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Peculiar Question Irregular semi-tone of scale. Hi folks, am hoping to achieve a formula or function that can be applied to the general following problem - any feedback appreciated ;

Given two values that are known ;

(A) Fundamental = 1.07 (B) Some note = 1.4437

Where both (A) and (B) are the rational numeric values representing two musical intervals occurring within the same 7 note musical scale. In many cases where (A) and or (B) do not represent standard values of known musical intervals the only way both values may occur in a musical scale is to modify the values of the semitone and whole tone.

For example, if the standard values of the semitone and whole tone are applied to the production of a scale based on the given fundamental (A) ; 1.07 we can see that the value for (B) does not appear precisely on either a major or minor scale only an approximation.

Where standard values are used ; semitone = 1.059.... wholetone = 1.125 ....

Given input numbers, return the major_scale and minor_scale arrays. This is done according to the rules: MAJOR SCALE W-W-H-W-W-W-H MINOR SCALE W-H-W-W-H-W-W

major_semitones = [0, 2, 4, 5, 7, 9, 11]    # Cummulating sum of W-W-H-W-W-W-H
minor_semitones = [0, 2, 3, 5, 7, 8, 10]    # Cummulating sum of W-H-W-W-H-W-W

Scales based on standard semitone, wholetone values fail to return the proper value of (B) ;

[b]MAJOR MUSICAL SCALE FOR ( C# ) IS ;[/b] C# Eb F F# Ab Bb C 1.0700 1.2010 1.3481 1.4283 1.6032 1.7995 1.0099

[b]MINOR MUSICAL SCALE FOR ( C# ) IS ;[/b] C# Eb E F# Ab A B# 1.0700 1.2010 1.2725 1.4283 1.6032 1.6985 1.9065

THE QUESTION IS ; How can modified values for non-standard semitone and whole tone be derived that provide the "best fit" for (A) and (B) to occur within the same scale?

In this example the correct modified values for irregular semitone and whole tone values which produce a music scale where both (A) and (B) occur are ;

WHL TONE = 1.127844 SMI TONE = 1.062

The resulting scale yields (B) at the position of the tritone ;

W 1.07 W 1.20679308 H 1.361074335 W 1.445460943 W 1.630254452 W 1.838672702 H 1.036867988

But how can we derive via a standard formula ;

WHL TONE = 1.127844 SMI TONE = 1.062

When only given (A) and (B) ?

MarnixKlooster ReinstateMonica
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AstroD
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  • The question is hard to understand, partly by the wording, partly for the inconsistent ordering and formatting (round to 4 decimals, then 9 without trailing zeroes, then 3...), partly because suddenly 'mod 2' is applied (1.0099 is used for its double 2.01199, same for 1.036867988 for 2.073735975), partly because the 'solution' yields a 2.058... octave), partly because of musical confusion (those are not the major/minor scales for C#, but enharmonic equivalents). – MarnixKlooster ReinstateMonica Nov 27 '22 at 17:44
  • It feels like all tags except [music-theory] are incorrect on this question. – MarnixKlooster ReinstateMonica Nov 27 '22 at 17:45
  • But this seems an honest question, and I seem to understand it. So I'm not voting to close, and might even try an answer. – MarnixKlooster ReinstateMonica Nov 27 '22 at 17:46
  • About your question: Why do you consider 1.4283 to be "only an approximation" of 1.4437, but apparently 1.445460943 from your solution is not? About your solution: How did you arrive at this? And have you considered using W=1.127294..., H=1.061741...? That reduces the octave ($W^5 \times H^2$) from 50 cents too large to 45 cents, and makes the F# in the C# major or minor scale exactly 1.4437. – MarnixKlooster ReinstateMonica Nov 27 '22 at 18:10
  • Thanks Marnix - I should have expressed this better ; Two notes are considered ; a root note from a 7 note scale and one other note that may be any note associated from the scale degrees from 2 to 7. The two notes are expressed as rational numeric values between 1.0 to 2.0. In some cases these two notes will not fall on a standard major, minor scale that uses a semitone value of 1.059 and a whole tone of 1.125. By modifying the value of the semi tone between a range of 1.05 to 1.09 we can generate a scale where both these values will occur on the same scale. – AstroD Nov 27 '22 at 18:23
  • This scale may turn out to be either major or minor structurally. How do we determine the optimal modified semitone value to get the closest match to the two provided values? Im hoping to end up with a function or formula – AstroD Nov 27 '22 at 18:23
  • If possible Im hoping to arrive at a solution that does not use cents - regarding the solution mentioned, I havent found one. I simply used a spreadsheet to toggle semitone values until I discovered a modified semitone value that returned a near match on a generated scale for both values. Id like to be able to auto-generate a modified optimal semitone value that will facilitate both values of any given root and other scale value – AstroD Nov 27 '22 at 18:27

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I'm assuming from the way you phrase your question, that your whole tone is always exactly twice a semitone.

In general you will only be able to get an approximation, in the sense that 5 whole tones and 2 semitones (or 12 semitones) never exactly add up to an octave ($\;2\;$). The best approximation to equal temperament tuning will be $$ n = [12 \log_2(B/A)] \\ H = (B/A)^{\tfrac 1 n} \\ W = H^2 \\ $$ where $\;[\dots]\;$ is rounding to the nearest integer.

This always makes $\;A H^n = B\;$, by design: in essence your question was really simply, "How can I find an $\;H\;$ so that multiplying $\;A\;$ by it a couple of times, I get exactly $\;B\;$?"

(Also, if rounding were removed, the result would always be exactly $\;H = 2^{\tfrac 1 {12}}\;$, the equal temperament semitone.)

The above gives you a semitone $\;H\;$ that is $\;1200 \log_2 H - 100\;$ cents off of an equal temperament semitone.

In your specific example of $\;A = 1.07\;$ and $\;B = 1.4437\;$ this gives $\;n=5\;$ semitones (so the interval that 'is' $\;B\;$ is the fourth, not the tritone) with $\;H=1.061741\dots\;$ and $\;W=1.127294\dots\;$, where $\;H\;$ is almost 4 cents too large, and the octave is almost 45 cents too large.

MarnixKlooster ReinstateMonica
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