Pythagorean Ratios
Tuning the world: the 3-limit system
At the end of the first chapter, we studied the concept of interval and learned to identify them according to their place within the major scale, measuring from above and below, and identifying them as combinations of smaller intervals. A thumbnail summary of this procedure is:
- A major second is a fundamental step or whole tone.
- A minor second is a half step or semitone.
- A minor third is a major second plus a minor second.
- A major third is two whole tones.
- A perfect fourth is a major third plus a semitone.
- An augmented fourth or diminished fifth is three whole tones and possesses a very dissonant quality when played or sung alone.
- A perfect fifth is a major third plus a minor third.
- A minor sixth is a perfect fifth plus a semitone.
- A major sixth is a perfect fifth plus a whole tone.
- A minor seventh is a perfect fifth plus a minor third.
- A major seventh is a perfect fifth plus a major third.
- An octave is a perfect fifth plus a perfect fourth.
This process is very helpful when it comes to identifying intervals on the written page, but it doesn't tell us much about why our music is arranged in just this way. To explore that more deeply, we must go back to the tale of Pythagoras and the blacksmith shop, and specifically to what legend tells us Pythagoras did upon discovering the harmonious nature of low integer relatioships.
In the above list, the most important bullet involves the octave as being a combination of a perfect fifth and a perfect fourth. This fundamental relationship is in fact the starting place in music theory and deserves a discussion all of its own.
This brings us to a topic that would have delighted Pythagoras, for it is as basic to music as his Pythagorean theorem about the square of the hypotenuse of a right triangle being equal to the sum of the square of the opposite sides is basic to mathematics. This is the discussion of the arithmetic, the harmonic and the geometric mean. We need to see that the above definition of the perfect fourth and perfect fifth is a bit backwards. Yes, you can think of these intervals in those terms. But the perfect fourth and perfect fifth are more elemental than that. They are first on the "Periodic Table" of intervals, you might say. They are the hydrogen and helium of intervals! They came first and form the foundation of all other intervals.
That is because the perfect fifth and the perfect fourth are mirror images of each other, and are born from the attempt to find the mean tone between the two notes of the octave. Let us explore this more deeply.
The three means, the monochord and Perfect fifths and fourths
As the legend goes—and it matters not that this legend is apocryphal: it is a teaching story that transcends its questionable historical veracity—Pythagoras returned home from the blacksmith shop inspired by his discovery of harmonic ratios and set out, like a good proto-scientist, to test out the veracity of his hypothesis. He is said to have created the monochord, an instrument with one string (or pehaps many strings) and a movable bridge afixed to a resonating chamber with ruled measurements cut into the wood. (Imagine perhaps a cello where one is able to take the large bridge that sits dow by the sound holes on the instrment and move it up and down the length of the string.) He was then able to test whether, and in what way, the measured weights of the blacksmith hammers, being at ratios of 1:2:3:4 to each other, created harmonious relationships with each other.
To make the mathematics simple, he divided the length of the string into 12 equal units. From a starting point, he moved the bridge to various points along the length, plucking the lengthening, longer end of the string and discovering the relationship between the various tones. Out of this experiment, he concluded that his experience in the blacksmith shop was indeed valid, and that harmonic ratios based on a lowest prime number of 3 (i.e 1—4) did indeed produce the most harmonious tones, and that these tones could be said to also represent fundamental ratios of both geometry and astronomy. Behold: a multi-disciplinary science was born!
Well, this is the way the story was told during the Renaissance, at any rate. The importance of this story to the present day is that these stories formed the basis for a musical theory which continues to the present day. I think it is instructive to explore the origins of this theory, so let us look more deeply at how the Pythagorean tradition was taught in the Renaissance and subsequently understood for all the centuries since.
If we take a string and divide it by 12 parts, and find the first octave by identifying the ratio of 2:1 as that of the octave, we are left with the second part of the string, measured by the numbers 6 through 12. From here, a series of simple mathematical procedures can be inacted to codify the fundamental ratios of the perfect fifth and the perfect fourth, which in turn added up to the complete octave and which also create the fundamental tone.
This was accomplished by finding both the arithmetic mean and the harmonic mean between the integers 6 and 12, according to the following formulae:
- The arithmetic mean between two points is a simple
averaging of numbers, as we are all accustomed to. It is expressed by
the formula (a + b) / 2. (Take any two numbers, add them and then divide
by two to get the average).
This creates an arithmetic series such as 2, 4, 6, 8 or 3, 6, 9, 12.
- In the case of musical ratios, finding the
arithmetic mean between 6 and 12 gives us the number 9, which sounds
the perfect fifth below the octave tone.
- The ratio of 9:6 reduces down to the ratio 3:2. 3:2 is therefore the fundamental ratio of the perfect fifth.
- In the case of musical ratios, finding the
arithmetic mean between 6 and 12 gives us the number 9, which sounds
the perfect fifth below the octave tone.
- The harmonic mean between two points uses the formula
2ab / (a + b).
To put this in slightly convoluted english, we can say: the difference
between the first and second term in a series is related to the difference
between the second and third as the first tem is related to the third.
- In the case of musical ratios, finding the harmonic mean between
6 and 12 gives us the number 8, which sounds the perfect fourth below
the octave tone. Thus: 2(12X6)/(12+6) -> 144/18 = 8.
- The ratio of 12:9 reduces down to the ratio 4:3. 4:3 is therefore the fundamental ratio of the perfect fourth.
- In the case of musical ratios, finding the harmonic mean between
6 and 12 gives us the number 8, which sounds the perfect fourth below
the octave tone. Thus: 2(12X6)/(12+6) -> 144/18 = 8.
This relationship is tradionally diagrammed as:
If you look carefully at the (somewhat ragged) curved lines, you'll observe:
- The curve of the fifth becomes the curve of the fourth, regardless of whether you begin at the 12 or the 6. Thus a perfect fifth and a perfect fourth is equal to an octave.
- The descending fifth and the ascending fifth, created by the difference between the harmonic and arithmetic means, creates a difference tone of 9:8. This ratio, 9:8 is the Pythagorean whole tone.
To visualize this a little more clearly in terms of an instrument and a string length, imagine that we measure this on the low C string of a cello and mark the cello fingerboard as divided into 12 equal lengths (this happens to conveniently work out to be 2.25 inches, since the cello string is 27 inches long). If you take the 1 as being at the bridge and the 12 as being at the nut (near the tuning pegs), you could visualize it this way, with the top of the diagram being the top of the cello's string, and the bottom of the diagram being where the cello bridge stops the strings vibration, represented by a curved line:
The 6 is at the exact centerpoint of the string, fairly high up towards the body of the cello. The 8 and 9 are on the fingerboad, creating the fourth and the fifth below the octave C, with a whole tone between them. Thus we can see clearly the correctness of the Pythagorean ratios when applied to a modern instrument. The low C string equals to 12, the octave C equals the point of the 6, and the fourth and fifth between them equals the F and the G on the C string. We could then write this in musical notation as (moving it up an octave for clarity):
Viewed in this way, we can now look at the concept of the tetrachord.
Filling in the Tetrachord
The tetrachord is formed by the harmonic mean above and below a given octave note. You can also think of it as the overlappying arithmetic mean, above and below the given octave—it amount to the same thing! While in common practice and much popular musics, we tend to fill in the tetrachord by either a major or minor pattern, there are in fact four patterns possible within our notated music, and it's useful to look at all four possibilities:
We begin to see now a scale pattern emerging. Within the lower tetrachord, we can fill in patterns of tones and semitones that bring four distinct emotional and expressive patterns. Here they are, played very slowly so that you can savor and understand the difference between them:
We can observe:
- The major pattern contains a major second plus a major third above the tonic note.
- The minor pattern contains a major second plus a minor third above the tonic note.
- The phrygian pattern contains a minor second plus minor third above the tonic note.
- The middle eastern pattern contains a minor second plus a major third above the tonic note.
These four patterns form the basic pattern available in the tertrachord system.
- The terms "major" and "minor" are the standard terms for a scale pattern using a major or minor third, respectively.
- The term "phrygian" refers to one of 6 modes that we will soon be learning, and is defined by the lowered 2nd and the lowered 3rd.
- The final tetrachord is not often used in Western classical music, except in the special case of the harmonic minor scale, which we are not yet ready to discuss. On these pages I will refer to this as the "Middle Eastern" tetrachord, because it is a sound we associate with Middle Eastern music and other musics that developed outside of or alongside of the mainstream European tradition.
I strongly recommend you play these the next time your are on an instrument, whether you play guitar or you are a wind player. You can also play them on the keyboard in Practica Musica. Get used to how the tetrachord can be divided up by a variety of steps. Much of what we will do next is rooted in this concept.
When you combine them with the same patterns in the upper tetrachord, various modes and scales are created. Here is how some of the combinations could be played out. The terms in paranthesis are the formal names of the modes that are created. We will covering this in more detail soon, so for now, just begin to gain some familiarit with them.
It's worth playing back the soundfile and hearing how these modes sound. We will be doing more work with this soon.
In the next section, we will investigate further how these can be created using low-integer ratios.
The Geometric Mean and the Tritone
Before we leave the concept of the mid- or mean-point between the notes of the octave, we should also consider the geometric mean. This is expressed as the square root of the distance between the two points. As we understand from Pythagorean arithmetic, if we take the hypotenuse of a right triangle and to determine the distance of that hyptonuse when the opposite sides are both equal to 1, this becomes the square root of 2, which is an irrational number. Harmonically, this square root of 2 is equal to the tritone, which is the mid-point between the octave (6 half-steps or 3 whole steps in each direction). It is interesting, and all that needs to be said here, that this interval of the tritone, which is going to concern is a great deal when we get into harmony, is based on an irrational number, whereas the perfect octave, perfect fifth and the perfect fourth are based on ratios that can be expressed using low prime (3-limit) integers, whereas the ration of the tritone cannot.
The Three Means
We can summarize the three means best by now conceiving of a string that has a theoritical length of 2 feet, as you might find on a harp, or as we might have on our monochord and we'll assign it arbitrarily the note C.
- The arithmetic mean divides the string by a+b/2, or 1+2/2 = 3/2. This gives us the ratio of the perfect fifth.
- The harmonic mean divides the string by 2ab/a+b = 2X1X2/1+2 = 4/3. This gives us the ration of the perfect fourth.
- The geometric mean is √ab or √1X2 which is the square root of 2.
The point of this information can be stated thus:
- The perfect fifth and the perfect fourth are the fundamental intervals of Western music. They are based on lowest-prime number ratios (1:2:3:4) are born as two separate answers to the same question: what is the mean between the 2:1 ratio of the octave. The overlapping of the two means creates a difference tone ratio of 8:9, which gives us our third fundamental interval: the tone (whole tone). From these three ratios, all other notes of the various modes can be created.
We need to explore a bit more about harmonic ratios now and begin to construct some scales based on those ratios. This deserves a new page.
NEXT:3B_PythagoreanScales