Is there an "easy" formula for calculating the species and quality of the musical interval between two notes?

Question

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?

Answer 1

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.

Answer 2

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.

Answer 3

(For the record, total musical noob also trying to make convinient formulas to avoid remembering stuff - this is for intervals within an octave)

(given x is the lower/root pitch note wise, regardlesss of letter) Example:

f(x,y) = 
    if Y>X
        return dist(X,Y) + 1 # +1 is to handle starting at 1 (A=1)
    else 
        return 8 - dist(X,Y)

    f(C,F) -> Y>X -> dist(X,Y) + 1-> Y - X + 1 -> 6-3+1 = 4th Interval
    f(F,C) -> X>Y -> 8 - dist(X,Y) -> 8 - (6-3) -> 8 - 3 -> 5th interval

    If Y>X but x is of lower letters, say A-D you can use 

    f(B,G) --> (A = B - 1 -> G - 1 = 6th, less hassle)
    Any bigger X just count it, its max 3 steps if Y>X    
    

    
    **Note sum is always 9 together.** 

For X>Y, We can internalize this table to calculate more easly:

dist(x,y) + 1	Quantity
2	7
3	6
4	5
5	4
6	2
7	2

1. 2<->7
2. 3<->6
3. 4<->5

Example:

F(G, D) -> dist(G,D) -> (7-4+1) = 4 --> pair of 4<->5 [Always equal 9 together] --> 5th Interval

Not sure this is easier at all, but I learned it quiet thoroughly ^ _ ^

Interval Quality: I don't have a convinient formula for the quality but I use this:

Major Second: Accidental matches, except root E/B where it’s raised one
Major Third: Accidental raised one, except root F/C/G where it matches
Perfect Fourth: Accidental matches, except root F where it’s lowered one
Perfect Fifth: Accidental matches, except root B where it’s raised one
Major Sixth: Accidental matches, except for root A/E/B where it’s raised one
Major Seventh: Accidental raised one except for root F/C where it matches

Always calculate the major/perfect interval and then apply any minor/aug/etc, super efficient :)

(source: https://www.musical-u.com/learn/spell-intervals-fast-tips-tricks/)