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As a computer scientist, I think of each element of musical analysis as a computation we have chosen to do on the music for some reason. I'm not saying I understand how humans decide what the meter is, but I know that we hear a piece (our input) and somehow determine the metrical structure (our output). There are possible analyses that we care about, such as the interval between pitches (defined as the result when you subtract one MIDI number from another), and those we don't, such as the product of pitches (defined as the multiplication of two MIDI numbers). At first glance, from a purely theoretical perspective, the two analyses seem equally valid, and yet obviously interval is a much more important feature to specify than product.

Can we think of a way to formally show that intervals are more important than pitch products? One way I can imagine justifying this belief is by proving through experiments that we perceive of two instances of the same interval as very similar, and don't think of two sets of notes with the same product as necessarily similar at all. What other experiments could we run that would provide real reason to believe that we are right in studying intervals and not products?

Or rather, can we think of aspects we can point to of our body of existing music that confirm that we should care about interval structure rather than pitch structure? We can say that, given the corpus of music that exists, we are used to a given interval being predictive of what will happen next - after a large leap up, we expect a small leap down - which is not true of the product of pitches. Are there other statistical or mathematical properties of existing music which "prove" that the idea of an interval is more useful than the idea of the product of a pair of pitches?

Edit: Realized "product of pitches" was unclear, as addition in frequency space is multiplication in pitch space - I meant that, while MIDI notes are a useful concept and multiplication in the frequency space is a useful concept, multiplication of MIDI notes is not.

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    Sorry, what's the "product of pitches"? – Dekkadeci Aug 8 '18 at 0:21
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    Can you better define the product operation you have in mind? The idea of product/division (of fundamental frequencies) is actually embedded in the idea of intervals (not the addition). – Allan Felipe Aug 8 '18 at 0:33
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    ...and I still don't see how this would be useful, or that this helps to identify qualities that we can hear. Or maybe I am just not seeing your point yet.... – David Bowling Aug 8 '18 at 1:25
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    I’m afraid I don’t follow this question very much all. Melodic intervals are not predictive. They are artistic. You wouldn’t look at one corner of a painting and see some blue there and then predict that there will be some particular color in the middle of the painting. Also, nothing in music theory is defined based on any MIDI note numbers. MIDI is just a way to assign numbers to music, it has no bearing on the actual structures or concepts that make up music. – Todd Wilcox Aug 8 '18 at 3:02
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    @alephzero The question does not talk about voltage or amplitude multiplication. I have all kinds of ways to multiply voltages in my non-portable home rig. That’s very different from multiplying frequencies. – Todd Wilcox Aug 8 '18 at 12:09
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We have to be a bit careful with the word 'interval' - there are a number of different ways of talking about relative pitch, and the word 'interval' might have different nuances of meaning across those. But in the context of your question, I would say that we care about intervals because:

  • A given interval between two notes corresponds to a certain frequency ratio between two notes

  • We know from experience that a given frequency ratio between two notes is heard subjectively as more or less 'the same difference' in pitch, whatever the absolute frequencies of the note pairs. This is perhaps unsurprising given that the way the human ear is structured seems to correspond to the notion of 'octave equivalence' - the way that each time you double frequency, you've subjectively 'come full circle' to the next octave.

  • Experiments have also shown the same frequency ratio between two notes will create a similar quality of consonance or dissonance between two notes - again unsurprising as the ratios of harmonics between the notes will be the same (simplistically speaking).

  • We may learn to associate certain emotions with particular intervals

The bottom line here is that a particular interval is significant because it corresponds to a particular objectively measurable frequency ratio between two different notes. Methods we use to denote pitch such as Midi note numbers or note names are concerned with giving us a way to easily spot this ratio - the distance from C to E, or a difference of 4 between two midi note numbers, should always correspond to the same frequency ratio. However, the absolute values used in those pitch systems are, to an extent, arbitrary; Just as we could have note names "starting" at A rather than C, MIDI note numbers could all be 3, or 12, or 1000 higher than they are now and they would have the same utility.

The difference between two MIDI note numbers therefore relates to an objectively-measurable real-world concept (that of frequency ratios) which also tends to be perceived in a subjectively consistent way.

Multiplying midi note numbers would make about as much sense as multiplying temperatures on the Fahrenheit scale. For a start, the scale is somewhat arbitrary, and secondly, even if you convert the temperatures to an absolute scale (such as Kelvin), it's unclear what real-world concept that multiplication relates to. (And furthermore, even the offset Farenheit scale relates objectively to temperature; pitch is to an extent a subjective concept, which makes it even harder to imagine what the multiplication would 'mean').

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    The analogy to multiplying temperatures is perfect. I know that multiplying frequencies didn't make any sense, but I couldn't explain why. Nicely done. – Wayne Conrad Aug 9 '18 at 18:17
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    @WayneConrad glad someone understood it :) I always find these 'deep' questions make me want to see if I can still articulate anything... – topo morto Aug 9 '18 at 20:13
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Look at it this way: if you are considering how two tones relate to one another, the frequency ratio specifies what interval you hear (an octave, a fifth, and so forth) but not the absolute frequency. The product, on the other hand, specifies how high or low the combination is, but not the interval. So for instance if you have a 5/4 ratio between the two tones, it will be heard as a major third, no matter how high or low. But if you have a product of 20 between your two tones, the interval could be anything, but the pitch of one tone would constrain the pitch of the other. If you start with a pitch of 5 for the first tone, the second will be 4. If you raise the 5 to 6, then the second will go down to 3 1/3. And so forth.

The thing is, though, I doubt very much if this product relationship is audible, unless you have perfect pitch, so it's very unlikely to have any musical significance whatsoever.

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Midi note numbers are used for describing an abstraction of pitch in the form of a logarithmic fundamental frequency ratio with 12 steps per power of 2. This abstraction has been made in order to capture essential properties of sounds used as Western music notes.

That means that only operations sensibly related to the underlying abstraction are likely to deliver any useful results.

Your question is akin to "how does the meaning of an English text change when we replace each letter in the alphabet with its next letter?". You may find some interesting patterns like when using any unrelated structured material as a random number generator, but there is no inherent meaning conveyed by the change.

If you want to experiment with sound using various mathematical operations, you are much more likely to get interesting results by working on the original signals rather than an abstraction like Midi note events. Analog synths (and in effigy their digital cousins) make experimenting on that kind of thing comparatively accessible.

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One point might be that the Fourier Analysis of functions (notes in the case being considered) is additive. One combines notes to make chords so a major chord (to use a simple example) has the basic frequency of Sin(4x)+Sin(5x)+Sin(6x) when analyzed (ignoring higher harmonic things and inharmonicities of physical objects). One rarely uses Sin(4x)*Sin(5x)*Sin(6x) as this isn't what is heard nor does it mathematically break sounds into sums of orthonormal functions.

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