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I've observed that the harmonic series which are the most consonant pitches of the fundamental such as the note A for this example, with A's first harmonic series being (A) E and C#.. Do not correlate with the consonance to dissonance ratios. In the ratios it would be A,E and D..

  • Harmonic Series - A (root) E (fifth) C# (Third)
  • Interval Ratios - A (root) E (fifth) D (Fourth)

It is also worth mentioning that the following harmonic after the third (C#) is a 7th (G).. While in interval ratios the order would be Unison, Octave, Fifth, Fourth and a Major Sixth.

What is going on here?

Also, could you folks confirm the validity of this interval ranking shown in the image below? Image Source: https://www.pnas.org/content/112/36/11155

enter image description here

Thank you.

Seery
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    Look into David Cope's concept of "interval strength", which is similar to what you're trying to do here. (My answer would be that harmony is only loosely related to harmonics, as exemplified by the eleventh harmonic, which is right in-between two notes.) – Your Uncle Bob Jul 14 '19 at 03:50
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    Where are you getting your information about what are "the most consonant pitches of the fundamental"? Are you saying that because they're the earliest harmonics? – Нет войне Jul 14 '19 at 08:19
  • @topo morto yes, I'm getting it from the harmonic series of a fundamental pitch although the order of consonance for ratios differ quite a bit. Why process (harmonic or ratios) provide the actual harmonic consonance order? – Seery Jul 14 '19 at 16:59
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    BTW I think it would really be worth reading and fully understanding that paper I linked to - at least I think it would be good for me, and probably good for you too! – Нет войне Jul 14 '19 at 19:43
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    What do you mean "confirm the validity of this interval ranking"? They asked a bunch of random people, and took the average ranking. That means it's valid for the average person from that group. There's a chance that the average person in another group will agree, unless the group is non-random, like only trained musicians or fans of a particular genre. But mr. Blue Dot likes minor sevenths as much as major thirds, while mr. Cross put major thirds first and minor sevenths in 10th place; those two will never agree, and I don't know how you're going to make music that appeals to both of them. – Your Uncle Bob Jul 16 '19 at 01:42
  • @YourUncleBob I'm asking about validity because i see different sources with the same intervals hold different ratios but maybe your conclusion of the source of their information is why. I really don't understand how if consonance is dictated by two waves constructive interference, this whole consonance dissonance stuff can be so vague and not at all precise. I've been weeks trying to find the universally accepted levels of consonance for intervals within an octave and i'm still at it.. – Seery Jul 16 '19 at 20:46
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    I don't think you can really have universally accepted rules when human perception and preference comes into play, anymore than you can determine the world's favourite colour. – Your Uncle Bob Jul 16 '19 at 21:19
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    Btw, this old question just popped up in the active questions list, and may interest you: https://music.stackexchange.com/questions/30531/what-are-the-characteristic-intervals-or-scales-of-13-limit-harmony-and-beyond – Your Uncle Bob Jul 16 '19 at 21:43
  • @YourUncleBob thank you Bob, ill check it out now. – Seery Jul 16 '19 at 21:58

2 Answers2

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It's because the fourth is the inversion of the fifth. E is the third harmonic of A, and C# is the fifth harmonic of A, but A is the third harmonic of D.

So the ratio of the ascending fifth (the interval, that is, in this example A to E) is 3:2. The ascending major third (for example A to C#) is a ratio of 5:4, and the ratio of the ascending fourth (A to D) is 4:3. Notice that the odd factor is in the denominator here, because A is an overtone of D rather than D being an overtone of A.

phoog
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    Yep. Another way of looking at it is that the series of most consonant intervals (in the classical music sense) is obtained (with the usual temperament factors) from the harmonic series, but not from 1/1, 1/2, 1/3, 1/4, 1/5, etc, but rather 1/2, 2/3, 3/4, 4/5, and so forth. – Scott Wallace Jul 15 '19 at 10:17
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    @ScottWallace it's more common to express the ratios for ascending intervals rather than descending, but, regardless, the powers of two in the numerator or denominator can also be seen simply as adjusting by octaves. Whether the major third is the distance between the fourth and fifth harmonics or the distance between the fundamental and the fifth harmonic, adjusted by two octaves, is not particularly significant, is it? I rather think the whole idea that the scale is "derived from" the harmonic series is seriously overrated. We don't use intervals based on the 7th or 11th harmonic. – phoog Jul 15 '19 at 13:33
  • @phoog in what way do ratios for ascending intervals differ from ratios of descending intervals? Also, when creating inversions of chords, does the level of consonance/dissonance of a chord shift and if so how does this process work? Thank you! – Seery Jul 15 '19 at 22:55
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    @Seery they are each other's reciprocals. For example, an ascending major third is 5:4 while a descending major third is 4:5. Inversions of chords may be more dissonant to the extent that the fourth is sometimes regarded as dissonant, that is, for reasons of melody or voice leading, but otherwise, that is acoustically, inversions are more or less equally consonant and dissonant. That is, the second and seventh are dissonant, while the third and the sixth are consonant. – phoog Jul 16 '19 at 00:01
  • @phoog i'll ask you also, if we say that the most consonant intervals are P1,P8,P5,P4.. How is it that a chord with P1,P5 and P4 sounds more dissonant compared to a chord of P1,P5 and M3 (C Major Triad)? – Seery Jul 16 '19 at 00:12
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    @Seery It depends on the context and on the listener. If a piece in G major or G minor ends with an open fifth (G-D-G), that doesn't sound more dissonant to me than a major or minor triad. Does it sound more dissonant to you? Or are you talking about a chord like G-C-D? That is more dissonant because you have to look at all the intervals, not just relative to the root. G-C-D has a perfect fourth, a major second, and a perfect fifth. G-B-D has a major third, a minor third, and a perfect fifth. The major second in G-C-D is more dissonant than any of the intervals in G-B-D. – phoog Jul 16 '19 at 01:11
  • @phoog in regards to this quote "an ascending major third is 5:4 while a descending major third is 4:5." would that imply that the descending major third is more consonant than the ascending based off its lower ratio? – Seery Jul 16 '19 at 20:50
  • @phoog "If a piece in G major or G minor ends with an open fifth (G-D-G), that doesn't sound more dissonant to me than a major or minor triad. Does it sound more dissonant to you?" No it doesn't it sound more dissonant to me, i would agree with you. – Seery Jul 16 '19 at 20:51
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    @Seery if the two tones are sounding at the same time there is no ascending or descending. Your question is analogous to comparing which relationship is closer, that between a parent and child or that between a child and parent. – phoog Jul 16 '19 at 20:52
  • @phoog excuse my novice-ness. I'm assuming that the ratios presented in the graph i posted are applicable to our standard equal temperament tuning, yes? – Seery Jul 16 '19 at 20:56
  • Let us [continue this discussion in chat](https://chat.stackexchange.com/rooms/96270/discussion-between-phoog-and-seery). – phoog Jul 16 '19 at 21:01
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The harmonic series is not in strict order of pitch consonance with the fundamental. This can be seen from the fact that there are harmonics that are two, three, four.... octaves above the fundamental, all of which would themselves be very consonant with the fundamental, and yet have harmonics either side of them representing pitches that are less consonant.

Consonance and dissonance are subjective sensations which aren't necessarily even particularly well-defined. Plomp and Levelt's 1965 paper discusses some of the difficulties in pinning down what consonance really is, and discusses how their curve of interval consonance was calculated. I thought it was obtained directly from experiments on people, but on reading the paper, that isn't the case - it's derived from experiments using simple tones (sine waves).

Нет войне
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  • i read a portion of the paper and found it to confirm concepts i've come across thus far. I've added an image in my post of a cons/disso curve i have found.. Could you confirm its validity? I apologise if i seem to be going in circles here. Also i came across compound intervals regarding 9ths,10ths,11ths,12ths,13th,14th and 15ths being extensions of 2nd,3rd,4ths,5ths and so on.. If for example a 2nd is dissonant and a 5th is consonant, how does that measurement of conso/disso correlate with the 2nd and 5ths compound intervals 9th and 12th? Thank you sir! – Seery Jul 15 '19 at 23:01
  • @Seery this is a topic that interests me too! I think I'd need to do my own revision on the derivation of that dissonance curve to really be able to make a stronger statement on the extent of its validity... I will aim to do so at some point! – Нет войне Jul 15 '19 at 23:08
  • Thank you! I also see that the consonance degrees differ from image A to image B once you reach Minor 6th on image B and compare it to the degree order of image A.. Why is this? I've been over a month possibly just on trying to validate this topic and its unbelievably vague in certain areas of the topic. Who would have thought that sonics that sound so perfect in its song format, can be so imperfect in its mechanics! Possibly one of the few arts thats perceived perfection doesn't match its actual mechanics. – Seery Jul 15 '19 at 23:17
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    @Seery I'm a bit confused as to what you mean - I can't see *anything* in image A in your post that says anything about consonance degrees. I think image A just tells you what's on the X axis on image B. – Нет войне Jul 15 '19 at 23:22
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    @Seery as to your point about imperfection - a lot of that is because it involves human beings with all their vagaries! (and I don't think music is alone in this) - The graph you've posted, which represents actual listener assessments, is what I *thought* the Plomp curve represented - in fact his curve for musical tones was calculated from his curve for simple tones. – Нет войне Jul 15 '19 at 23:36
  • Cheers for validating. Music is not alone in this although its imperfection may be uncommon in other arts. I will be studying mixed martial arts (another passion of mine) after this project is complete and i can move on and i have reasons to believe mma will hold perfect mechanics as it's basically the anatomy of the human body in its movement limits which are the basis of jiu jitsu and movements generating maximum force such as in striking. Thank you again topo! – Seery Jul 15 '19 at 23:47
  • @Seery the mechanics might be perfect, but I suspect there's a lot of psychology involved in actually winning a fight! BTW you remind me of these guys: https://www.bbc.co.uk/news/uk-scotland-glasgow-west-48879158 ; https://hcmf.co.uk/laura-jayne-bowler-pieces-never-just-piece-music/ – Нет войне Jul 15 '19 at 23:58
  • Absolutely psychology is also an element. The link won't load and a google search didn't bring up the article! Also a question, if we say that the most consonant intervals are P1,P8,P5,P4.. How is it that a chord with P1,P5 and P4 sounds more dissonant compared to a chord of P1,P5 and M3 (C Major Triad)? – Seery Jul 16 '19 at 00:10
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    @Seery it's like we said in another question - you can't just look at the interval from the root to every other note - you have to look at the intervals between every note and every other note. The minor third between M3 and P5 is more consistent than the second between P4 and P5 – Нет войне Jul 16 '19 at 00:53