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I'm currently reading Henry Cowell's New Musical Resources and On page 98, Section 6,(TEMPO) in part 2 (RHYTHM), is a paragraph of for instances. Directly, I'm grappling with the SECOND FOR INSTANCE ...

"By the correct application of a change of tempo, a change of the rhythmic key of time or metre can be made.

For instance, if a quarter-note at M. M. 96 equals (C), it is evident that if the tempo is changed to M. M. 72, the quarter note will be of different value, namely the equivalent of (G). In this way the key tone of the whole time or metric system can be changed at will, and many simplifications of practice can be made. if, FOR INSTANCE, the rhythmic chord of C (in the the ratio of 4:5:6) has been struck, and it is desired to strike the chord of (F), this can be done in the key of (C) only by the use of three-sixteenths notes against three twentieths notes against eighth notes. If, however the tempo be changed from M. M. 96 to M. M. 64, the chord of (F) can be expressed by quarter notes, since by means of the change of tempo the key will have been changed to that of (F) "

I understand the striking of the Rhythmic (C) Chord and it's ratio (5:6:4) and my grasp of the Whole note Series (3/16 . 3/20 . 2/8) concept is fairly solid. the bit I'm slightly confused by is the striking of the Rhythmic (F) chord in the Rhythmic key of (C).

How is it that the Ratio for the (F)Chord (3:5:2) .. and, I'm assuming this Ratio after a good stare at the Scale Rhythm Chart on Page 99. [C:C-1:1 . C:C#-14:15/(C:Db-15:16) . C:D-8:9 . C:Eb-5:6 . C:E-4:5 . C:F-3:4 . C:Gb-5:7 . C:G-2:3 . C:Ab-5:8 . C:A-3:5 . C:Bb-8:15 . C:C-1:2], . .arrived at using the Overtone Ratios and or the Simple Overtone Ratios?

Also .. in the ratio relationship of (3:5) why , again [according to the chart on p.99], is the (C) given as the (3rd)partial and the (A) as a (5th)partial in regard to the Overtone Series?

Lastly..

The Changing of the Tempo from MM96 to MM64 with the Key change to (F), again out of confusion I'm guessing the Tempos are related by 16 as a base? and the Rhythmic Chord of (F) expressed by quarter-notes, against fifth-notes, against sixth-notes(4:5:6) in the rhythmic key of (F) is now the Tonic key conducted by the tempo MM64?

This Post is mostly Directed at Pat Muchmore, but I will be super grateful to anyone willing to help.

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    Well I'm looking forward to the answer to this one. I've read it four times and I still don't understand what it says!
    – JimM
    Mar 11, 2019 at 16:34
  • @JimM My thoughts exactly :) I bet it will end up being a good question, though.
    – user45266
    Mar 11, 2019 at 17:09

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I don't own this book, so I unfortunately can't consult the page references you gave. But I'll try to offer some insights, at least until Pat gets here :-)

If, however the tempo be changed from M. M. 96 to M. M. 64, the chord of (F) can be expressed by quarter notes, since by means of the change of tempo the key will have been changed to that of (F) "

This is done because of the ratio between C and F. Since C is a perfect fifth above F, C is a 3:2 relationship to F. Another way of thinking of it is that, in the harmonic series of F, F is the second partial and C is the third partial. So he's able to switch from "the key of C" to "the key of F" by creating tempos in a 3:2 relationship of 96:64. The music at MM=64 is thus in the key of F.

Also .. in the ratio relationship of (3:5) why , again [according to the chart on p.99], is the (C) given as the (3rd)partial and the (A) as a (5th)partial in regard to the Overtone Series?

He has apparently switched to the F overtone series of F, F (an octave higher), C, F, A; hence C and A are the third and fifth partials.

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  • All very well and good, but the point is..?
    – Tim
    Mar 11, 2019 at 18:27
  • @Tim Cowell was just theorizing some possible avenues of composition, that's all. Ideas like this one can be traced back at least to Helmholtz.
    – Richard
    Mar 11, 2019 at 18:30
  • PDF of the book can found here (link to PDF is contained on this page, but it takes a long time to open, at least on my phone): ubu.com/historical/cowell
    – b3ko
    Mar 11, 2019 at 20:57
  • @ richard cheer for the insight... i have another fairly straight forward conundrum. on page 100 in cowell's book(see attached PDF) 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) ?
    – M G Easter Jnr
    Mar 28, 2019 at 0:22

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