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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, but I am not sure why. 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)]).

Furthermore, 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)].

Why is this?

Thanks!

-Burch

closed as unclear what you're asking by jjmusicnotes, Tim, Richard, Dom Jul 12 '17 at 12:45

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.

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    Are you familiar with the concept of Pythagorean tuning? If not, I recommend you check it out; your 1.4238 is a "comma" of sorts. – Richard Jul 5 '17 at 16:22
  • or huygens-fokker.org/scala to mess around with scales and intervals – thrig Jul 5 '17 at 20:01
  • @thrig offtonic.com/synth is a great webapp for the same purpose – Some_Guy Jul 5 '17 at 21:10
  • could you include what definitions you're using for the diatonic and chromatic semitones in the question please? And also make it clear which decimals correspond to which rationals. Thanks :) – Some_Guy Jul 5 '17 at 21:18
  • Thanks for the feedback! Here is a summary of the terms that I am using: *the diatonic semitone = 1.0535 *the chromatic semitone = 1.0679 *a perfect fifth = 3/2 *a perfect fourth = 4/3 *raising a note by six fifths = [(3/2)^6] *lowering a note by three octaves = [1/(2^3)] * raising a note by six fifths, then lowering it be three octaves = [(3/2)^6/(2^3)] = 1.4238 -Burch – Burch Jul 5 '17 at 23:06
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I think I answered my question, at least to my satisfaction. My detailed thinking is below. In general:

(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.

Thanks!

-Burch

DETAILED THINKING:

(1) Pythagorean tuning within an octave (i.e., raising the base note and each subsequent note by either a fifth (3/2) or lowering it by a fourth (3/4) such that it occurs within an octave) produces the following circle of fifths and fourths (based on C):

C=1f (base frequency);

G=1.5000f by raising C a fifth;

D=1.1250f by lowering G a fourth;

A=1.6875f by raising D a fifth;

E=1.2656f by lowering A a fourth;

B=1.8984f by raising E a fifth;

F#=1.4238f by lowering B a fourth;

D♭=1.0679f by lowering F# a fourth;

A♭=1.6018f by raising D♭ a fifth;

E♭=1.2014f by lowering A♭ a fourth;

B♭=1.8020f by raising E♭ a fifth;

F=1.3515f by lowering B♭ a fourth; and

C=2.0273f by raising F a fifth (i.e., 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). The following six CS and DS pairings are apparent: 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).

(4) Each note in (2) above can be determined from its half-step counterparts by incrementally moving up/down the circle of fifths/fourths in accordance with (1) above. For example, D♭ can be determined from C by raising C by 3 fifths and lowering it by 4 fourths (raising C by a fifth generates G; lowering G by a fourth generates D; raising D by a fifth generates A; lowering A by a fourth generates E; raising E by a fifth generates B; lowering B by a fourth generates F#; and lowering F# by a fourth generates D♭). More generally:

(note)=(counterpart)x[(1.5)^n]x[(0.75)^m]

(6) The equation in (4) produces the following values for each counterpart (with the arrows indicating the progression from either the lower note to the higher note or vice versa):

C -> D♭: n=3; m=4; CS

D♭ <- D: n=2; m=3; DS

D -> E♭: n=3; m=4; CS

E♭ <- E: n=2; m=3; DS

E -> F: n=3; m=4; CS

F <- F#: n=2; m=3; DS

F# <- G: n=2; m=3; DS

G -> A♭: n=3; m=4; CS

A♭ <- A: n=2; m=3; DS

A -> B♭: n=3; m=4; CS

B♭ <- B: n=2; m=3; DS

B -> C: n=3; m=4; CS

(7) From (6) above, the following generalizations may be made:

(a) Each chromatic semitone (“CS”) arises from producing the higher note (“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 (“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.

  • Okay, there are a lot of numbers here, but what is the larger point? It seems as though you've gone through an awful lot of work to "prove" something that's already been very well documented and established by many thousands of people over a couple hundred years. You do know that we already have the wheel? Jokes aside, I've voted to close the question as the question (and answer) are both quite vague and are unhelpful to other readers. If you could clarify your point / question / answer, that would be wonderful. – jjmusicnotes Jul 10 '17 at 1:22
  • @jjmusicnotes what's your point here? they're not trying to "prove" an existing fact, just trying to understand how and why tuning systems are derived. The maths above certainly isn't the most concise or clear way to explain why the intervals have the relationship they do, but you can hardly fault someone asking about the derivation of specific intervals in a tuning system for then not giving the most transparent answer to what is actually conceptually quite difficult for most musicians. – Some_Guy Aug 2 '17 at 21:04

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