Since ancient times music has been understood as number made audible. Music brings to the sense of hearing the numerical order of nature. The mind recognises in the beauty of harmonies this order of nature, which is also a pattern in the soul. In music we hear the qualities of the numbers, not simply their quantities, and it is these qualities that move the heart. There is a correlation between the nature of the soul and the nature of the world, and the harmonies of music derive from this correlation, and when we hear music that springs from this nature it tempers the mind and brings it to its natural order and rest. This is why music can restore the mind and heart.
Since harmonies are relationships of numbers, and since beauty shines most clearly in what is unified and whole, the most simple, unified and whole numbers produce harmonies, while complex numbers produce discords. Discords have a place in music, but only when they are resolved to harmonies. The telos or pull of a discord is the harmony it naturally resolves to. Music therefore moves from simplicity and elaborates in complexity, and then returns to simplicity. This is the natural motion of music. It moves in a circle and returns to where is began. This is reflected in another way. Music begins in silence, manifests in sound, and finally returns to silence.
The basis of music is the scale. The musical scale is produced through simple ratios. For example, if we begin with a single note and call that note 1, and if we add to that note other sounds of the same pitch, also 1, we have a unison. This is expressed as a ratio of 1/1 - that is 1 to 1. If we have this same 1 and add to it another note, 2, this brings the simplest harmony ratio of 2/1, which is an octave. What this means is that for every time the first note (1) vibrates once, the second note (2) vibrates twice. It is twice the frequency. This shows us something interesting about numbers. The smallest numerical step, as here from 1 to 2, produces the largest interval, the octave. All the other notes of the scale fall between these two notes or this interval. They are in fact simple divisions of the interval of the octave, dividing the octave only by 2 or by 3.
For example, if we divide the octave by half we arrive at the fifth note of the scale. To put this in numerical terms, half way between 1 and 2 is one and a half (1½). This ratio of 1/1½ may be rounded to whole numbers, which makes a ratio of 3/2. This means that for every time the lower note vibrates twice, the higher note vibrates three times. This interval is called the perfect fifth - it is the notes C and G in C major.
This interval when divided produces the ratio 5/4, which is the major third, the notes C and E in C major. We may halve that interval again, and this produces an interval of 9/8, which is a major second, or the notes C and D in C major.
Notice that we have derived from this simple initial doubling of 1 to 2, and then the division of that by halves, the notes C D E G C. Notice also that these notes contain among themselves other intervals. The interval between D and G, and between E and C' (the higher C). The ratio between D and G is 4/3. The ratio between E and C' is 8/5.
Suppose we divide the octave by one third, instead of one half. This gives us the note F in C major, an interval of 4/3 again - a perfect 4th from C upwards to F, or an interval of 3/2 from F to C'. If we divide the interval between F and C' we have a major 3rd again, an interval of 5/4, giving us the note A in C major. If we halve the interval between A and C' we obtain B, which completes all the notes of the scale, giving us a ratio of 9/8 once again.
So that we can see all the intervals in a very simple way, we may make a map of all the notes of the scale. This map can be based on the number 24 since this number allows us to express all the ratios in whole numbers related to one another.
We pointed out earlier that all these ratios could be expressed in their smallest whole numbers. For example, if we look at the ratio of 48/24 we see 48 is twice 24, and therefore it is a smallest whole number ratio of 2/1. Similarly with 36/24. 36 is half as much again as 24, so is a simple number ratio of 3/2. See how this works: 12 goes twice into 24 and three times into 36. That is 3/2. One more example: 27/24. If we divide these numbers by 3 we see that 3 goes 8 times into 24 and 9 times into 27, thus giving us the smallest whole number ratio of 9/8. So to reduce any of these intervals to their smallest whole numbers we simply find the greatest common divisor. The larger that greatest common divisor is, the smaller the numbers in the ratio expression. For example, 24 and 48 are divisible by 12 as the greatest number, and this produces a ratio of 2/1 - the smallest ratio. Again, 24/27 cannot be divided by a greater number than 3, and this produces a ratio of 9/8 - a large ratio.
Take the interval 45/48. Which is the greatest number both these numbers are divisible by? It is 3. 3 goes 15 times into 45 and 16 times into 48. This produces a ratio of 16/15. Notice that there is no common number that will divide 15 and 16. Therefore 16/15 is the smallest expression of this ratio between SI and DO.
Again, take 27 and 40, the interval RE and LA. This is not divisible by any common number at all, therefore its smallest expression is 40/27.
Those who know a little about harmony may be surprised by this ratio because it is normally regarded as a perfect 5th, like the interval between DO and SOL. It is not, however, a perfect 5th in the natural tuning, which is what these numbers give us. It is in fact a discord, and a very curious one if you listen to it. This interval cannot be heard on the piano or any other "tempered" instrument, where the tunings between notes are all slightly compromised, save for octaves, so that one can play in any key without retuning the instrument.
The best way to explore these ratios is to reduce each of them to their lowest common ratio expressions. If this is done it will be noticed that the number 7 never occurs in any ratio. This includes 27/24 because that is 9/8. One finds that the numbers involved in all the different ratios are 1, 2, 3, 4, 5, 6, 8, 9, 15, 16, 27, 40. One will also notice that all the harmonious ratios employ the numbers 1 to 6, and that the discords employ all the higher numbers. The number 6 is said by the Pythagoreans to be the number of full manifestation, the limit number. In music this is evident in that it is the limit of the harmonious sounds. Also there are only 6 harmonies in music.
This number 6 may be seen to be a limit number in music in another interesting way. Suppose we add octave upon octave, starting from our original number 1. We saw that octaves are doubling. This gives us a sequence of 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, and so on. Now if we add up the digits in these numbers we find a recurring sequence of 1, 2, 4, 8, 7, 5. This recurs every six octaves, no matter how far it is extended. If we add up that recurring sequence another interesting fact about number occurs. The sequence added together makes 27, and 2 and 7 added together makes 9. The number 9 is the "last" or unbreakable number. All multiples of 9 add up to 9, and every number that adds up to 9 is divisible by 9. This law of number holds even if we change the starting number. Suppose we start from 3 instead of 1. This will give us the sequence 3, 6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 6144 etc. This sequence gives us the series 3, 6 recurring, which also always adds up to 9, and each series of 2 octaves and 4 octaves and 6 octaves adds up to 9.
If we had started with 5 instead of 3, we would produce the first sequence again. Thus we see that the law of octave sequences expresses principles about number. This "doubling" of the octave is seen in nature as the process of the division of cells in living organisms.
So far we have considered only the ratios of two notes together. What of three? Those who know a little of harmony will realise that no more than 3 notes can be put together to make consonances. Those notes may of course be repeated in higher or lower octaves, but this does not give us more than three notes in any consonance. According to Pythagoras, three is the number of manifestation - not the limit of manifestation, which is 6 as we have seen. This is evident in music in several ways. We saw earlier that if we take 1 and join another 1 with it we have a unison. Nothing "new" has been born. Again, if we double 1 and have the octave, we only have the same note again, only an octave higher, a ratio of 2/1. Again nothing new has been created. But when we halve the octave and make the three notes DO, SOL and DO' sound together we have the "first" real harmony, and the first triad of sounds, and these are produced with the numbers 1, 2, and 3. Pythagoras also says that numbers are male and female, and that to create something there must be a "marriage" of male and female numbers. This is the case here. The number 1 is neither male nor female, but "prior" to gender. Number 2 is female and 3 is male. All even numbers are female and all odd numbers are male. So it is only when we introduce the first female and the first male numbers that we get the first harmony. We notice also that all the consonances in the octaves are combinations of female and male numbers, or even and odd numbers. It may seem strange to us that numbers have the qualities of gender, yet the harmonies confirm this is so.
It may also seem strange to us that numbers express qualities, yet music shows that this is so. We might say that music specifically expresses numbers as qualities. And to this we may add that our response to harmony springs from an innate knowledge of these mysterious yet precise qualities. Music, however, is not explained by number. No composer composed music according to a numeric formula. Rather it might be more true to say that music explains number, because it is the qualities in the sounds that bare the numbers forth meaningfully and according to their nature.
We have looked here at the mathematical proportions of the musical scale. If these are considered further a great deal more can be seen in them than we have elaborated here. For example the numbers will show how the harmonic series arises, and also how these ratios can be applied to geometry. For instance, the triangle with line proportions of 3, 4 and 5 which, if made into strings, produces the minor chord, or the same as the three highest pitched open strings on the guitar. There is ample material here to experiment and explore with that will lead to all kinds of insights into both number and music.
© Joseph Milne 1996
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