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An interval holds two pitches (C an G). Those two pitches have a fundamental frequency which represents their pitch names, along with their harmonic series/overtones.

When we turn that interval into a ratio (2:3) which demonstrates that intervals level of consonance from the two pitches wave cycle synchronicity, does that ratio account for the two pitches harmonic series also or just the fundamental?

If the ratio just accounts for the fundamental frequency, is that sufficient enough to establish two pitches wave relationships without taking into account their overtone relationships also?

Thank you.

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    I think the original Just ratios are taken to be from the harmonic series. The harmonics present will depend on the situation so it's hard to say whether their existence should matter when defining an interval. – ggcg Sep 22 at 19:32
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    If the ratio between G and C is 3:2, then the ratio between their n-th harmonic is also 3:2, at least theoretically; in practice, some instruments (e.g. guitar, piano, harp, pizzicato strings) have harmonics that aren't precise multiples of the fundamental frequency. – Your Uncle Bob Sep 22 at 19:39
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    @Seery n = any integer. If e.g. E is 330Hz and A is 220Hz and their ratio is 3:2, then e.g. their 17th harmonics are 5610Hz and 3740Hz, which are also 3:2. – Your Uncle Bob Sep 22 at 20:35
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    @Seery The harmonics of plucked strings are not random multiples, they are 2x, 3x, 4x, ... but they gradually go slightly "out of tune" the higher up they go, depending on the string's thickness and stiffness. See e.g. newt.phys.unsw.edu.au/jw/harmonics.html – Your Uncle Bob Sep 22 at 21:47
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    The only reason why you care about ratios is the overtone spectrum: A 2:3 ratio matches every second overtone of one note up with every third overtone of the other. Basically, the more overtones match up, the more consonant the interval sounds. From the octave (every overtone of the high note matches up) all the way to the tritone (irrational ratio, nothing matches up). The simpler the ratio, the more overtones match up, the more consonant the interval sounds. – cmaster Sep 23 at 5:51
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When we say that the pitch ratio between notes is 2:3, that ratio only expresses the ratio of the fundamental frequencies. However, there will of course be lots of other ratios between the harmonics of those notes which may be relevant to the perceived consonance.

Let's consider two notes each with 3 partials:

One note has a fundamental at 100Hz, and harmonics 200Hz, 300Hz. The other note has a fundamental at 150Hz, and harmonics at 300Hz, and 450Hz.

This would mean that there are actually a number of ratios going on there:

100:200 (=1:2)
100:300 (=1:3)
100:150 (=2:3)
100:450 (=2:9)
200:300 (=2:3)
200:150 (=4:3)
200:450 (=4:9)
300:150 (=2:1)
300:300 (=1:1)
300:450 (=2:3)
150:300 (=1:2)
150:450 (=1:3)

Have I missed any out? anyway, you can see that even with just 3 partials in each sound, there are a whole bunch of ratios that contribute to the overall level of consonance. If we look at the unique simplified ratios, ignoring inverses and the unison, there's still:

1:2
1:3
2:3
2:9
4:3
4:9

Imagine how many more ratios there are in a sound with more harmonics.

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It is fundamentals only. Apart from the mathematical problem (how to reduce a long series of overtone coefficients into a simple ratio) just the fundamental is accessible to normal tuning. The harmonics are called tone color since they are specific to an instrument. Even for piano a different octave will exhibit different overtones.

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I'll make this an answer, because you can't embed a picture in a comment.

Two notes with harmonic overtones, fundamentals in 2:3 ratio

There are two notes, with six partials each, a total of 12 separately sounding partials, many frequency pairs. Clearly only some of the frequency pairs have a 2:3 ratio.

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    Note that every second overtone of the high note matches up with every third overtone of the low note. This perfect match is actually why a fifth has such a characteristic sound to it. And it's also the reason why you hear it immediately when a fifth is slightly too large or small: In that case, the overtones do not match up anymore, making the sound quite dissonant immediately. – cmaster Sep 23 at 5:43
  • Perhaps the graph would be clearer with a logarithmic scale on the frequency axis, since we're comparing ratios? – NobodyNada - Reinstate Monica Sep 23 at 17:59
  • @NobodyNada An extra layer of math stuff to explain, so... no. :) Here you can see the frequencies clearly: 100, 200, 300, 400, 500, 600. – piiperi Reinstate Monica Sep 23 at 18:09
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    Don't all the frequency pairs have a 2:3 ratio? 100:150, 200:300, 300:450... Or have I misunderstood your graph? – Andrew Leach Sep 23 at 21:27
  • @AndrewLeach 100:450, 400:450, 450:500, 100:750, 100:900, ... – piiperi Reinstate Monica Sep 24 at 3:03
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It's only the ratio between fundamentals. Of course, corresponding harmonics have the same ratio as their fundamentals.

The spectrum of overtones of a note depends not only on the fundamental but also on the instrument being played. Flutes have very little sound energy in their overtones; they are about as close as one can get to a pure sine wave with orchestral instruments. Clarinets lack even numbered intervals (clarinets have no octave key; it's a twelvth key.) (Because of irregularities, the clarinet does produce some even overtones.

A piano is so tightly strung (not to meant pianists), their overtones are generally sharper that the overtone series would indicate.

Taking overtones into account would complicate things without explaining much. However, Helmholtz did discuss dissonance with respect to overtones of intervals but didn't really explain things fully.

  • "It's only the ratio between fundamentals. Of course, corresponding harmonics have the same ratio as their fundamentals." This seems to be the general answer. I'm aware of instruments having different amplitudes in their overtones. Your writing on it was interesting to read, thank you! – Seery Sep 22 at 23:43
  • “A piano is so tightly strung” – in particular, it uses quite thick strings. That's really what causes the inhamonicity, because those string have a non-neglectable bending springyness. The thickness does also bring up the mass and that's why they have so much tension, but both are separate physical mechanisms. – leftaroundabout Sep 23 at 8:44
  • Pianos are tempered instruments too, which means their notes are an approximation for every key. – marcellothearcane Sep 23 at 20:01
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Those two pitches have a fundamental frequency which represents their pitch names, along with their harmonic series/overtones.

That is not necessarily true. This web page has an example of a sound with all of the first ten harmonics missing, but is still heard as being at the fundamental pitch. (Scroll down to the section "Pitch is virtual fundamental frequency".)

Pipe organ builders (and organists) have known for centuries that the perceived fundamental pitch of a "note" is not necessarily the same as its lowest frequency component.

MRI scans of brain activity have shown that there are two different mechanisms for pitch recognition, labeled "fundamental pitch" and "spectral pitch", and in individual subjects one or the other method is more dominant. See https://www.nature.com/articles/nn1530 (unfortunately, behind a paywall).

All this can be summarized as "any simple theory based on overtone ratios is wrong".

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I believe it's usually just the fundamentals, because how far in the overtone series would you be willing to go to analyse each pitch or interval set? Depending on the timbre of the sound, or the room you're in, certain overtones might resonate, and others might not. This is acoustically, though. In electronic music, you may have other ways of measuring and analyzing these things.

  • But the fundamental frequency is in itself the first harmonic in the series,so would it not be useless to ignore all the others that absolutely do play a role in how consonant two pitches are? – Seery Sep 22 at 19:55
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    @Seery We do take into account the other overtones. They are why there is a "lower interval limit". music.stackexchange.com/questions/77173/lower-interval-limits – Your Uncle Bob Sep 22 at 20:43
  • I've understood this question i asked with your comment below my post. Thanks again Bob. – Seery Sep 22 at 21:08

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