Isn't that picture only for this particular timbre with the given six partials? sethares.engr.wisc.edu/images/image2.gif For e.g. pure sine waves it would look different.

@piiperi yes - we've mentioned that before in conversations with Seery, but you're right to point it out! Again, the remedy for that is to would to be to calculate one's own curve for the timbres in question, as endolith has done here: music.stackexchange.com/questions/64910/…

Moreover, "If this timbre is sounded at various intervals, the dissonance of the intervals can be calculated by adding up all of the dissonances between all pairs of partials." ... what is this based on? I think that's just speculation and an assumption which might be a slightly false premise. Intuitively, at least the lowest pitch has special importance and I find it suspicious that you could simply take the sum of all the dissonances and claim that to be the total dissonance. And how about if there are more than two sounding notes? The whole interval ranking picture feels shady. ;)

@piiperi Yep, I hear you completely! This definition/notion of consonance is only one of the possible definitions, and it may well be it isn't the most 'correct' or useful for all (or any?) purposes. It is a quantified definition, and so can be discussed in the same terms as concepts like interval 'ranking', 'sall!

@topomorto in the ranking i provided in my post edit, it states the ratios of for example a P5 3:2 (The root must perform 3 cycles and the 5th 2 cycles for their crest to coincide creating consonance within a 3 second time frame). With that understanding and knowing that the difference between JI interval consonance/dissonance relative to the root and 12tet consonance/dissonance relative to root, being reasonably similar, would my JI ranking be applicable to my 12tet compositions. This is all intervals consonance/dissonance relative to the root and not other intervals.

@Seery approach this curve with caution. For example, it shows that the minor sixth and tritone are equally dissonant, which is probably about as close to an objectively false proposition as you can get, even in equal temperament.

@phoog would a sine wave graph be more appropriate to accommodate different harmonics for different instruments?

@Seery To me, it would make no sense to use a JI ranking as it would only be valid for chords built on the root - in any case, I don't have access to the equivalent JI graph, but if you have one you can do the same exercise yourself. "do the vertical lines going across the graph represent the notes of a scale in order" - C'mon, you must be an expert in this diagram by now! (But yes! :). Feel free to @ me in the main site chat.

@phoog I totally agree about approaching the graph with caution, for all sorts of reasons - but for the sake of discussion, why do you think the idea that the minor sixth and tritone are equally dissonant is particularly surprising? They both sound 'kinda clashy' to me.... but no huge difference...

@topomorto is the chord G-B-D-F dissonant because there is a tritone between the B and the F? Yes. Is the chord G-B-D-G dissonant because there is a minor sixth between the B and the G? No.

@topomorto a JI version of that graph can be made by moving the vertical lines to the left or right. The m3 and M3 lines move right and left, respectively, to coincide with the local minimum points that they are near. I suspect that the lack of a local minimum corresponding to the m6 has to do with the lack of higher partials in the analysis. In particular, there is no eighth partial, which coincides with the fifth partial of the m6.

@phoog Re. the chords - maybe, but maybe GBDG partly sounds more consonant because D-G is more consonant than D-F, and G-G is more consonant than G-F. GBDF is just a 7th chord, so not a strong dissonance to many ears. Just my personal thoughts, and I'm not disputing that a tritone is usually seen as very dissonant - in fact I drew the purple line to show the equivalance (according to this model) because I thought it might be surprising!

@phoog re. the graph - I reckon we need a widget on the site to plot variations on this graph!

@topomorto I suppose what I'm getting at is that B-G is perhaps rather dissonant in B major, but not at all in G major (even, I would argue, if there is no G sounding below the B). Context matters. Also read my answer regarding melodic considerations: there is a lot of confusion between melodic tension and harmonic dissonance; the two are related but not the same.

@topomorto "It would make no sense to use a JI ranking as it would only be valid for chords built on the root" When you say this, do you mean that since interval ranking is based on root to note intervals, that once you add one interval to a chord, the interval ranking would be irrelevant from lets say a major 3rd to a second interval? If this is the case, is there any formula applicable to overcome this issue?

@Seery Let me put it another way - in any just intonation, the overall comparative level of dissonance for each interval is potentially different depending on which note in the octave you start measuring the interval from. I guess that may or may not affect your ranking. I've suggested elsewhere that you might want to put the idea of the ranking aside, and use the curve values directly instead. That seems even more of a good idea with non-ET temperaments.

@topomorto my curiosity around the interval ranking is pretty satisfied at this point, i simply wanted a yes or no answer if the JI ranking was applicable to 12tet sonically and the answer seem to be yes it is. If you could answer this following question it would be of great context to me.. If i use the interval Major 3rd (root and major 3rd) in building a chord, are you saying that from that point onward it would be useless to use the interval ranking for the next interval in the chord as **the following interval would be measured from the root note and not the major 3rd we had established?

@topomorto i also @ you at the practice room chat!

Apologies topo, it just dawned on me that the curve is in 12tet so evidently it is applicable to 12tet composition. I am so in depth in this topic juggling multiple questions and information at once for the sake of time that i lost track of logic. Thank you!

@Seery yes, the vertical lines on this graph correspond to 12-tet intervals.

"If i use the interval Major 3rd (root and major 3rd) in building a chord, are you saying that from that point onward it would be useless to use the interval ranking for the next interval in the chord as the following interval would be measured from the root note and not the major 3rd we had established?" - as we've said elsewhere, you need to consider the interval from every note to every other note. So if you are considering a chord with 4 notes, you have 6 intervals to take account of.

Thank you for all of this.