# Best guide to consonance?

It is said that lower fractions = greater consonance. 3/2, The Perfect Fifth, being the smallest fraction, has the highest consonance. However it is also said that 12TET has nice fifths, because they are close to Just fifths. In terms of cents difference, this is true. Just is 701.95 cents, while 12TET is 700. In terms of pitch, many people would not be able to hear a difference. Yet as a fraction, one is 3/2, while the other, unrounded, is 1498307077/1000000000. In terms of consonance, or how nice they sound, this would suggest a huge difference. So what is the better guide to consonance here, the small fraction, or the cents difference to Just?

• Actually, since 7 times the 12th root of two, the equal tempered fifth, is an irrational number, your fraction is rounded. Not by much, but necessarily. Jan 7 '20 at 9:53
• Maybe it's simplification, but to me it's the blend of two notes, sonically. Never mind what the mathematical representation is.
– Tim
Jan 7 '20 at 9:56
• @Tim - agreed. We can't hear the mathematics directly. Jan 7 '20 at 11:32
• as a ratio isn't the unison 1:1 the highest consonance? Jan 7 '20 at 21:12
• @MichaelCurtis, correct. I could have said 3/2 has the highest consonance after the unison/octave, but I didn't think it was necessary for people to understand the question. Jan 7 '20 at 23:55

The guide to consonance here is the amount of difference between the tempered interval and the just interval. How complex the fractional difference is, is not audible; how close the interval is to a small-integer ratio is audible.

Of course, perception of consonance is also influenced by other factors, including the source of the tones, cultural and personal tastes, etc.

• Thanks for your answer. How then do you determine which is the best Just interval to compare it to? For instance, in 5 Limit Just the Tritone is often 45/32, but in 7 Limit Just it may be 7/5. Do we accept 7/5 as the gold standard because it is the smallest fraction possible for that interval? If so, the question is somewhat circular. We are saying smallest fractions are best, whilst knowing some large fractions may at times sound the same. Therefore smaller fractions are our guide, but not definitively best. If instead we accept 45/32 as better than 7/5, then why? Jan 7 '20 at 10:46
• @ItHertz - good question, and I doubt there's a definitive answer. To my ears, 7/5 is more consonant than 45/32, but I hear 45/32 as being more within a scale, which brings its own sensation of consonance: it's "fifthier" than 7/5. Reportedly, most people also prefer major thirds that are a bit sharper than 5/4, but not quite as high as equal tempered, as harmonic intervals. I suspect this is also mediated at least a bit by feelings of scale motion. But this is all very subjective after a certain point. Jan 7 '20 at 11:29