# How to define “simple” when considering consonance and interval ratios

My understanding is 'simple' ratios for intervals are considered acoustically consonant and this is often used to explain is considered musically consonant.

So, a unison 1:1 (1.000) is simple - it's just 1 to 1 - and therefore consonant.

The octave and perfect fifth seem easy enough to understand 2:1 (2.000) and 3:2 (1.500) respectively. The octave deemed more consonant than the fifth, because the fifth is not a whole number.

But when moving on the the perfect fourth and major third we get 4:3 (1.333) and 5:4 (1.250) respectively and the notion of "simple" becomes unclear to me.

Is the fourth not simple, because the decimal part repeats?

Is the third less consonant than a fifth, because the remainder of the fifth - the .5 part - is closer to 1 than the third's .25?

It seems like numbers involving basic division by half are considered simple and consonant - the ones with decimal vales of .0 .5 .25 rather that .333.

But then consider this list...

m3 6/5 (1.200) m6 8:5 (1.600) M6 5:3 (1.667) M7 15:8 (1.875)

I get that the m3 and m6 are 'simpler' as they only go out to the first decimal place as apposed to the M6 and M7 which go out past the tenths. But should they be considered more, less, or the same simplicity as the major third (1.250)?

Especially unclear to me is how the M6 is simpler than the M7 - (1.667) compared to (1.875). In what numeric way can one be simpler than the other?

Can someone explain the acoustic or numeric meaning of simple with these ratios?

Please let me know if I'm conflating acoustic and numeric ideas.

• Note that some traditions call "simple" intervals any interval up to an octave. This contrasts with "compound" intervals, which are more than an octave. So a ninth is a compound interval, the compound equivalent of a 2nd, which is simple. This doesn't address your question, but I felt it was important to share since the terminology is the same. – Richard Oct 2 '18 at 23:43

'Simple' in this context just means that the numbers involved in the ratio (expressed as an integer ratio or fraction, not as a decimal) are small.

If you want a rough-and-ready idea of 'how simple' a ratio is, you could just take the smaller of the numbers in the ratio.

So 1:1 and 2:1 are both very 'simple' - the smallest number is '1'.

2:3 has a smallest number of 2; 5:3 has a smallest number of 3, and so on.

If we look at a graph of disssonance over the octave... ...we can see that the unison and octave (lowest number 1) are very consonant, the fifth (lowest number 2) is also consonant but less so, and so on.

In reality there are other factors that affect consonance and dissonance, so things aren't quite as neat as this - but I hope this gives the basic idea.

• It sounds like it's vaguely related to Zarlino's senario. – Richard Oct 4 '18 at 1:30

Slightly overlapping topo morto's answer but I would like to point out that the number of decimal places in the ratios is almost certainly irrelevant. That 4/3 requires recurring decimals but 5/4 does not is just an artefact caused by our use of base 10. If we chose another base, it may different. Despite the decimal representation, most mathematicians would regard 4/3 as simpler than 5/4 (as topo describes).

The idea that the simplicity of the ratio determines consonance is attractive but mathematically problematic. If it were exactly so then any interval other than an octave on a well tempered instrument would be highly dissonant as the ratio would be irrational. Even on an instrument capable of just temperament, the tiniest pitch error would transform a perfect fifth into a terrible dissonance.

Topo's graph shows that simple ratios are part of the explanation but it also shows that they are not the full story. A similar graph based on the ratios alone would be a horrible totally discontinuous mess.

I am a mathematician and I like to spot connections between maths and music but I resist the temptation to overdo it and force fit the maths onto music.

• Definitely true to say that "simple ratios" aren't the full story when it comes to consonance/dissonance - factors like the harmonic structure of the notes in question, issues of tuning and temperament (as you mentioned), and biological/psychoacoustic considerations as to how the ear works all play a part. – topo Reinstate Monica Oct 3 '18 at 12:07
• @topomorto gave me the answer re. simplicity, but you express exactly my personal feelings on the topic. A mathematical explanation of consonance/dissonance seems nice but doesn't hold water. Slightly imperfect but acceptable instrument tuning is a great example to the contrary. – Michael Curtis Oct 3 '18 at 16:34