1

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.

Thanks!

-Burch


DETAILED THINKING:

(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)].

closed as unclear what you're asking by leftaroundabout, Richard, Tim, ttw, Dom Jul 5 '17 at 14:41

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

  • 2
    As much as I like any discussion about frequency ratios, I'm voting to close this one because it's completely unclear what you mean. Please get to the point as to what relationship you mean, why you reckon it is there in the first place. And avoid writing out those decimals, they're just noise; use fractions instead. If you're talking about some “triangle”, then perhaps a sketch would also be useful. – leftaroundabout Jul 4 '17 at 19:02
  • (a) Each chromatic semitone (“CS”) arises from producing the higher note of a half-step counterpart (“HN”) by raising the lower note (“LN”) by three fifths and lowering it by four fourths. In other words: HN=(LN) x [(1.5)^3] x [(0.75)^4]=(LN) x (1.0679) OR CS = (HN)/(LN) = 1.0679 (b) Each diatonic semitone (“DS”) arises from producing the lower note of a half-step counterpart (“LN”) by raising the higher note (“HN”) by two fifths and lowering it by three fourths. In other words: LN=(HN) x [(1.5)^2] x [(0.75) ^3]=(HN) x (0.9492) OR DS = (HN)/(LN) = 1.0535. -Burch – Burch Jul 9 '17 at 21:44
  • Ok, and perhaps there's something really interesting going on there, but I'm not going to through all these cryptic symbol definitions, and I doubt anybody else will. Again, best sketch out your ideas in a graphic, and most of all phrase a clear question that can unambiguously be answered. – leftaroundabout Jul 10 '17 at 12:00