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On page 100 in Henry Cowell's book (PDF here) there is an explanation centered around C-C#

(C#) contained with in a whole note has four and two-seveths vibrations per second and it will be found that a seven-thirtieths note, which is the corresponding time-value of (C#) will be contained in a whole note four and two-sevenths times , in other words, four seventh-thirtieths notes...

What is a seven-thirtieths note? how does it relate to this Ratio (14:15)?

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Keep in mind that this is Cowell at his most theoretical. As such, there aren't notational analogues for some of the things that he suggests.

This "seven-thirtieths" note is a theoretical entity. Just as a quarter note is really a "one-fourth" note (that is, it receives one-fourth the value of a whole note), a "seven-thirtieths" note will last seven-thirtieths of the duration of a whole note.

The reason that C♯ is a "seven-thirtieths note" is because the ratio from C to C♯ is the same as the ratio of 14 to 15. And if C is a "one-fourth" note, it becomes a simple question of math to determine what C♯ will be: we multiply 1/4 by 14/15 to get 14/60, which reduces to 7/30, and thus our "seven-thirtieths" note of C♯.

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  • Richard your reply to the "seven-thirtieths note" is duly appreciated, I have another question, regarding Slonimsky. what method is Slonimsky using to generate the 7 chord progression that belongs to pattern 10? ( @ the top of page 4.Thesaurus of Scales and Melodic Patterns lapetitedistribution.org/archive/Nicolas_Slonimsky.pdf) My understanding so far is that the pitches that divide the octave C-F#-C into equal distances are ornamented with 2 notes in between ( interpolation) and a Cmajor chord outlines satb, with the ascending form of pattern 10 in the s.
    – M G Easter Jnr
    Commented Jul 31, 2019 at 17:08

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