Amen, amen

I went looking the other day for information on chordal harmony in Bohlen-Pierce tuning, and was surprised at how little I found. Basically this: http://ziaspace.com/NYU/BP-Scale_research.pdf. Walker presents a number of chord progressions, created “by ear”, which I find rather hard to understand. Her first two, for instance, start and end on C-F#-A# (using BP pitch names), and she calls that “I”, but it certainly isn’t the tonic in the lambda mode which she emphasizes up to that point, nor in any of the other modes in her Figure 7… unless I’m misunderstanding something, which is very possible. (Her “piano rolls” in Figs. 11–18, have horizontal lines corresponding to the lambda mode… and then her initial and final chords don’t sit on those lines.) And I don’t see much of anything else online.

Okay, so what can I come up with? Understand, I’m not particularly knowledgeable about music theory. I did pass a college course on harmony, but that was decades ago. On the other hand, it seems to me that at this stage what Bohlen-Pierce needs is not a Mozart nor a Bach, but something more like a William Billings, a tunesmith whose harmonies are simple, straightforward, comfortable. The tuning is challenge enough for the listeners; let’s not burden them with tricky modulations, suspensions, and so forth. I’m no Billings, of course, but that’s beside the point.

It’d be silly to expect and require BP harmony to be structured in one-to-one correspondence with conventional harmony. But on the other hand, where such correspondences or at least analogs can be found, might as well investigate them. So, okay, I-IV-V7-I; pretty basic: Can we come up with something like it in BP tuning, and specifically in the lambda mode?

First, let’s make note of our triads. In conventional tuning a major triad is three notes in a 4:5:6 frequency ratio. (Our tuning is based on the primes 2, 3, and 5, but 2:3:5 spans more than an octave; shift the 5 down an octave to 5/2 to put it between the 2 and the 3, or equivalently shift the 2 and 3 up to 4 and 6.) In Bohlen-Pierce the obvious analog is 3:5:7. (Though note that 5/3 is arguably BP’s perfect fifth analog, and it’s not the interval between the top and bottom notes as the perfect fifth is in a 4:5:6 triad.) Bohlen calls this the “wide triad”, and uses “narrow triad” to refer to 5:7:9, but that (if you are willing to invoke “tritave equivalence”) is just an inversion of 3:5:7. And if you’re not willing to invoke “tritave equivalence”, then why are you calling notes a tritave apart by the same name?

On the other hand, if 7/5 on top of 5/3 is regarded as an analog of the conventional major triad, then there’s 5/3 on top of 7/5 as an analog of the minor triad; that’s 15:21:35.

Anyway, let’s focus on the wide triads for now, and see what we can do along the line of I-IV-V7-I.

IV, I, V are separated by perfect fifths, the basis interval (along with the octave) of conventional music. We might then consider three wide triads separated by either 5/3 or 7/3 for an analogous progression. But not so fast. The most important feature of the resolution from V (or V7) to I is the leading tone moving to the tonic; I may not remember much from my harmony class, but I do remember Prof. Curran would get very sarcastic if you tried to go anywhere else from the leading tone! So let’s include that feature. That means we need to know which chords in lambda contain the leading tone. Here’s a chart, from Bohlen’s site (reference is C lambda):

Wide triads Narrow triads
I C – G – A i C – F – H
ii D – G – J
III E – J – C iii E – H – A
V G – B – E v G – A – C
VI H – C – F
VII J – D – G vii J – C – E
VIII A – E – H
ix B – E – G

We see that the only wide triad containing the leading tone is the V, which is 5/3 above I. So far so good, and analogy to conventional harmony suggests our other chord be 5/3 below I or 9/5 above, which is VI. So I-VI-V-I, and if we want to push the analogy even harder, introduce an extra note into the V chord to play the role of conventional harmony’s seventh.

Well, but this gets messy. In V-I the leading tone (B) should resolve to C, and the G in the root (assuming the V is in root position) also resolves to C. If there’s a doubled G in the V it should stay there. Then what resolves to A? The remaining note in the V chord is E, which is half a tritave away from A! Well, E is analogous to the dominant of the V triad (maybe!) so we can omit it in favor of, um, whatever’s going to serve as the analog of the seventh. That ought to resolve to the A, and preferably should do so downwards, in the opposite direction of the leading tone to tonic. Except that would mean it’s another B! So we need it to be J instead, and both it and the B resolve upwards. Deal with it. J is two lambda steps above G, so we’ll call this chord V2.

But conventionally, the seventh in the Vserves partly to link to the IV as a common tone. We were going to use a BP VI as the analog of the conventional IV, but there is no J in the VI chord. If we want the J as a common tone then the preceding chord has to be a III or VII. VII shares a G with both I and V, though, which may be too static. So let’s use III, which shares a C with I. Our complete progression then is I-III-V2-I. In the notation recommended by Bohlen (click to listen):

And that, it seems to me, would be a good candidate for a stereotypical “A-men, a-men” in the Bohlen-Pierce hymnal.

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