Musical notes can be mapped onto many different spaces. The two I find useful so far are:
— Harmonic space, the space of the lattice, organized by harmonic connections (ratios of whole numbers).
— Melodic space, the space of the scale, organized by pitch, or frequency.
Both maps show the location of a note relative to a reference tone, the Tonic, the “do” of do-re-mi.
Distance on the lattice could be measured by the number and length of the connections to the tonic, sort of “how many Tinkertoy sticks away are we?”
How to measure distance in melodic space?
One of my favorite music theorists is Alexander Ellis. Ellis was an interesting character, a researcher in phonetics, and the prototype for Professor Henry Higgins of George Bernard Shaw’s Pygmalion (My Fair Lady). He wrote a huge appendix for Helmholz’s foundational book about psychoacoustics, On the Sensations of Tone, in which he laid out a version of the harmonic lattice that is very much like the one I’m using. The appendix was published in 1885.
Ellis proposed dividing each equal-tempered semitone into 100 equal parts, called cents. This gives 1200 cents to the octave. Cents have caught on almost universally as a way to describe and compare pitches of tones.
Cents are a logarithmic unit. Logarithms form a bridge between addition and multiplication. When you add logarithms, you are multiplying in the real world. Adding 1200 cents is the same as multiplying by 2. When you add one cent, you are multiplying by a small number, the same number each time. It’s the 1200th root of two, in fact, a very small number, about 1.0006. Multiply by 1.0006, 1200 times, and you get 2.
The ratios themselves show what the pitch of a note will be, and there’s a formula for translating from harmonic space (ratios, the lattice) to melodic space (cents, pitch). It is great fun, if you’re a geek like me, to plug this formula into a spreadsheet and start exploring the musical spectrum.
For any ratio, b/a, the pitch in cents is:
1200 x log2(b/a)
That’s log to the base 2. A good straightforward explanation of logarithms can be found here. They are a handy concept in the study of perception, since many human senses, including visual brightness, loudness and pitch, work in a logarithmic way. A 100-watt amplifier sounds louder than a 10-watt amp, but it’s nowhere near 10 times as loud. Maybe three times as loud, subjectively? A 10-watt amp is louder than a 1-watt by about the same amount. I have a 1-watt Vox tube amp that the neighbors have yelled at me about. For something to sound “twice as loud,” it has to be moving something like 4 or 5 times as much air.
So let’s run that formula. The untempered major third is a ratio of 5/4.
log2(5/4) = 0.32
x 1200 = 386.3 cents
The ET major third is at exactly 400 cents, 14 cents sharper. This is a clearly audible difference — the ear can distinguish a difference of about 5-10 cents.
Cents give us a language for comparing pitches, and quantifying the differences between them.