mohaha / ptolemy
			\| 13/8 = Golden suhajira
			}}

	== See also ==		== See also ==

	* Graham Breed's page on 7L 3s (and 3L 7s to an extent)		* Graham Breed's page on 7L 3s (and 3L 7s to an extent)

Ganaram inukshuk: a space

2024-02-18T20:21:05Z

a space

← Older revision		Revision as of 20:21, 18 February 2024
Line 2:		Line 2:

	{{MOS intro\|Scale Signature=7L 3s}}		{{MOS intro\|Scale Signature=7L 3s}}

	7L 3s represents temperaments such as [[mohajira]]/[[mohaha]]/[[mohoho]], whose generators are around a neutral 3rd. Mohaha and mohoho form a [[Chromatic pairs\|chromatic pair]] consisting of a [[Mohaha7\|seven]] and [[Mohaha10\|ten-note scale]].		7L 3s represents temperaments such as [[mohajira]]/[[mohaha]]/[[mohoho]], whose generators are around a neutral 3rd. Mohaha and mohoho form a [[Chromatic pairs\|chromatic pair]] consisting of a [[Mohaha7\|seven]] and [[Mohaha10\|ten-note scale]].

Ganaram inukshuk: /* Scale tree */

2024-02-18T20:20:50Z

Scale tree

Show changes

Ganaram inukshuk: /* Intervals */

2024-02-18T09:22:30Z

Intervals

← Older revision		Revision as of 09:22, 18 February 2024
Line 7:		Line 7:
	==Intervals==		==Intervals==
	:''This article assumes [[TAMNAMS]] for naming step ratios, intervals, and scale degrees.''		:''This article assumes [[TAMNAMS]] for naming step ratios, intervals, and scale degrees.''
	Names for this scale's [[degrees]], the positions of the scale's tones, are called '''mosdegrees'''. Its [[Interval\|intervals]], the pitch difference between any two tones, are based on the number of large and small steps between the two tones and are thus called '''mossteps'''. Per TAMNAMS, both mosdegrees and mossteps are ''0-indexed'', and may be referred to as '''dicodegrees''' and '''dicosteps'''. Ordinal names, such as mos-1st ~~for the unison~~, are discouraged for non-diatonic MOS scales.		Names for this scale's [[degrees]], the positions of the scale's tones, are called '''mosdegrees'''. Its [[Interval\|intervals]], the pitch difference between any two tones, are based on the number of large and small steps between the two tones and are thus called '''mossteps'''. Per TAMNAMS, both mosdegrees and mossteps are ''0-indexed'', and may be referred to as '''dicodegrees''' and '''dicosteps'''. Ordinal names, such as mos-1st instead of 0-mosstep, are discouraged for non-diatonic MOS scales.
	{{MOS intervals\|Scale Signature=7L 3s}}		{{MOS intervals\|Scale Signature=7L 3s}}
	Intervals of interest include:		Intervals of interest include:

	* The '''perfect 3-mosstep''', or the scale's dark generator, whose range is around that of a neutral third. ~~Its inversion,~~ '''~~the~~ perfect 7-mosstep''', ~~has a~~ range around that of a neutral sixth.		* 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 '''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 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; ~~and its inversion, the~~ '''minor 6-mosstep''', whose range coincides with that of a perfect 5th.		* 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.

	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.		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.
Line 23:		Line 25:

	=== Quartertone and tetrachordal analysis ===		=== 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 ~~is recommended~~ for step ratios in which the small step approximates a quartertone.		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 note 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 [[tetrachord\|tetrachordal scale]]. ~~The perfect~~ fourth ~~can be traversed by~~ 3 ~~t's~~ and ~~a q~~, ~~or 2 t's and a T.~~		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 (''T''-''t''-''t''), or an augmented step, small tone, and quartertone (''A''-''t''-''q'').

	~~I (~~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~~
	==Scale tree ==		==Scale tree ==
	{{Scale tree\|7L 3s}}		{{Scale tree\|7L 3s}}
Line 137:		Line 117:

	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).		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).

			== See also ==

			* Graham Breed's page on 7L 3s

Ganaram inukshuk at 05:57, 18 February 2024

2024-02-18T05:57:09Z

Show changes

Ganaram inukshuk at 05:16, 18 February 2024

2024-02-18T05:16:28Z

Show changes

Ganaram inukshuk: Test page for proposed rewrite

2024-02-18T05:14:24Z

Test page for proposed rewrite

New page

{{MOS intro}}
Graham Breed has a 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:

t q t t t q t t q t
== Names==
This MOS is called '''dicoid''' (from [[dicot]], an exotemperament) in [[TAMNAMS]].

==Intervals==
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.

The "large step" will fall between 171 cents (1\7) and 120 cents (1\10), ranging from a submajor second to a [[sinaic]].

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.

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.

Note: In TAMNAMS, a k-step interval class in dicoid 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.
{| class="wikitable"
!# generators up
!Notation (1/1 = 0)
! name
!In L's and s's
!# generators up
! Notation of 2/1 inverse
! name
! 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="6" rowspan="2" | Generator
! colspan="2" | Cents
! rowspan="2" | L
! rowspan="2" | s
! rowspan="2" | L/s
! rowspan="2" | Comments
|-
! Chroma-positive
! Chroma-negative
|-
| 7\10 || || || || || || 840.000 || 360.000 || 1 || 1 || 1.000 ||
|-
| || || || || || 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 ||
|-
| || || || || || 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 || 1.428 ||
|-
| || || 19\27 || || || || 844.444 || 355.556 || 3 || 2 || 1.500 || L/s = 3/2, suhajira/ringo
|-
| || || || || || 69\98 || 844.698 || 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 ||
|-
| || || || || || 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 dicoid<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 || Hemif/hemififths
|-
| || || || 29\41 || || || 848.780 || 351.220 || 5 || 2 || 2.500 || Mohaha/neutrominant
|-
| || || || || || 75\106 || 849.057 || 350.943 || 13 || 5 || 2.600 || Hemif/salsa/karadeniz
|-
| || || || || 46\65 || || 849.231 || 350.769 || 8 || 3 || 2.667 || 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 || 851.163 || 348.837 || 11 || 3 || 3.667 ||
|-
| || || || 22\31 || || || 851.613 || 348.387 || 4 || 1 || 4.000 || 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 || → 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|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 (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

[[Category:10-tone scales]]