The familiar 12-note scale is also called the chromatic scale. There are many ways to generate an untempered version of such a scale.
The oldest way seems to be Pythagorean tuning. When you multiply a frequency by 3, you get a new note, an octave plus a fifth above. Multiply by 3 again, and you get another new note, and so on. Dividing by 3 gives you another new note, the perfect fourth, dividing by 3 again delivers another new note, and so on. Do this enough times and you can generate a 12-tone scale.
This scale forms the central, horizontal spine of the lattice.
The central six notes of the Pythagorean scale are highlighted. If you keep extending to the left and right you get all 12 notes, and then even more.
Trouble is, once you’re more than two or three steps away from the center, the notes are definitely dissonant. The ratios just get so big that the ear can no longer hear them as harmony. The Pythagorean major third, for example, is the next note to the right of the 6+, just off the central lattice. Its ratio is 3x3x3x3, or 81/1. The octave reduction trick allows you to divide by 2 until it’s in the same octave as the tonic, so the ratio becomes 81/64.
Harmony is built from small, whole number ratios, and these are not small numbers. Here is what a Pythagorean major third sounds like:
Ah. Using the prime number 5 allows a more consonant chromatic scale, with much smaller ratios. Here it is on the lattice:
All of these notes have been covered in previous posts. Each one has a unique personality. Some are more consonant, some more dissonant, but all twelve have small enough numbers in their ratios to be perceived as harmony by the ear.
Here is the scale, compared with the 12 notes of equal temperament.
When I first drew this scale, on graph paper, I was startled by how narrow the major-minor half steps are, compared with the equal-tempered versions. Take a look at the b3 (minor third) and the 3 (major third). They are only 70 cents apart, not the 100 cents of the piano scale. Same with the b6-6 and b7-7. It’s such a small move in pitch, such a large move harmonically!
The Mozart movies I made a month ago or so illustrate the major-minor pair quite well. The blandification of equal temperament has obscured a beautiful detail of harmonic music. So much is gained with ET, and so much is lost or obfuscated.
Here’s how that tuning compares with the equal tempered scale:
The black numbers show the pitches of 12-tone equal temperament. They are equally spaced, like inches on a ruler.
The red numbers show the tuning of the untempered major scale. They are spaced in the way they naturally turn out when you generate them with small whole number ratios. As is so often the case with the natural world, they don’t line up too well with the nice human grid lines we love.
The way I see it, when you play in equal temperament, you’re playing the grid lines on the map.
When you sing or play the untempered notes, you are visiting the actual territory.
I’ve read that it’s not possible to combine the two, but I disagree. It’s a matter of showing the ET instrument who’s the boss. My favorite example is Ray Charles. Here’s a video from 1976. He’s playing the piano, laying down those grid lines, and the rest of the band is too, but when he sings, his voice owns the sound, and the sound becomes him. A great, dominant singer will infuse the whole combo with that soul.
Another great example is Ella Fitzgerald. Want some goosebumps? Check this video out.
In harmonic space, the clearest name for each note is its ratio — 5/3, 3/2, 4/3, etc. Precise and unambiguous. But the ratios don’t give a very good idea of how to go about playing or singing those notes.
There are a few instruments, such as the left hand keyboard on the accordion, that are organized for harmonic thinking.
The rows are arranged in fifths, and the first two rows are a major third apart, just like the lattice.
For most music making, though, we need to know the pitch of the note. Instruments and voices tend to live in the world of melodic space — scales and pitches.
Here’s what the simplest, most harmonically consonant major scale looks like on the lattice:
One can convert the ratios to pitches using the formula for cents. The ET values are in parentheses.
1 = 1/1 = 0 cents (0)
2 = 9/8 = 204 (200)
3 = 5/4 = 386 (400)
4 = 4/3 = 498 (500)
5 = 3/2 = 702 (700)
6 = 5/3 = 884 (900)
7 = 15/8 = 1088 (1100)
Note that the 1, 2, 4 and 5 are very close to their ET equivalents. Most ears would be unable to tell the difference.
The third, sixth and seventh, however, are all noticeably flat. Or perhaps I should say their ET namesakes are noticeably sharp.
I think this explains a lot about rock music, which depends heavily on power chords (roots and fifths with no thirds) and 1-4-5 chord progressions. The notes of a 1-4-5 power chord progression are 1, 4, 5 and 2!
As Eddie Van Halen said in his terrific Guitar Magazine interview, “Really, the best songs are still based on I-IV-V, which is so pleasing to the ear. Billy Gibbons [of ZZ Top] calls me now and then, and he always asks, ‘Eddie, have you found that fourth chord yet?’ [Laughs].”
Of course the I-IV-V is inherently satisfying, it’s that great rocking chair between reciprocal and overtonal territory, in the most consonant part of the lattice, as close to the tonic as you can get (harmonically).
But I think the special appeal for rock music is that, in ET (and guitar is essentially an equal-tempered instrument), the 1, 4 and 5, and also the fifth of the 5 (the 2) are all in tune. Whatever one might think of ZZ Top’s simplicity (and some do scoff), it’s undeniable that they are fiercely in tune, and harmonically their music strikes a deep chord in the psyche, pun intended.
If the bass and rhythm guitar stick to those roots and fifths, the voices and lead guitar can play all the other notes, because they can be bent and wiggled until they sound right.
This works just as well in traditional country music. Let the bass player nail down those roots and fifths, and the voices (and the fiddle) can sing in-tune harmonies as sweet as you please.
Even though I love just intonation, I have a couple of problems with the term itself.
One is grammatical. It’s a noun, and sometimes I want an adjective, as in “the just intonation version compared with the equal tempered version.” Kind of awkward. How else would you say this? “Justly intonated”? “The version in just intonation”? I haven’t found a construction that satisfies me.
The other reason is cultural. If you search “just intonation,” and start reading, you will get the distinct impression that just intonation is something avant-garde, esoteric, on the fringes. It’s as though equal temperament is the basic system of music, and just intonation is a modification of it. The word “microtonal” has similar connotations.
In fact, equal temperament is the newcomer, a development of a few hundred years ago that facilitated the flowering of a particular kind of music in Europe, and has spread, I think, because it makes so many things so much easier.
Equal temperament is built upon just intonation, not the other way around. If I put my music in the “just intonation” or “microtonal” category, I’m in great company — Harry Partch, Ben Johnston, Kyle Gann. These composers are exploring the edges of just intonation, picking up the trails that were abandoned when such music as Ars Nova was superseded by the slow growth to dominance of tempered scales. Ars Nova is amazing music, terribly neglected now. I like it better than either earlier or later European music — some of it sounds like jazz or bluegrass. Check out this exquisite piece by the group Ensemble PAN, performing some of the last of such music, from early 15th century Cyprus.
I’m not a classical composer, I’m a folk-pop singer-songwriter. I’m interested in such things as modulation, and exploring the edges (especially the world of the prime number 7). But my interest in JI comes from wanting to play music that is more accessible by virtue of being in tune, and thus having a more direct route to the heart and soul. My interest is in communication, and in musical joy. Untempered music simply speaks more directly to my heart.
Think of Ladysmith Black Mambazo on Paul Simon’s Graceland album. I get goosebumps even listening on these tiny computer speakers. Untempered music is not avant-garde at all. It’s the ancient miracle of resonance and joy that happens when we hear in-tune harmony.
Of course I still need a noun, and I’ll continue to use “just intonation” when it’s the word that works. But I have my adjective. I’m calling my music “untempered music.”
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.
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