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The Pythagorean system of tuning produces, via succeeding 3/2 multiples of a base note (e.g., C), the following set of successive tone ratios:

C to D: 9/8 D to E: 9/8 E to F: 256/243 F to G: 9/8 G to A: 9/8 A to B: 9/8 B to C: 256/243

However, in order to obtain the value for F, one must either drop down a fourth (i.e., multiply C by 2/3) or continue around the circle of fifths for eleven steps. Would it instead make sense to begin the scale at F or, if it begins at C, to end it on F-sharp? In other words, would it be better, on a keyboard, to start the scale with four white notes rather than the current three? If so, the ratios would be as follows:

9/8 9/8 9/8 256/243 9/8 9/8 256/243

This seems to be where the math leads. Any thoughts are welcome. Thanks!

-Burch

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    @Richard - C>F# is the dreaded tritone, quite dissonant. That's probably a good enough reason in itself to not be the 4th degree of a major scale... Music seems to work better, and is more listenable when the notes don't jar. Although the aug4 is useful to modulate to the dominant.
    – Tim
    Jun 16, 2017 at 21:49
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    @Tim "C>F# is the dreaded tritone, quite dissonant" So is F>B, but nobody has tried to exclude that from a major scale, AFAIK. Anyway, the OP is talking about Pythagorean tuning, which makes tritones a different interval from equal temperament. In some temperaments tritones are not dissonant at all!
    – user19146
    Jun 17, 2017 at 0:50
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    Isn't this essentially the idea presented by George Russell in The Lydian Chromatic Concept? I believe he took fifths to be fundamental, and then noted that the Lydian mode can be constructed by stacking fifths.
    – user39614
    Jun 17, 2017 at 11:57
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    All, thanks for the helpful feedback! David hit the nail on the head with his reference to Russell's Lydian Chromatic Concept of Tonal Organization, which is new to me. Per the Wikipedia entry: "Russell believed that dominant function was the driving force behind all harmonic motion. Russell focuses on the Lydian mode because it can be built with fifths. For instance, to construct a C lydian scale one could list the first seven tones on the circle of fifths starting with C, the desired Lydian Tonic. This process would yield C, G, D, A, E, B, F♯." Thanks again! -Burch
    – Burch
    Jun 17, 2017 at 12:58
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    The bottom line is that however you slice the math, if you adopt the basic idea of Pythagorean tuning to Just intervals with small integer ratios, somewhere you have to "lose" a factor of 81/80 to close the octave correctly. The "traditional" way to do this is have two different sizes of whole tones, 9/8 and 10/9, not to get rid of the perfect 4ths and 5ths between C and F.
    – user19146
    Jun 17, 2017 at 14:16

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What would be the advantage?

F# is the tritone in C, which sounds quite dissonant to most people, so that's a pretty good reason, possibly the primary one.

Keep in mind that you're talking about using the Lydian mode instead of the Ionian mode or major scale, and some pieces do use the Lydian mode. The Ionian mode/major scale is more popular, probably because of the way it sounds.

Personally, I love the fourth scale degree being 5 half steps above the tonic, and I like having a whole step between the fourth and fifth degrees.

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