Talking about Tuning
I'm scheduled to teach "anything" to my seminar on Wednesday. I though I should tackle tuning. This talk is just written and the sound samples and diagrams have not yet been generated.
what is Just Intonation?
Lou Harrison said that "Just Intonation is the best intonation." An intonation is a type of tuning. Just tuning is a tuning that uses fractions. In just intonation, pitches are set using whole number ratios. To understand this, let's look at the harmonic series.
The fundamental is the base frequency. (sound sample)
The first overtone is twice the base frequency. It's relationship to the base frequency is 2/1. In other words, the base frequency * 2 = the first overtone. this makes a perfect (or just) octave. (sound sample)
The second overtone is 3 times the base frequency. It makes an octave plus a fifth. this fifth is perfectly in tune with the base frequency and the first overtone. (sound sample) but is an octave too high to use in a scale between those two notes. We can divide it by 2 to make it an octave lower. this new pitch, the fifth between the base frequency and the octave, is related to the base frequency by a ratio of 3/2. (sound sample) The three comes from it's place in the harmonic series. The two comes from dividing it down to be in the first octave.
2's and octaves
All notes in the first octave, will be between the ratios of the base frequency, which is 1/1 and the octave, which is 2/1. If something is too small, it is below 1/1, and if it's too large it's over 2/1. We can transpose it to the correct octave by multiplying or dividing by 2.
Therefore 2's are very important for transposition, but they don't change a pitch, except by octave. If the base frequency is C and we multiply it by 3, we get a G. If we multiply it by 3/2s, we also get a G, but in the first octave. so 2's are "for free" you can multiply and divide by them any time you need to change the octave and you will still have the same note as before, just an octave higher or lower.
the inversion of a fifth is a fourth. So we can invert the fifth fraction to get a fourth. the inversion of 3/2 is 2/3. But 2/3 is too small. It is not between 1/1 and 2/1. We can multiply it by 2, to get 4/3, a perfect just fourth. (sound sample)
You can do this with any tuning fraction. Invert it to find the inversion, then multiply or divide it by 2 to put it in the correct octave.
3/2, the perfect, just fifth is a ratio made up of small, whole numbers. Small numbers sound more in tune because they are lower in the harmonic series. 3/2 is the most in-tune sounding note that you can get aside from the perfect octave.
There's a story that pythagoras was walking by blacksmith shop and heard very harmonius sounds. After experimenting with the smiths, he discovered two excellent intervals, 3/2 and 9/8.
9/8 is a major second and since it still has small numbers, it sounds really good. (sound sample)
From this, he hypothesized that good rations were made up of powers of 3 over powers of 2 and their inversions. You know that the circle of fifths will eventually take you through all 11 notes in an octave. According to pythagoras, you can use this to tune all the notes. First, turn the first two strings as a perfect 3/2 fifth. Then tune from the 3/2 to the next fifth, a 9/8. then tune from the 9/8 to the next fifth, the 27/16. Notice that everyone of these ratios is a power of 3 over a power of 2.
You can create a chart of these (pass out handout) called a tuning lattice. a lattice of powers of 3 over powers of 2 is called a Pythagorean tuning lattice. The line on your handouts at the top is a pythagorean tuning lattice. Below that, is chart of them in oder of the notes in the scale. Notice that E, the third is not a small number ratio. It is 81/64. This was considered ok at the time because thirds weren't considered consonant. Notice also, that the octaves don't line up. The octave, instead of being 2/1 is 243/128.
this is a sound sample of the tuning lattice going around the circle of fifths. (sound sample). This is a sound sample of it climbing the scale diatonically and then chromatically (sound sample). And this sound sample shows the difference between 243/128 and a 2/1 octave.
That last example demonstrates why mixing in other numbers than just three is a good idea. People often use 5's, 7's and sometimes higher prime numbers like 11's. Your tuning system draws it's name from the largest prime number that you use. A tuning that used 2's and 3's is a 3-limit tuning. One that uses 2'3, 3's and 5's is a 5-limit tuning.
(if I have time)
(Draw on blackboard) this is a tuning lattice of 5's. This note 5/4 is a just third. The ratio has much smaller numbers than the pythagorean third. This is the pythagorean third (sound sample). This is the 5-limit third (sound sample). there's almost a quarter-tone difference between them (sound sample).
When you are drawing lattices, every new prime number gets a new axis. so a tuning lattice could be thought of as an N-dimensional array, where N is the number of prime numbers. this one, with 5's and 3's is a two dimensional array. If we added 7's, we'd need a new axis and we'd have a three dimensional array.
We can add notes to our lattice that are multiples of 5's and 3's. (draw on blackboard) All of these notes are the note right below it multiplied by 3/2s. Remember before, that multiplying fractions was raising them. Like 3/2 * 3/2 makes a note a fifth above. So because all of these notes are multiplies by 3/2s, they are all a fifth higher than the notes below them.
this is useful for two reasons. One is that 6/5 is the smallest numbers ratio we've yet seen for d#, the minor third. (sound sample). So adding lines like this helps us find extra ratios. the other thing that it's good for is transposing. All of these notes are the same as the ones right below it, but raised by a fifth. So you might use this when you modulate to a new key.
Uses of Lattices
composers use lattices like these to figure out what tunings they want to use in their piece and then manipulate them for key changes and transpositions. they then use this to give instructions to instrumentalists or to program them into synthesizers.
One composer that uses tuning lattices to figure out how to tune her instrument is Ellen Fullman. She uses larger prime numbers to tune the many strings of her Long String Instrument. This track is based on a sweep of the harmonics of a C chord. (sound sample)