: Harmonic Series Phase Difference? Good day stack family, I'm trying to figure out the phase degrees within the harmonic series (pictured below) to understand consonance and dissonance within the harmonic series through

In general, for a combination of harmonics, there are no points on the string where the amplitude is zero.

In any case, you don't "hear" the motion of the string at individual points, but the sound created by the whole instrument vibrating.

I don't know what you mean by "...to understand consonance and dissonance within the harmonic series through the phase of constructive/destructive interference" but I suspect you have a fundamental misunderstanding about something. For a vibrating string, all the harmonics are consonant with each other, since they are all multiples of the fundamental frequency. So I can't guess what you are trying to ask here.

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I came across your question while studying quantum decoherence. But with regard to musical consonance and disonance, it is best to HEAR these relationships and not to visualize them with numbers, scale letters, or diagrams.

Keep in mind that the scale you play is tempered so that the harmonic series does not evolve forever. Tonic, Fifth, third, etc. circle back to multiples of the tonic again. That is almost multiples because tempering alters them to limit the size of the scale.

You need to hear this and not visualize it which will only get in your way.

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I would suggest an experiment that might shed some light on your question. If you can get your hands on an oscilloscope, plug a mike or instrument into the input and watch the signal of your selected tonic on the scope. Then play the tonic in combination with each interval of your selected scale and watch how each interval modulates the tonic you have selected. Possibly, you can see the effect each interval has, and maybe that will give you the insight you seek. Sometimes seeing with your eyes can help you understand what you are hearing with your ears.

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As noted in a comment, the human ear has no phase discrimination. None of this matters to the concept of consonance or dissonance.

In fact, the concept of "in phase" or "out of phase" is only stable for waves that have the same frequency. If they have different frequencies, their relative phase will be constantly changing.

Consider two sine waves, one with a frequency of 200Hz and the other with a frequency of 100Hz.

At t=0, they are in phase, both at 0 degrees, if you will. They return to the state at t=10ms, after the first wave has completed two cycles and the second has completed one. Look at their phases in degrees at 1ms intervals:

t 100Hz 200Hz
------------------
0 0 0
1 36 72
2 72 144
3 108 216
4 144 288
5 180 0
6 216 72
7 252 144
8 288 216
9 324 288
10 0 0

These two waves will be in phase for an instant 100 times every second, the same as the difference between their frequencies. They will also be 180 degrees out of phase with the same frequency.

This phenomenon can give rise to beats in tones that are very close in frequency. For example, waves that differ in frequency by 2 Hz will beat twice a second as constructive and destructive interference alternate with that frequency. When the difference in pitch is greater, the phenomenon can result in something known as a "difference tone" or a "resultant," but, like overtones, it does not depend on phase.

(Anyway, with overtones, the difference tones end up reinforcing the fundamental, as in the example above, or the other overtones.)

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Consonance and dissonance is not caused by the misalignment of harmonics of a single tone but between two separate tones. You cannot look at the harmonics of an A 440 and say look the 7th harmonic is out of alignment so the A 440 is a dissonant tone. Rather you look at two tones and judge whether the combination played together are consonant or dissonant. Helmholtz provided a physics based theory for why we judge intervals as being consonant or dissonant and this theory is based on comparing harmonics and seeing if the difference in frequency is less than the critical band for a human to distinguish the two. When this happens a human cannot tell that there are two distinct tones are hears a muddied sound. The result is frequency dependent . For example a minor second is very distant. Even the fundamentals are close enough to sound muddy. This will be more of an issue in the bass register. A minor second played somewhere st 3000Hz may be less dissonant than the same played at 30Hz. Now a Major 7th is a large interval! So why the dissonance? Because the fundamental of the 7th is close to the first harmonic of the One. You can get a description of the formula and procedure for the analysis from most books of the physics of music. One example is Rigden, Physics and the Sound of Music.

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From your comment:

Would this statement be true? Consonance is created when two or more frequency waves peak and drops are in sync.

I think "are in sync" might not be the best phrasing, because might imply they need to stay in sync to be consonant. That statement would then only be true when talking about waves of the same frequency (which could be seen as consonant, but trivially so).

A better statement might be "consonance is created when waves come into sync at regular intervals", where by "come into sync" we mean they both hit the same given amplitude level (such as both crossing through zero). This would agree with another common description of consonant frequencies as having 'simple ratios' between them; the "coming into sync" happens at the lowest common multiple of those ratios.

You can see that all your waves are in sync at point '0' on your X-axis, and have come back into sync at '1' (one cycle of the fundamental). Then they'll be back together again at '2' and '3' and so on. So in that sense they are all 'consonant' with the fundamental.

(As mentioned in other answers and comments - we don't usually talk about partials being 'consonant' with each other; rather, we talk about them being 'harmonic' (or 'inharmonic'). The word 'consonance' is more usually reserved for talking about the mixing of complex tones that are, themselves, composed of multiple harmonics).

When looking at the harmonic series transverse waves succession, their crests don't line up most of the time at stated degrees

As just mentioned, the fact that they don't line up at all points in time isn't the same as what people mean by the waves being 'in sync' in your statement about consonance.

Two waves will only stay in sync all the time if they are the same frequency (and therefore trivially consonant).

so how would you measure constructive and destructive interference at that point?

Constructive and destructive interference is something that you normally talk about when you have two waves of the same frequency. Depending on the phase difference between them, they may interfere constructively (resulting in a higher overall amplitude) or destructively (lower amplitude than either wave in isolation).

In your diagram, the waves are of different frequencies - any "cancellation" between waves caused by them being on opposite points of their cycle is only going to be momentary - i.e. it isn't akin to the amplitude of the resultant wave 'over time' being the result of constructive or destructive interference. This is because all the waves in your diagram have different frequencies.

I'm trying to figure out the phase degrees within the harmonic series (pictured below) to understand consonance and dissonance within the harmonic series through the phase of constructive/destructive interference.

I honestly don't think that makes much sense! Consonance and dissonance aren't a simple function of phase relationships, because whenever you have waves of different frequencies, their phase relationships are always changing. Likewise, constructive/destructive interference are something that you only observe with waves of the same or very similar frequencies, so it's not of much relevance when talking about waves of different frequencies.

"consonance is created when waves come into sync at regular intervals" is this how consonance is truly produced? Like a root note of a scale and its first octave are at most consonance within the entire scale because of the wave activity of 2:1?

You can look at this either in the frequency domain, or the time domain.

In the frequency domain, two pitches will be consonant if their frequencies have simple ratios.

In the time domain, two pitches will be consonant if their wave periods have simple ratios.

If we plot two consonant pitches (here, 100 and 150 Hz, or 3:2) together....

...we can see that the phases do come into alignment at the same amplitude at regular intervals, but I don't think it's very helpful to think of that as "how consonance is truly produced". I think it's easier to think about simple ratios. The 'phases coming into sync' is a kind of side-effect.

I'm trying to seek the structure of most consonance/dissonance in a scale.... If you could express to me the order of consonance in the 7 pitches of a scale....

If you are most interested in musical notes, rather than sine waves, things get more complicated, because you have to consider the frequency ratios between all harmonics in each note. This produces a curve a bit like this, from sethares.engr.wisc.edu/consemi.html:

However, bear in mind that the precise curve, as well as being somewhat subjective, depends on the harmonic structure of each note - see Dissonance: why doesn't the roughness curve have a dip for complex intervals like 7/6?

while confirming the answer stems from wave alignment between pitches

hmmm.... honestly, I don't think that's a helpful way to think about it. Thinking about simple ratios between frequencies will help you much more, I think.

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