Tortoiseshell Music Home of composer Nathan Curtis

The past couple of weeks, I've been putting a lot of work into this site, and I still have plenty more to do before I'm ready to unveil it to the public. I enjoy working on the site for the most part, but it's also occasionally stressful. Diving headfirst into web design, barely knowing HTML and having no prior experience with CSS, PHP, SQL, and other related acronyms is a bit of a challenge, even when I have a great set of tools to work with. So, to clear my head, I thought I'd work on composing for a little bit.

For a while, I've been intrigued by just intonation, and I'm finally dipping my toe into the water. Let me say up front that just intonation (JI) is one of the most mathematically fertile areas in music. You get number theory: JI is all about the comparison of various products and ratios of whole numbers, and the reason why we historically had to make compromises in our tuning systems in the first place is tied to the Fundamental Turkey of Arithmetic. You get group theory: some tunings can be constructed to be isomorphic to the free abelian group on n elements, with a set of relations. You get linear algebra: that same tuning is also isomorphic to a set of n-dimensional lattice points bordered by some hyperparallelepiped. You get topology: either way, when you're actually looking at where that group or lattice lies in the pitch continuum, you homomorphicaly map it to the unit circle. It's possible to do a lot of music without every dirtying your hands with mathematics, but not if you're a modern composer working in JI.

Basically, just intonation is the practice of tuning musical intervals according to (preferably small) whole number ratios like 3/2, 4/3, 5/4, and the like. Most musical instruments today, and consequently most music today, is tuned according to the idea that every half-step -- the distance between two immediately adjacent notes on a piano keyboard -- should be of equal size, and this results in every half-step having a ratio of the 12th root of 2, which is irrational. So in order to perform music in just intonation, contemporary musicians have a limited range of options:

1) Make their own instruments, specifically constructed to be in tune in some variety of JI. This is the route most famously followed by Harry Partch, and was at first the only option JI composers had.

2) Compose for synthesizers, or other electronic instruments whose pitch can be absolutely controlled. In terms of available pitch inventory, this is the most versatile option, though obviously a more recent development

3) Use or adapt existing musical instruments to produce pitches in JI. This method can be implemented with varying degrees of success. Fretless stringed instruments have the ability to produce notes across the entire pitch continuum; all the performer needs to learn is where to put their fingers. Keyboard instruments can be retuned, but this is a major undertaking, and the instrument is essentially limited to playing in a single 12-pitch JI scale indefinitely. Nonetheless, some notable works have been written for justly-retuned piano, most notably The Well-Tuned Piano by La Monte Young. Woodwind instruments can approximate JI pitches with nonstandard fingerings and embouchure adjustments, but the performer must learn a new fingering and adjustment for each note. For brass instruments, it is possible to get a certain set of JI scales by properly tuning the individual valves on a trumpet, horn, or tuba, and the trombone, like the strings, can play any desired pitch with little difficulty. It's difficult for an individual keyed brass player to change their scale in the midst of a piece, but with, say, a brass quintet, it's possible for the different players to achieve a variety of related tunings between them.

That last bit is the route I'm taking. I've got a good idea of a tuning that works for any individual instrument, and by using instruments which are naturally pitched in different keys -- one trumpet in C, the other in Bb, a double horn in F and Bb, a trombone which can play anything but most naturally gravitates toward Bb -- I can get multiple overlapping scales for a good bit of harmonic variety. And I think that the tuba, with longer tuning slides, might have enough leeway for me to give it a slightly different tuning. I'd have to try it out with an actual tubist to be sure. Also, different tuba players favor instruments in a surprising variety of keys -- Bb, C, Eb, and F -- so I should probably nail down a particular brass quintet and find out which key their tuba is in, before commiting too much to paper.

In the meantime, I've been doing some preliminary exercises, to get myself used to working in JI. The first order of business is notation. Actually, that's the second order of business, but I already took care of the first -- figuring out a JI scheme for (most of) the instruments. But while I know what pitches I can get from each instrument, it isn't as clear to me, from the start, what available harmonies I will have spanning disparate tunings, and one way for me to figure that out is to get everything written out on paper. After all, it's a little easier for me to deal with the notes B, C, and D than it is to deal with the ratios 15:16:18.

Now, just intonation uses a different set of pitches than equal temperament, so some notational changes are necessary. However, our modern equal temperament, and its concomitant notational system of naturals, sharps, and flats, evolved in steps from older tunings based on whole-number ratios -- just intonation is not merely a recent innovation. Accordingly, the system I am using, devised by Ben Johnston, is in some ways merely an extension of traditional notation.

The first step is to define the natural pitches, those without accidentals. In Johnston's notation, the chords C-E-G, F-A-C, and G-B-D are all tuned as perfect major triads, in the ratio 4:5:6. If you work out all the pairwise intervals involved, you see that F-C, C-G, G-D, A-E, and E-B all form perfect fifths in the ratio 3:2, as one might expect from looking at the keyboard and counting semitones (each of those intervals is 7 semitones wide). However, D-A, which also looks like it should be a perfect fifth, is decidedly not -- those notes are in the ratio 40:27, which is in fact a pretty severe dissonance. So, right from the start, some of our assumptions are shaken up. But we're only just beginning!

Next, we have to define the accidentals, and there are a lot of them. We do have sharps and flats, though. We said that C-E-G was a 4:5:6 major triad; can we make C-Eb-G a minor triad? Sure! A purely tuned minor triad has ratios of 1/6:1/5:1/4, or 10:12:15. Now, that means that, if the frequency of Eb is 12/10=6/5 times that of C, and the frequency of E is 5/4 times that of C, then the frequency of Eb is (6/5)/(5/4) times that of E, or 24/25. So a flat sign (b) lowers a pitch, any pitch, by a factor of 24:25. Conversely, a sharp (#) raises a pitch by an interval of 25/24. To deal with the fact that D-A wasn't a perfect fifth, we use the symbols + and - to indicate altering a pitch by 81/80 and 80/81, respectively. To get pitches relating to the seventh harmonic of the overtone series, we use 7 to lower a pitch by the interval 35/36, and L (an upside-down 7) to raise the pitch by the interval 36/35. There are also accidental symbols incorporating numbers with factors of 11 and 13, but they don't come into play in the tuning system I'm using, so I won't go into them. Any of these symbols can be combined -- compared to C, a Bb7 is 3/2 (C-G) x5/4 (G-B) x24/25 (b) x35/36 (7) = 7/4, which is precisely the seventh harmonic of C (two octaves removed, but we tend to ignore octaves -- or powers of two, from a mathematical perspective -- when comparing intervals). We can even repeat accidentals on a single note, just like the occasional double-sharps and double-flats in traditional notation. We have lots and lots of accidentals!

Now, in my case, I can't just go around writing #s and +s and Ls willy-nilly. I have a very definite set of pitches that I can work with, and I have to make the accidentals work with what I got. Since the JI notation is centered around the key of C, I started out with the C trumpet. I worked out all the ratios I was going to get in a single partial, applied those ratios to the available harmonics in the overtone series, and for each resulting ratio, factored it into a product of the appropriate accidentals. But I'm going to be working in Finale, so I have to define a bunch of expressions to attach to notes for the weird accidentals. But I don't just have to define +, -, 7, and L. I want to be able to hear the resulting pitches and intervals accurately, and since these pitches are not covered by the equal-tempered scales, I have to define pitch adjustments. Usually, intervals in general -- JI, equal-tempered, and anything in between -- are measured in cents. There are 100 cents in a semitone, and 1200 cents in an octave. However, Finale defines microtonal pitch adjustments in terms of the pitchwheel, and after some experimentation, I was able to determine that there were 8192 equally-spaced pitchwheel divisions in an octave, at least as my computer played it back. But wait, there's more. Since the naturals in Johnston's JI notation are not the same as their equal-tempered counterparts, I have to define separate accidentals, with separate pitch adjustments, for each note. The difference between JI C#L and equal-tempered C# is different from the difference between F#L and equal-tempered F#, so they need to be different Ls. I even need to make invisible markings to apply to the diatonic pitches, to make sure they're in tune. For every note in my desired scale, I had to go back to the ratio, look it up in Kyle Gann's Anatomy of an Octave, use Google's built-in calculator to convert from cents to pitchwheel increments, and define an accidental just for that pitch, complete with a description telling me which pitch it's defined for. Sounds like a lot of work? It is. Fortunately, I can save all my JI accidentals in a library, so it'll get easier as I go along.

Now, that was for the trumpet in C. Other instruments have the same overall shape of their scale, but built on different starting pitches. Also, some of these instruments -- the Bb trumpet and horn -- are transposing instruments, which means that the notes on the page are a certain transposition away from the actual sounding notes. Working in equal temperament, this isn't a big deal at all -- I got used to reading most of the standard transpositions in high school, due to my general interest in band music. In JI, or at least my current flavor of JI, it wreaks havoc on the system. I have a different set of pitches, some of which overlap with the pitches of the trumpet. They're certainly related to the trumpet's pitches -- in fact, if I'm working on the F side of the horn, they're just the same pitches as the C trumpet, transposed down a perfect fifth. But because of the transposition, the written notes, including accidentals, are the same as the trumpet. When the actual pitches do coincide with some of the C trumpet's, I can reuse the already defined markings -- after all, the same pitches are going to have the same pitch adjustments. Except that sometimes the actual written accidentals aren't the same: the pitch that was a B in the trumpet is not an F# in the horn, but an F#+ -- again going back to the fact that some of the "fifths" in the diatonic JI collection were not actually perfect fifths. Similarly, when defining new pitches, I have to not only do all of the above calculations and lookups, but I also have to keep reminding asking myself, "was this a written A+ but a sounding D, or a written A and a sounding D-?" It's almost enough to make my head go 'splode. And I haven't even gotten to the Bb side (technically the Bb- side, if you're being JI-precise about it) of the horn yet. I'm half afraid that that will make my head go 'splode.

Maybe I should go unwind, and work on debugging the online store.