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Structures of scales
In the program of the minor World music composition in practice there is a module where scales are studied, especially from North India. As this minor has the aim to incite and inspire students to create their own compositions, their own music, so they are challenged to also create their own scales. For this purpose the following collection of 135-5-6=124 scales was created. I chose to present them visually as circles (to a background of a circle of seconds) in order to demonstrate their inherent structures, as my personal circles of scales collection. It can be seen as similar to Peter Schat’s Tone Clock (to which I contributed some minor corrections in 19841), which starts out focusing not on melody but on harmony, more specifically: on triads.
The categorizing code
Each of the scales is coded with a series of numbers and letters, describing its main melodic and harmonic features. This code is explained here by example:
7-3310-21(HWHWWH)-2 (in fact the harmonic minor scale from one of its points)
- 7 is the number of tones in the scale; it also gives the number of tones to begin from (which in this case leads to 7 completely different scales, sharing the same structure
- 3310 is a sequence of four numbers describing the scale steps, here is means first 3 semitone steps, then 3 hole tone steps, then 1 minor third/augmented second step, and finally 0 major third/diminished fourth steps
- 21 describing perfect fifths and fourths as possible harmonic framework, there are 2 connected perfect fifths and fourths and 1 extra unconnected perfect fifth or fourth
- (HWHWWH) the sequence of seconds between larger intervals can vary and needs to be described, for this purpose this is coded in H (half step) en W (whole step), reading the circle always clockwise
- 2 the final number describes possible harmonic tension, as it counts the number of tritones embedded within the scale
The circular representations
Here follows the collection of all scales, which is not intended to be theoretically complete, and which for now stops at the known octotonic scale(s).
The drawings are color-coded, on the edge of the circle:
- red line: semitone step
- dark red line: whole tone step
- purple line: minor third or augmented second step
- clear blue line: major third step
The lines on the inside of the circle indicate harmonic possibilities of the structures:
- dark brown triangle: fifth-fourth to a tone skeleton (German: Quint-Quart Gerüst – Hindemith), giving clear possibilities of harmonic stability; the top of such a triangle is a bolder dot, as it is a likely central tone in harmony created with such a structure
- ochre line: “loose” fifth or fourth, giving some possibility of harmonic stability
- gray line through the center of the circle: tritone, giving possibilities of harmonic tension
The scales in notation
When I drew the circles, I deliberately turned the underlying circle by 15° so no visually fixed starting position would be suggested. In the same vein, the scales as presented in notation below start from a relatively random point, then presenting the “degrees” which are not really degrees, because the I is not necessarily a center of tonal gravity.
With this in mind, here follows the presentation of the scales in notation.
All of these scales have two consecutive minor third jumps, which embeds a rather exposed triad in the scale.
There is 1 tone with a fifth-fourth-skeleton, this is seen in mode I.
This is the pentatonic scale structure of which III is normally presented as “the pentatonic scale”. We now know better, as the are many many more possible pentatonic scales.
- I is known in North Indian music as Raga Durga2
- II is the minor pentatonic scale, but also known as the pentatonic blues scale
- III is the major pentatonic scale and known as Raga Bhupali in North Indian music
- IV is known as Raga Mamad Sarang in Indian music
- V is known as Raga Malkauns in North Indian music
There are 3 tones with a fifth-fourth-skeleton, seen in modes I, II and IV.
- II is known as Raga Shivranjani in North Indian music
- IV is known as Raga Hindol in North Indian music
- I is known in Indian music as Raga Hamsa Dhwani
- I is known as Raga Abhogi in North Indian music
All parallel modes (“inversions”) of this are one and the same: this is the whole tone scale. It is one of the completely symmetrical scale structures, and nice as a contrast to diatonic scales, but all by itself rather bleak and unexpressive.
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To be continued: the scales in repertoire?
As this represents original research, here published for the first time, this page will change in due time: at least it will become expanded with more specific remarks, conclusions and explanations. Also it is my intention to find and list repertoire examples for each of the 124+14=148 scales, IF they exist…
Oscar van Dillen ©2015-16