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.
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'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.
When comparing the relative consonance of intervals sporting different ratios, whichever ratio has the lower smaller value is the more consonant. This is because the higher note (represented by the higher number) agrees with the lower note sooner, that is in fewer cycles of the lower note. For example, The major 6th (5:3) is more consonant than the major 3rd (5:4), because the lower note of the major 6th needs only 3 cycles to line up with 5th cycle of the upper note, whereas the major 3rd requires 4 cycles to line up with the 5th cycle of the upper note. If the smaller numbers of both ratios are the same, then whichever ratios upper number is smaller is the more consonant interval. For example, the perfect 4th (4:3) is more consonant than the major 6th (5:3). A perfect 4th (4:3) has fewer disagreements (the first three 3 of the upper notes 4 cycles) than does the major 6th (where the first 4 cycles of the upper notes 5 cycles disagree). Note that the ordering of consonance between intervals exactly follows the ordering of intervals in the harmonic series. The harmonics of the major 6th (5:3) appear sooner than those of the major 3rd (5:4), because the 3rd harmonic (bottom note of major 6th) comes sooner than the 4th harmonic (bottom note of Major 3rd).
The more complex ratios employed by equal temperament do not pose the problem many argue it does. A study of primate ears by Kadia and Lang determined that the ears of primates respond most robustly to intervals of simple ratio (as described above). Significantly, this study also determined that as a tempered interval approaches the size of a perfectly tuned simple ratio the ear gradually responds more robustly. This observation establishes that intervals near the size of a simple ratio do stimulate the ear more robustly than those intervals less close in size to consonances. So mildly tempered consonant intervals are still perceived as consonances by primates.
When comparing the relative consonance of intervals sporting different ratios, whichever ratio has the lower smaller value is the more consonant. This is because the higher note (represented by the higher number) agrees with the lower note sooner, that is in fewer cycles of the lower note. For example, The major 6th (5:3) is more consonant than the major 3rd (5:4), because the lower note of the major 6th needs only 3 cycles to line up with 5th cycle of the upper note, whereas the major 3rd requires 4 cycles to line up with the 5th cycle of the upper note. If the smaller numbers of both ratios are the same, then whichever ratios upper number is smaller is the more consonant interval. For example, the perfect 4th (4:3) is more consonant than the major 6th (5:3). A perfect 4th (4:3) has fewer disagreements (the first three 3 of the upper notes 4 cycles) than does the major 6th (where the first 4 cycles of the upper notes 5 cycles disagree). Note that the ordering of consonance between intervals exactly follows the ordering of intervals in the harmonic series. The harmonics of the major 6th (5:3) appear sooner than those of the major 3rd (5:4), because the 3rd harmonic (bottom note of major 6th) comes sooner than the 4th harmonic (bottom note of Major 3rd).
The more complex ratios employed by equal temperament do not pose the problem many argue it does. A study of primate ears by Kadia and Lang determined that the ears of primates respond most robustly to intervals of simple ratio (as described above). Significantly, this study also determined that as a tempered interval approaches the size of a perfectly tuned simple ratio the ear gradually responds more robustly. This observation establishes that intervals near the size of a simple ratio do stimulate the ear more robustly than those intervals less close in size to consonances. So mildly tempered intervals are perceived as consonances by primates.
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.
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