Heptatonic Scales

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Earlier Attempts to Synthetic Scale Formations
Since the early twentieth century certain theorists and composers sought to expand the number of available synthetic scales for composition. In 1907 the Italian composer Ferruccio Busoni (1866–1924) as part of an examination of the tonal system based on what he called “the two Series of Seven,” meaning the major and minor keys in relation to the chromatic scale, attempted to write all the possible heptatonic (seven-tone) scales in his book Sketch of a New Esthetic of Music. Busoni states:
That some few have already felt how the interval of the Series of Seven might be differently arranged (graduated) is manifested in isolated passages by Liszt, and recently by Debussy and his following, and even
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Mason published an article explaining through the use of algebra of matrices, another system to obtain all the heptatonic pitch collections that confirmed Barbour’s results, that is, 462 scales. In his paper Mason provides all the Busoni scales written in letter notation, but he fails to provide the complete Barbour list. It is important to note that included in the 462 pitch collections are the major scale and its modes, and the harmonic and melodic minor scales, that is, the non-synthetic heptatonic scales. The total number provided by Barbour and confirmed by Mason includes all the possible seven-note scale formations (synthetic and non-synthetic).
From the list of 462 heptatonic scales, there are a limited number of them that contain a different intervallic relationship. All these scales are going to be associated to a parent scale, for example, the Dorian mode is related to the major scale; they both share the same interval relation and, although inverted, in essence they are equal. Since each heptatonic parent scale derives seven modes (like the major scale), the total amount of parent scales that result from the list is the product of 462 divided by 7, which is 66 (i.e., 66 parent scales and 396 related modes = 462 heptatonic

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