# What determines the correlation between musical scales?

I was reading something about the distribution of diatonic scales (Transpositions of C major) in the first movement of Mozart's "facile" sonata. With some calculations, it was stated that C major (duh) contributes the most, then G and D major, and then a little bit of D# major.
The article says "F# major has the strongest negative coefficient, which makes sense in that F# may be thought of as among the least correlated scales with C major."

I'm not an expert in music theory by any means. I know how all major scales can be created from transposing another major scale (keeping the same intervals and stuff). I want to know what this correlation means. Does it simply mean that they have very few (2) pitches in common?

Several people mentioned the circle of fifths so I thought I would edit the post and add it here:

• What happened to the contributions of F major, G minor, D minor, and A minor? (Those are key centers I found in that Mozart movement's development and early recapitulation. I'd easily believe that their scales contribute more than the poorly musically spelled "D# major".) Commented Dec 28, 2020 at 12:15
• Yes yes (Actually I don't know) but let me explain what the paper did, it decomposed the pitch distribution in the piece based on the major diatonic scales. It doesn't say anything about the minor scales. F major had a positive coefficient so yeah it does contribute. I'm assuming if it created the transformation with the minor scales it would end up with your answer.
– Zara
Commented Dec 28, 2020 at 14:41
• Here's the decomposition vector. The i-th element shows how much the i-th major scale contributes where the 1st one is C major: (216, 6, 76, 64, -38, 9, -63, 85, 49, -17, -22, 24)
– Zara
Commented Dec 28, 2020 at 14:44
• What does "contribute" mean here? Commented Dec 29, 2020 at 10:10
• @phoog Using the same vocabulary as the paper here. I'm assuming it just means it occurs more? Like notes from that scale show up more. But as ttw pointed out, "Counts don't show locality." so I don't really believe this decomposition has that much value in the end.
– Zara
Commented Dec 29, 2020 at 11:36

The term correlation refers to a mutual relationship between two or more things. This is not a musical definition but the definition found in a standard dictionary.

It is synonymous with connection, link, or relationship. The use of the term depends on how you define a possible relationship among a set of things. In terms of major scales or key signatures one can define two as being related if they share notes, or have notes in common (but this is not the only way to define a relation).

It has been pointed out that Keys like F and G are different compared to C by only one note. These are referred to as compatible keys and are, no surprise, a 5th or 4th away from each other. C is the 5th of F and G the 5th of C. At any point on the Circle of Fifths wheel the keys on either side of that point share this relationship with the key in question.

Consequences related to this (in my point of view) include the strength of movement from from IV-->I, and V-->I, and the ease with which one can modulate key without creating a jarring effect for the listener. In contrast, trying to modulate from C to F# is possible but would require several changes strung together. A book by Max Reger called Modulation goes through the process of "smoothly" walking from any major or minor key to C, or some home key.

• I'd say scales are related, but correlation is a statistical term that describes similar effects that must not be interdependent but they may have a common source. Now in music I think the common tetrachord or intervals could be considered as a kind of correlation. Commented Dec 28, 2020 at 16:03
• I think both ideas have validity. If I run a correlation algorithm on any 2 major scales looking for the intervals w-w-h-... they will all come up 1. The same will find the tetrachord twice in each, and in every other mode too. If correlate is judged by having same or similar patterns and you include wrap around effects all modes will correlate with Ionian 100%
– user50691
Commented Dec 28, 2020 at 17:29
• IMO, you can remove the "in my point of view". My understanding is that this is a generally accepted aspect of tonal theory. Commented Dec 29, 2020 at 5:52
• I use that phrase because it is not clear whether the things that follow are truly a consequence of the antecedent IMO
– user50691
Commented Dec 29, 2020 at 11:39

I want to know what this correlation means. Does it simply mean that they have very few (2) pitches in common?

This question is unanswerable without access to the full paper and this is not possible without subscription. The paper is mathematical in nature and we can presume that somewhere in the full text there is an explanation of the specific type of correlation being discussed - either this or there is a generally accepted meaning for the term in the specialised field to which the paper contributes. I suggest you search the paper from the beginning to find the first use of the term correlate/tion and see to what it refers.

Talking about correlation makes no sense unless you state the metric. You can't say that A is correlated with B unless you define how you are measuring them both and over what period.

In one sense, all major scale are correlated 100%. For example the correlation between relative pitches of scales, measured logarithmically, will be totally linear, regardless of key.

I wonder if the author is talking about Hamming distances between strings of 1s and 0s. https://en.wikipedia.org/wiki/Hamming_distance

Incidentally when discussing music, one has to distinguish between absolute and relative pitch. Otherwise you end up in the wilderness.

• That was the first time "correlation" was mentioned. Before the Mozart example, it was mostly just talking about, how for example a minor scale can be written as the sum of some other major scales.
– Zara
Commented Dec 28, 2020 at 14:19
• " the sum of some other major scales" this seems rather odd phraseology. What is meant by the "sum" of two scales? Is it the set-theory intersection of the scales or the union? (or something else). I'm beginning to wonder about the credentials of this paper. Commented Dec 28, 2020 at 14:27
• Sorry, I should have explained in more detail. The paper represents each scale in a scale vector of 12 elements. (0 or 1) Where 1 shows the occurrence of a pitch. It uses a linear transformation (Kind of like the Fourier transform) to decompose a scale as the sum of basis elements (diatonic scale vectors in this case) of the transformation. So sum of vectors. (That represent scales)
– Zara
Commented Dec 28, 2020 at 14:34

Check out the 'circle of fifths'. As a key moves to its neighbour, there is one changed note. Look at key C - neighbour key G gains F♯ instead of F♮, on the other side, key F gains B♭ instead of B♮. All other notes stay as is. And that small change continues as we go round the circle.

Now find key F♯ on the wheel. Opposite!

And - while key C and key F♯ may have two pitches in common, they have only one note name (B) in common.

• I think the question is close to the question of related chords: music.stackexchange.com/questions/91711/… Commented Dec 28, 2020 at 10:22
• Have you actually seen the term "correlation" used in this sense? If not then this interpretation would be just speculation. Commented Dec 28, 2020 at 10:39
• @ piiperi: Who has used the term correlation in this thread? Commented Dec 28, 2020 at 11:26
• @AlbrechtHügli The original poster. The question is about the meaning of the word correlation in the context of musical scales. Commented Dec 28, 2020 at 11:43
• I think they used the term "correlation" vaguely. I searched around the web a lot and didn't find anything. Maybe the paper was using the word loosely. But I think Tim's answer and ggcg's answer make sense.
– Zara
Commented Dec 28, 2020 at 14:26

I read the article, however, I'm not sure of how I would use the results. The author represents (as noted in another answer) a major scale by seven 1s in a twelve-vector. Each scale consists of the pattern 101011010101 and its rotations written as a circulant matrix. This matrix happens to be non-singular (has an inverse) so that an arbitrary 12-vector can be written as a linear sum of each row of the matrix. The rows are identified as scales and in the Mozart case, the total counts for each note are transformed into this basis. The numbers show how much of the piece can be described to each scale.

I don't particularly like this interpretation as it can take negative numbers (needed because the note d can be considered coming from the scales of C and E minus G which is nice descriptively but not so useful musically. The author is assuming that big the numbers show "how much" of a piece "derives" from a given scale.

Another problem to me is the consideration of minor scales; this method needs 4 different minor scales (one equivalent to the Dorian mode and one equivalent to the relative major.) That's not how minor scales are used. Scale steps 6 and 7 are mutable, not "borrowed" from another tonality. Likewise, the paper's suggestion of the F# scale appearing could be due to a D major or D seventh appearing as a secondary dominant. Counts don't show locality.

Like most mathematically-oriented music stuff; I don't find much correlation between the math and the music.

• Thank you so much. I'm not an expert but I agree with your points. Problems like that are pretty common when it comes to math/music papers.
– Zara
Commented Dec 29, 2020 at 5:12