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Let's number the scale steps of the major scale $1,2,\ldots 7$, i.e., label them from the tonic upward mod $7$ and then add $1$. With that numbering scheme, let the lowest note of a given diatonic interval be $x$ and the highest note $y$.

Is there an "easy" or at least "manageable" formula/algorithm for the function $I(x,y)$, where $I$ takes values $1,2,\ldots,7$ according as the interval between $x$ and $y$ is a unison, second, third, etc.?

For example, $I(6,4)=6$.

As a further question, could we then modify the formula for $I(x,y)$ into one for $J(x,y)$, where $J(x,y)=(I(x,y),Q)$, with $Q$ the quality (major, minor, perfect, augmented, diminished) of the interval according to some convenient numerical labeling?

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  • $\begingroup$ The formula for $I$ depends on what operators you allow. The binary $\bmod$ operator would be useful (written % in many programming languages). For the "quality" component, however, it seems to me that a lookup table would be far simpler than a formula. $\endgroup$ – David K Apr 16 '20 at 2:51
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For the intervals part, I think that by definition the number-name of the interval is simply the distance, the difference $\;y-x\;$, between scale steps $\;x\;$ and $\;y\;$.

Only with two complications: Intervals are numbered starting from $1$, not $0$; and sometimes one doesn't want to distinguish between a sixth, a 13th, or an inverted third.

That leads to the $\;+1\;$ and the $\;\text{ mod }7\;$ in the formula from the earlier answer $$ I(x,y) \;=\; (y-x)\text{ mod }7 + 1 $$

Now for the quality, I have found the following.

First, we calculate the number of fifths in the interval, which is $\;F(y) - F(x)\;$, where $$ F(x) = 2m - \left\lfloor (m+4)/7 \right\rfloor\times7 - \left\lfloor m/7 \right\rfloor\times7 \text{, where } m = x-1 $$ is the number of fifths (modulo octaves) from the tonic to $\;x\;$. (Note that $\;\left\lfloor \dots/7 \right\rfloor\times7\;$ rounds down to the nearest multiple of $7$.)

Then the quality is $$ Q(x,y) = h(k) - h(-k) \text{,} \\ \text{where } k = F(y) - F(x) \\ \text{ and } h(k) = \max(0, \left\lfloor (k+8)/7 \right\rfloor) $$ where the quality is encoded as

Q(x,y) quality name
... ...
$-3$ doubly diminished
$-2$ diminished
$-1$ minor
$ 0$ perfect
$ 1$ major
$ 2$ augmented
$ 3$ doubly augmented
... ...

There is more behind this, but I cannot expand on that now. For now, suffice it to say that $\;h(k)\;$ indicates how 'major' the $\;k\;$-fifths interval is; and a perfect interval is, in some sense, one that is both major and minor.

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  • $\begingroup$ Thank you, this is exactly the sort of thing I was looking for! $\endgroup$ – Kim Fierens Jun 2 at 15:40
  • $\begingroup$ @KimFierens You're welcome! As an additional remark, things become simpler if you take intervals as the basic concept, and build scale steps on top; then you only need to use $\;I(1,y)\;$ and $\;Q(1,y)\;$, making the formulas simpler. And unlike scale steps, intervals can be added together, although not via the standard arithmetic-- it helps to write an interval $\;y\;$ as (nrOfFifths, nrOfOctaves), with nrOfFifths = $\;F(y)\;$ describing the interval quality per this answer; and as (nrOfMajorSeconds, nrOfSharps), where nrOfMajorSeconds describes the interval 'degree'; and add elementwise. $\endgroup$ – MarnixKlooster ReinstateMonica Jun 2 at 18:42
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Although you don't say explicitly what $\ I(x,y)\ $ means when $\ x>y\ $(i.e. the index of the lower note is greater than the index of the higher note) , I assume from your example that it is the degree of the interval from the note represented by $\ x\ $ to the one an octave higher than that represented by $\ y\ $. In that case, if $$ I(x,y)\stackrel{\text{def}}{=} y-x\hspace{-0.5em} \pmod{7}+1 $$ then the interval between $\ x\ $ and $\ y\ $ will be a unison if $\ I(x,y)=1\ $, and an $\ n$-th if $\ I(x,y)=n\ $ with $\ n=2,3,\dots,7\ $.

For the qualities, I doubt if there's any simpler way to represent them than in a table something like the following: $$ \begin{matrix} \,_{\large x}\backslash^{\large y}&1&2&3&4&5&6&7\\ 1&u&2&3&p4&p5&6&7\\ 2&m7&u&2&m3&p4&p5&6\\ 3&m6&m7&u&m2&m3&p4&p5\\ 4&p5&6&7&u&2&3&a4\\ 5&p4&p5&6&m7&u&2&3\\ 6&m3&p4&p5&m6&m7&u&2\\ 7&m2&m3&p4&d5&m6&m7&u \end{matrix} $$ in which the numbers unaccompanied by a letter represent major intervals. While you could no doubt concoct some sort of numerical formula to represent the same information that's contained in this table, I doubt if it could be made any more succinct.

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