4

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 the phase of constructive/destructive interference.

I understand that phase degrees in a wavelength starts at 0°, moves to 90° at a quarter the wavelength, 180° at half wavelength, 270° at 3/4 the wavelength and 360° at the end of a wavelength.

enter image description here

My question is as follows..

When looking at the harmonic series transverse waves succession, their crests don't line up most of the time at stated degrees, so how would you measure constructive and destructive interference at that point?

Many thanks folks!


i see a lot of confusion here as to what i'm trying to ask. I will try to be more clear as it is fundamental that i receive this answer for the quest i am on.

I am a music producer.. I came to the conclusion that there must be a reason why certain notes sound better together than others and i turned to the purest form of music for the answer, physics.

What i am trying to get out of this question is as follows..

I want to look at a scale on my piano roll in my daw and know exactly, out of the 7 pitches in a scale, which notes hold the most consonance (stableness) within that scale and which notes hold the most dissonance (unrest) so i have complete knowledge and therefore control over my tension and release in my compositions as opposed to guessing what "sounds good". The closest answer i got was integer ratios which i have concluded hold that answer but i want to go a step further and dissect those ratios to understand it in an even purer form. That quest thus far led me to wave interaction as that is where the ratios stem from and which i had thought was a matter of crest/trough matching up between intervals. Can you guys assist me in concluding my question? Thank you

  • 4
    The phase differences between the harmonic components of a complex periodic sound wave are inaudible to humans (the human ear does not have a phase discriminating mechanism) so those phase differences are not considered or studied in the context of musical timbre or human hearing. Also, the harmonic series is inherently consonant with itself. “Dissonant” overtones or partials are not harmonic nor are they part of a harmonic series. They are often called *inharmonic partials”. Complex sounds composed of completely inharmonic components are aperiodic and often called “noise”. – Todd Wilcox Jun 15 at 1:55
  • 1
    The acoustics of musical instruments are a lot more complex than what that diagram suggests. If you really want to approach music via physics I'd recommend that you read e.g. Eric Heller's "Why You Hear What You Hear". press.princeton.edu/titles/9912.html – Your Uncle Bob Jun 15 at 5:51
  • @ToddWilcox - interesting comment. Care to add some and turn it into an answer? – Tim Jun 15 at 9:19
  • Thanks Bob, i checked out the book but didn't find the information i was seeking. – Seery Jun 17 at 19:38
5

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....

100:150

...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 http://sethares.engr.wisc.edu/consemi.html:

Plomp/Levelt curve

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.

  • I'm guessing from your answer that two notes that are very nearly in tune with each other will produce the beats we hear, as their consonance, due to being very nearly the same frequency waves, makes the volume pulsate. – Tim Jun 15 at 9:28
  • 1
    @Tim exactly - the 'beating' happens as we move from constructive interference (waves in phase) to destructive interference (waves out of phase) and then back again. – topo morto Jun 15 at 9:30
  • topo, "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? If so, how does phase then not dictate consonance if phase is the meeting of crest and troughs between two pitches? I'm trying to seek the structure of most consonance/dissonance in a scale and the degrees in between and not specifically the harmonic series if that makes more sense. – Seery Jun 17 at 19:58
  • topo, "Two waves will only stay in sync all the time if they are the same frequency (and therefore trivially consonant)." Yes understood, but given that each 7 pitches in a scale vary in wavelength per second, does that mean that the reason there is a most consonant pitch out of the 7 and a most dissonant pitch out of the 7 pitches in a scale, is due to crest and trough alignment? If you could express to me the order of consonance in the 7 pitches of a scale, while confirming the answer stems from wave alignment between pitches, id be immensely grateful as this is my question really. thanks – Seery Jun 17 at 20:03
  • 1
    @Seery I hope my edits are helpful! – topo morto Jun 17 at 21:00
7

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.

  • 1
    Would this statement be true? Consonace is created when two or more frequency waves peak and drops are in sync. The most consonance would be found in a frequency wave that is most in sync than any other harmonic of the fundamental. The most dissonance would be produced with a frequency wave that is most out of sync with the fundamentals frequency wave. Between those two opposites are a spectrum of consonance and dissonance based off wave synchronicity. im going to sleep and will reply to your reply soon. Thanks bro – Seery Jun 15 at 2:33
  • 1
    @Seery - I don't think your statement is completely true. Two waves where the crests of one correspond to the troughs of the other produce destructive interference that results in silence (which I'd argue is neither consonant nor dissonant) or a quieter note. Some would argue that destructive interference involves out-of-sync waves. – Dekkadeci Jun 15 at 6:41
  • 1
    It's really destructive interference as that can occur for a perfect unison (which is a consonant interval). This can happen if the two sources are 180 degrees out of phase, an experiment that can be done with speakers and a tone generator. For a minor second played with both sources in phase you get beating between the tones that are near each other. If I remember correctly it's the beating that relates to dissonance, not destructive interference. – ggcg Jun 15 at 23:12
  • ggcg, i understand misalignment is between two separate tones which is what my question revolves around. I have used the harmonic series under the basis that it holds intervals within itself but i should have been more specific. I'm trying to seek the structure of most consonance/dissonance in a scale and the degrees in between. How could i find this answer? If you suggest integer ratios, then what are those ratios based off exactly? Many thanks! – Seery Jun 17 at 19:45
  • Dekkadeci, you're absolutely right that 180 degree waves coming together cancel each other out creating silence. But lets say that a crest and a trough 180 of each other cause silence, if we slightly shifted them, it would no longer create silence but maybe dissonance, or would it simply just be a loss in amplitude at most? thank you – Seery Jun 17 at 19:49
3

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.

  • Well, lets say the fundamental frequency to the 5th harmonic has less consonance than the fundamental to the octave which is the 2nd harmonic. This consonance is produced by constructive interference between the two waves (f+h2). I'm trying to measure phase difference between the fundamental and its successive harmonics. – Seery Jun 15 at 1:31
  • Exactly @ggcg lets say the 2nd harmonic has a 90 degree phase against the fundamental, what about the 5th harmonic that doesn't land on 90 degree phase but somewhere between 0 and 90. How do you calculate its phase degree? – Seery Jun 15 at 1:58
  • "there are no points on the string where the amplitude is zero": it is zero at the fixed ends of the string. – phoog Jun 17 at 5:01
2

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.

1

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.)

  • thanks for your insight phoog. much appreciated. – Seery Jun 17 at 20:09

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.