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Posted by on Jan 29, 2013 in Background, Just Intonation | 0 comments

“Untempered” vs. “Just Intonation”

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are exploring the edges of just intonation, picking up the trails that were abandoned when such music as Ars Nova was superseded by the slow growth to dominance of tempered scales.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.”

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Posted by on Jan 28, 2013 in Just Intonation, The Lattice, The Notes | 0 comments

Cents

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.

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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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Posted by on Jan 18, 2013 in Background | 0 comments

Calendar Page

I’ve added a new page with places and times you can come hear and see this music live. If you’d like to be on my regular mailing list, please let me know through the contact page. I send out a few emails a month, mostly about upcoming shows, occasionally with other news.

I’m excited about the concerts with my friend Jody Mulgrew. And the SLO Down Pub showcase on January 24, with Loren Radis, and the Salty Suites, promises to be splendid. Hope to see you at a show!

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Posted by on Jan 16, 2013 in Just Intonation, The Lattice | 0 comments

Melodic Space, Harmonic Space

Throughout my musical education, I’ve been taught that music happens in a linear space. This is the space so beautifully laid out on the piano keyboard.

Piano-keyboard

Music teaching is organized around scales. In most Western music, the full scale consists of twelve notes, equally spaced. Other scales, such as the seven-note minor and major scales, are subsets of this full, “chromatic” scale. Due to octave magic, a mysterious and crucial aspect of our inner perception, when we get to the thirteenth note, we have multiplied the original note by two, and the sequence starts over again.

So, fortunately for musical analysis, melodic space can be described in one octave. It takes about ten of these octaves to cover the range of human hearing.

On the piano keyboard, melodies look the way they sound. When the pitch goes up, you move up the scale, and when the pitch goes down, you move down the scale. Short distances (the shortest is from one key to the next, a half step), feel short. Long distances (more than about three half steps) feel long. This is a good and useful space for visualizing melody.

Harmony, not so much.

Musical nomenclature, as I’ve pointed out before, has grown like an old city over the years. As music theory changes, bits and pieces of the old terminology are appropriated and redefined by new thinkers. The result is a cobbled-together mass that has a lot of weird contradictions and misleading names.

I think one of the most regrettable bits of confusion comes from the word interval.

The distance between two notes on the keyboard is called an interval. When my melody moves by an interval of a minor third, it has covered a distance of three half steps. When I move by a major third, I’ve covered four half steps. The major interval is bigger than the minor one — that’s why it’s called “major.” No problem! The move feels bigger when you sing it.

The problem comes when you start to think about harmony — two or more notes sounding simultaneously. The word “interval,” with the same connotation of pitch difference, is also used to describe the distance between harmony notes. Yet in the world of harmony, the interval, or pitch distances don’t make any intuitive sense at all.

For example, two notes a fifth apart (seven half steps) sound wonderful when played together. C and G are two such notes. They are closely related to each other, harmonically. So are C and F, which are a fourth apart (five half steps). These are the best consonances there are, except for unisons and octaves.

So what about the note in between them, an interval of six half steps?

Yep, none other than the dreaded tritone, the devil’s interval, definitely a dissonant note.

If the linear scale were the best way to think about harmony, wouldn’t the tritone be between the fourth and fifth in consonance? Why would three notes in a row, next-door neighbors on the scale, be so drastically different from each other harmonically? The scale gives no clue. You just have to remember.

Perhaps there is a more intuitive way to visualize harmony, one that puts harmonically related notes closer to each other, and puts the notes that are harmonically farther apart … farther apart?

I think there is indeed a harmonic space as distinct from a melodic space. This space can be illustrated on the lattice. It’s not a good model for melody — scales do a much better job. But it’s a great model for visualizing harmony — what you see corresponds intuitively to what you hear.

The interplay between these two spaces creates the beautiful dance that is harmonized music.

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Posted by on Jan 13, 2013 in Recordings | 0 comments

A couple of good recordings

Jody Mulgrew and I are putting together a few shows in the next few months, and for rehearsals, I’ve been recording versions of some of my songs. Jody and I blend like brothers, so the emphasis is on the harmonies. I especially like the way Flying Dream and Breakup Songs came out. That’s me singing the high part. I’m so looking forward to performing these with my friend. First show is at Steynberg Gallery in San Luis Obispo, on February 8. FileItem-57643-SteynbergGallery_fullWe’ll also be playing Bazaar Cafe in San Francisco on March 9. Here are the demos:

Breakup Songs BS Jody 2-8

Flying Dream FD Jody 2-8

Enjoy! If you’d like to be updated about these and other shows, please let me know through the contact page, and I’ll put you on the email list. I send something out a few times a month.

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