User:Ganaram inukshuk/7L 3s: Difference between revisions
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==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 | 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''', | * 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; | * 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. | ||
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=== 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 | 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]]. | 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''). | ||
Thus, the possible tetrachords | |||
a | |||
a t | |||
t | |||
t q | |||
==Scale tree == | ==Scale tree == | ||
{{Scale tree|7L 3s}} | {{Scale tree|7L 3s}} | ||
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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 | |||
Revision as of 09:22, 18 February 2024
| This page is an in-progress rewrite for a main-namespace page. For the current page, see 7L 3s. |
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 ¢.
Name
TAMNAMS suggests the temperament-agnostic name dicoid (from dicot, an exotemperament) for the name of this scale.
Intervals
- 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 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.
| 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 ¢ |
Intervals of interest 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.
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.
Theory
Temperament interpretations
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 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 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
| Generator | Cents | L | s | L/s | 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 (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 | ||||||
TODO: add scale tree entries from old scale tree.
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 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).
See also
- Graham Breed's page on 7L 3s