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(I will ask this question in musical terms, but this seems to be related to projecting integer vectors onto each other, which I'm unfamiliar with. Perhaps I'm just looking for some existing notation that I'm not aware of, I'm not completely sure.)

I'm given a music interval, e.g. a minor 6th which I'll write as $[\text{m}6]$. Each interval can be uniquely expressed as a linear (integer) combination of fifths and octaves, e.g $[\text{m}6] = -4[5] + 3[8]$.

(Note that $[5]$ and $[8]$ are linearly independent: there is no (integer) linear combination of them that equals to zero, that is, the unison / perfect prime interval $[1]$.)

My question is: How can that integer $-4$ be expressed in terms of $[\text{m}6]$, $[5]$, and $[8]$?

MarnixKlooster ReinstateMonica
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  • So, counting half-tones, you want to solve $8=x \cdot 7 + y \cdot 12$, to get $x=-4$ and $y=3$. That's just a linear diophantine equation: https://math.stackexchange.com/questions/20717/how-to-find-solutions-of-linear-diophantine-ax-by-c – Hans Lundmark Jun 05 '21 at 10:38
  • @MorganRodgers: [5] is a fifth, the musical interval spanning 7 semitones. And [8] is an octave, spanning 12 semitones. And [m6] spans 8 semitones. So the question is all really about $8 = (-4) \cdot 7 + 3 \times 12$. – Hans Lundmark Jun 05 '21 at 11:16
  • @MorganRodgers: No, not quite. See my my very first comment. The question is “how many fifths and how many octaves do you need to make a minor 6th?”. And it's a very standard problem in elementary number theory, which is why I voted to close as a duplicate. – Hans Lundmark Jun 05 '21 at 11:21
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    @MorganRodgers: The way I interpret is is: “The intervals are given, i.e., you know that you want to make a minor 6th, and you know that you want to make it out of fifths and octaves. The answer turns out to be to go 3 octaves up and 4 fifths down. How could one have known that the coefficient for the fifths would be $-4$? Is there a formula for that coefficient, which can be applied to other intervals, say making a major third out of fifths and octaves, or out of fourths and fifths?” But I agree that it's not very clearly formulated... – Hans Lundmark Jun 05 '21 at 11:26
  • Updated my question for clarity, based on earlier comments (for which thanks!). Note how, in the world I'm working in, an augmented fourth is different from a diminished fifth, and $12[5] \not= 7[8]$. So I don't think the Diophantine equation link is relevant? But I'm not sure. – MarnixKlooster ReinstateMonica Jun 05 '21 at 13:29
  • @MorganRodgers Let me try to rephrase. So $[m6]$ can be uniquely written as a linear combination $(-4, 3)$ of the independent 'base vectors' $([5], [8])$. What is the integer "$-4$" in that previous sentence called? And what can I write on the right hand side of $-4 = \dots$? If this were a vector space, it would be a projection, something with an inner product. But I don't see how that applies here? – MarnixKlooster ReinstateMonica Jun 08 '21 at 11:27

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