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For the most part, I understand the application of simple ratios when determining consonance/dissonance, but I still don’t really know what makes one ratio more ‘simple’ than another. What really defines how ‘simple’ a ratio is? One thing I’ve noticed is that if you add the two numbers in the ratio together, the smaller the total, the more simple it is, but why? Does anyone have a good understanding of what a simple ratio really means?

  • Yes, 'simple' here is equivalent to 'small numbers' – PiedPiper May 25 at 20:49
  • Why would small be simple? 2 is the simplest non-trivial ratio, an octave. In terms of consonance and dissonance bigger is better, or rather as far away as possible from the bottom not and its octave. I think the context might be rational rather than small or large. – ggcg May 26 at 12:35
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The basic idea is that (supposedly) interval rations sound less dissonant if the largest number in either the numerator or denominator has smaller factors that other cases.

Octave: 2/1 big factor 2 Fifth: 3/2 big factor 3 Just Third: 5/4 big factor 5 Pythagorean Third: 81/64 (or something close, I'm not sure) big factor 81

A few quick computations will show that one can't have all intervals in a scale close to good if major and minor chords should have their "ideal" ratios, 4:5:6 and 10:12:15.

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  • Why a "Pythagorean" third? There are other thirds with rational relation to the 1? – ggcg May 26 at 12:37
  • The "Just Third" is 5/4 and there are tempered thirds. Equal tempered thirds would hae a ratio of 2^1/4 to 1. These are just name used by the "temperamentalists." – ttw May 26 at 13:25
  • My point is that you are focusing on a definition of a third that supports the notion of "simple". It isn't clear to me whether this is the historical derivation of the term or just an example. If the former that is interesting and could be highlighted better. If the latter then it seems like speculation or cherry picking. The Just 3rd is "simple". It is not clear to me that the OP provide enough context to answer. – ggcg May 26 at 13:44

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