Musical Ratio and Musical Proportion
This course will touch on areas of “Music Math,” which adds a different perspective to the traditional conversation about music theory. I think it is very helpful from time to time to look at musical principles from a mathematical perspective, and it helps to link music theory to other areas of the Earlham curriculum.
Musical Ratio and Musical Proportion
Ratio as Relationship
What is a ratio in mathematics? Accordingto the excellent website Mathpage, “A Ratio is the spoken language of arithmetic. It is the language with which we relate quantities of the same kind.” A ratio is, therefore, a relationship between two quantities. If you have $10, and I only have $5, the fact that you have two times more dollars than me is the ratio 2:1. If a kind friend gives each of us $5, you are still richer than me, but the relationship, i.e. the ratio, of our money has changed. You now have $15, I have $10, thus you now have three dollars for every two of my dollars, which is the ratio 3:2.
If we invest our money and we each increase our sum tenfold, we are both happier people, but our relationship has not changed. Your $150 and my $100 remains in the relationship, in the ratio, of 3:2.
Proportion is related to ratio, and simple shows that two ratios are the same. No matter how much our money increases (or decreases), if the ratios remain the same, their proportions remain identical. This matters of course for music, since two notes an octave apart may have different frequencies, but as long as they remain octaves, they are in direct proportion to one another.
I know that all this is likely obvious and trivial, but it’s worth stating this clearly before we move on to the musical application of ratio.
Simple Musical Ratios
A musical interval is a mathematical ratio. As we saw in the legend of Pythagoras and the Blacksmith Shop on the previous page, the simplest ratios, when measured on the monochord, produce musical intervals that are most pleasing to the ear. When Pythagoras measured the hammers from the blacksmith shop, he discovered them to be in exact low-integer relationship to each other. When he then created the monochord and moved the bridge along the string, measuring the relationship of the two distances on both sides of the bridge, he took note of the ratios between the two sides of the bridge and discovered them also to be in precise low-integer relationships with each other.
This straightforward image is worth holding in your mind as you think of simple musical ratios. Here, we’ll derive these ratios a little bit more mathematically.
The simple ratios that I want you to know at this point are
Octave as 2:1.
Perfect Fifth as 3:2.
Perfect Fourth as 4:3.
Pythagorean whole tone as 9:8.
Let’s see how these ratios are derived.
Octave as 2:1 (or in Pythagorean terms, 12:6)
The octave, 2:1, is of course the most basic ratio, or relationship, in music. It occurs naturally when women and men, or men and children, sing together. It is fundamentally the relationship of one thing vibrating twice as much as another thing. There are a number of ways to think of this:
If you play a string, then stop the string at half its length, it will sound exactly an octave higher. If you play a string, then play another string that is exactly twice its length, it will sound an octave lower. Ignoring the physics of physical tension and string construction, this is how a harp is designed. A string twice as long sounds an octave lower. The piano, in this sense, is just a harp with hammers.
If you play a tube closed at one end, and play a similar tube that is twice its length, the second tube will sound an octave lower.
If you measure the tension on a tuned drum head, and then precisely double that tension, it will sound an octave higher.
Regardless of how you think of it physically, 2:1 is a fundamental principle of harmonic relationship. If a string or tube vibrates at frequency ƒ, then 2f will sound an octave higher, and
f/2 will sound an octave lower. This results in an exponential curve as regards frequency, but in a linear curve as regards our perception of octave relationship.
As the frequency of a pitch doubles in value, the musical relationship remains that of an octave. Thus for any given frequency, rising octaves can be expressed by the formula: f * 2^x, where x is a whole number.
So for a frequency of 27.5 (the lowest A on the piano):
An octave higher is 27.5 * 2^1 = A 55. An octave above that is 27.5 * 2^2 (two squared) or 27.5 * 4 = A 110. An octave above that is 27.5 * 2^3 (two cubed) or 27.5 * 8 = 220, and the octave above that is 27.5 * 2^4 or 27.5 * 16 = 440, which is the standard tuning note for the orchestra.
Thus octaves rise exponentially as 27.5, 55, 110, 220, 440, 880, 1760 and 3520, the fundamental frequency of the highest A on the piano. Every octave is twice the frequency of the previous octave.
For the Pythagoreans, they calculated the octave as being a ratio of 12:6, which is a proportion of the fundamental octave 2:1. This allowed for the remaining intervals to be calculated using only positive integers. Let’s look at how that worked.
What we will discover is that the arithmetic mean between the octave is 3/2, and the harmonic mean between the octave is 4/3. These are the intervals of the perfect fourth and the perfect fifth, respectively. When adding intervals together, you multiply their ratios, thus:
4/3 * 3/2 = 12/6 = 2, which is the octave.
This explains why, when adding intervals together that are inversions of each other, they result in the perfect octave, even though arithmetically, 4 + 5 = 9!.
Perfect Fifth as 3:2, from the Arithmetic Mean between the Octave
What I find useful and interesting is that for the earliest music theorists, the perfect fifth and the perfect fourth were both considered to be the mean tone, or the mid-point, between the octave. They were each derived, however, by two different mathematical calculations. The importance of this is to stress the centrality of the perfect fourth and the perfect fifth in all subsequent theories of music, and to demonstrate, mathematically, why they exist as reciprocal inversions of each other.
Taking 12:6 as the fundamental expression of the octave proportion, the Pythagoreans defined the perfect fifth as resulting from calculating the Arithmetic Mean.
The arithmetic mean, or midpoint, between two numbers is the what we normally mean when we say “take the average.” This can expressed algebraically as (a + b)/2. (For example, the arithmetic mean between 4 and 6 is 5, which is obvious, but can be expressed in a formula as (4 + 6) / 2 = 5).)
This is how the Pythagoreans derived the interval of the perfect fifth:
Divide the string of the monochord into 12 equal parts.
Take the octave to be 12:6
Take the perfect fifth to be the arithmetic mean between 12 and 6.
This results in the simple calculation (12 + 6) / 2 = 18/2 = 9.
Thus the ratio of the octave to the fifth is 9 : 6, which can reduced to 3 : 2 (by of course dividing each side of the ratio by 3).
Perfect Fourth as 4:3, from the Harmonic Mean between the Octave
The main point in discussing these musical means is that the Perfect Fourth was also derived from finding the mean, or midpoint between the octave, but instead of using the arithmetic mean, it made use of the harmonic mean.
What is the harmonic mean? Well, in a series, the difference between the first and second terms in the series is related to the difference between the second and third as the first term is related to the third. Algebraically, we could say:
a:c = (a-b) : (b-c) or, put another way–
b = 2ac/(a + c)
Since in the example above, the arithmetic mean between 4 and 6 was 5, the harmonic mean would be 2 * 24 / 10 = 4.8.
While a visit over to the Wikipedia page on the harmonic mean shows its complexity, (as does the page on the Pythagorean means in general), for the purposes of this specific discussion, we need only use this formula to find the harmonic mean between the 12 : 6 octave —
Using the formula b = 2ac/(a + c), we get:
2 * (12 * 6) / (12 + 6) = 144 / 18 = 8.
Thus the ratio of the perfect fourth is 8 : 6, which can be reduced down to 4:3 (by of course dividing each side by 2).
For millennia, this has been diagrammed in the following way:
We see here the octave as 12:6, reduced to 2:1
We see the perfect fifth as either 9:6 or 12:8, which can both be reduced to 3:2
We see the perfect fourth as either 8:6 or 12: 9, which can both be reduced to 4:3.
We can see, too, that the perfect fifth and the perfect fourth are in a reciprocal relationship with one another:
Up a fifth is the same as down a fourth
Up a fourth is the same as down a fifth.
Thus are derived the fundamental intervals of the perfect fourth and perfect fifth as the arithmetic and harmonic means, respectively, between the octave.
(As an aside, and about which we’ll have more to say later, the third Pythagorean mean, the Geometric mean, results in the diminished fifth, which is the irrational number √2 — the Devil in Music, indeed!)
Pythagorean Whole Tone as 9:8, or the space bewteen the two Means
Finally, we can discover the final, essential Pythagorean interval, which is the Pythagorean whole tone.
From the diagram above, we can see the space between the overlapping fourths and fifths. It is expressed here as the ratio of 9:8. This is the fundamental Pythagrean whole tone.
This Pythagorean whole tone can also be derived mathematically as the difference between the results of the arithmetic and harmonic means:
To find the difference between ratios, they must be divided.
3/2 divided by 4/3 is the same as 3/2 multiplied by 3/4 (to divide fractions, take the recipropcal of one fraction and multiply).
3/2 * 3/4 = 9/8, the Pythagorean whole tone.
It is also worth discovering this same Pythagorean whole tone if we add two fifths together, and then divide by an octave. Here’s the the math on that:
Two perfect fifths added together is calculated by multiplying their ratios.
3/2 * 3/2 = 9/4.
(If we use our modern language and think in the key of C major, then this would be C up to G, and then G up to the D, the octave and a fifth above the C).
To bring that 9/4 ratio down an octave, we must divide by 2, or multiply by 1/2.
9/4 * 1/2 = 9/8, our Pythagorean whole tone (C up to D in a major scale).
The Pythagorean Circle of Fifths
Since the Fifth was considered the perfect ratio (as was its reciprocal, the perfect fourth), by means of its being derived by the simplest of mathematical ratios, the Pythagoreans concluded that musical scales could, and indeed should!, be constructed solely through using Perfect Fifths and reducing them down by octaves.
If we start on a low C and go up by fifths, we would end up with the five notes of our basic C major pentatonic scale:
This stack of perfect fifths, all derived from 3:2 ratios (C, G, D, A, E), could be brought into a single octave, resulting in the pitches C, D, E, G and A.
Theoretically, this stacking of perfect fifths could be carried on until one reached B#. On the piano, and on other equal tempered instruments, B# = C, but in actuality, the math doesn’t quite work out. This results in the Pythagorean Comma, the subject of the following page.
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