@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 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 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.

Thank you, understood!

@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 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 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 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 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 No. These are acoustic ratios. They are associated with just intonation. Equal temperament is an approximation based on the ratio (or irratio, if you prefer) of the twelfth root of two. In equal temperament, only the octave agrees with the acoustic octave. The fifth is 2^(7/12) rather than 3:2 (1.4983, rounded, instead of 1.5 exactly). The major third is 2^(4/12) rather than 5:4 (1.2599, rounded, instead of 1.25 exactly).

Also, what do you mean by "lower ratio"? The ratio 16:15 is smaller than the ratio 3:2, but the latter is the more consonant interval.

Thanks for opening this chat. So in simple terms, the just intonation ratios i've been using this whole time literally serve no purpose in me composing in equal temperament. Also, when composers are creating compositions they are in fact not utilising just intonation interval ranking but instead equal temperament interval ranking? Could you tell me where to find or supply me with the interval ranking for equal temperament?

I was under the impression that the simpler the ratios/lower digits (2:1 versus 32:27) the more consonant the intervals