I have recently been struggling with the derivation of the diatonic and chromatic semitones. I have laid out my detailed thinking below. In short, there appears to be a relationship between the diatonic semitone and the perfect fifth and a relationship between the chromatic semitone and the perfect fourth. Specifically:
the diatonic semitone x 1.4238 = a perfect fifth (i.e., 1.0535 x [(3/2)^6/(2^3)] = (3/2)); and
the chromatic semitone x a perfect fourth = 1.4238 (i.e., 1.0679 x (4/3) = [(3/2)^6/(2^3)]).
The common factor in each of the above (1.4238) appears to correspond with raising a note by six fifths [(3/2)^6] and lowering it by three octaves [1/(2^3)].
I am not sure what these relationships mean and/or what they might explain. Any thoughts about what I am missing and/or where I went wrong would be much appreciated.
(1) Pythagorean tuning (i.e., raising the base note and each subsequent note by 3/2 and lowering each into the same octave) produces the following circle of fifths based on C: C=1f (base frequency); G=1.5000f; D=1.1250f; A=1.6875f; E=1.2656f; B=1.8984f; F#=1.4238f; D♭=1.0679f; A♭=1.6018f; E♭=1.2014f; B♭=1.8020f; F=1.3515f; and C=2.0273f (the portion of this last number following the decimal point equals twice the Pythagorean Comma).
(2) Reordering these notes into the circle of half-steps (i.e., lowest to highest frequency) produces the following scale (C Major): C=1f (base frequency); D♭=1.0679f; D=1.1250f; E♭=1.2014f; E=1.2656f; F=1.3515f; F#=1.4238f; G=1.5000f; A♭=1.6018f; A=1.6875f; B♭=1.8020f; B=1.8984f; and C=2.0273f.
(3) Calculating the ratio between each note and the preceding note produces the following: D♭/C=1.0679 (a chromatic semitone; “CS”); D/D♭=1.0535 (a diatonic semitone; “DS”); E♭/D=1.0679 (CS); E/E♭=1.0535 (DS); F/E=1.0679 (CS); F#/F=1.0535 (DS); G/F#=1.0535 (DS); A♭/G=1.0679 (CS); A/A♭=1.0535 (DS); B♭/A=1.0679 (CS); B/B♭=1.0535 (DS); and C/B=1.0679 (CS)
(4) The following six CS and DS pairings are apparent in (3) above: CS-DS (D♭/C and D/D♭); CS-DS (E♭/D and E/E♭); CS-DS (F/E and F#/F); DS-CS (G/F# and A♭/G); DS-CS (A/A♭ and B♭/A); and DS-CS (B/B♭ and C/B).
(5) Mapping the circle of half-steps at its points of intersection with the first pairing in (4) above (i.e., D♭/C and D/D♭ mapped at C and D) yields the following two triangles and sets of side ratios:
(a) triangle D♭-C-G-D♭ with the following side ratios: D♭/C=1.0679; C/G=1.3333; and D♭/G =1.4238; and
(b) triangle D-D♭-G-D with the following side ratios: D/D♭=1.0535; D♭/G =1.4238; and D/G=1.5000.
(6) Likewise, mapping the circle of half-steps at its points of intersection with the other five pairings in (4) above yields the following triangles and sets of side ratios:
(a) E♭/D and E/E♭ mapped at D and E: (i) triangle E♭-D-A-E♭ with the following side ratios: E♭/D=1.0679; D/A=1.3333; and E♭/A =1.4238; and (ii) triangle E-E♭-A-E with the following side ratios: E/E♭=1.0535; E♭/A =1.4238; and E/A=1.5000;
(b) F/E and F#/F mapped at E and F#: (i) triangle F-E-B-F with the following side ratios: F/E=1.0679; E/B=1.3333; and F/B=1.4238; and (ii) triangle F#-F-B-F# with the following side ratios: F#/F=1.0535; and F/B=1.4238; and F#/B=1.5000;
(c) G/F# and A♭/G mapped at F# and A♭: (i) triangle G-F#-D♭-G with the following side ratios: D♭/G=1.4238; G/F#=1.0535; and D♭/F#=1.5000; and (ii) triangle A♭-G-D♭-A♭ with the following side ratios: D♭/A♭=1.3333; A♭/G=1.0679; and D♭/G=1.4238;
(d) A/A♭ and B♭/A mapped at A♭ and B♭: (i) triangle A-A♭-E♭-A with the following side ratios: E♭/A=1.4238; A/A♭=1.0535; and E♭/A♭=1.5000; and (ii) triangle B♭-A-E♭-B♭ with the following side ratios: E♭/B♭=1.3333; B♭/A=1.0679; and E♭/A=1.4238; and;
(e) B/B♭ and C/B mapped at B♭ and C: (i) triangle B-B♭-F-B with the following side ratios: F/B=1.4238; B/B♭=1.0535; and F/B♭=1.5000; and (ii) triangle C-B-F-C with the following side ratios: F/C=1.3333; C/B=1.0679; and F/B=1.4238.
(7) Based on (5)-(6) above, I make the following observations:
(a) each triangle takes the form of one of the following: 1.0535 x 1.4238 = 1.5000 or 1.0679 x 1.3333 = 1.4238;
(b) in other words, (i) since 1.0535 corresponds with the diatonic semitone and 1.5000 corresponds with a perfect fifth, DS X 1.4238 = a perfect fifth and (ii) since 1.0679 corresponds with the chromatic semitone and 1.3333 corresponds with a perfect fourth, CS x a perfect fourth = 1.4238; and
(c) finally, if (b) is expressed mathematically, (i) 1.0535 x [(3/2)^6/(2^3)] = (3/2) and (ii) 1.0679 x (4/3) = [(3/2)^6/(2^3)].