For people that spend time in the world of new music, especially in academic circles, one of leading themes of the last several decades has been alternative intonations systems – that is, tuning systems other than the standard western scale known as twelve-tone equal temperament. The study of alternative tunings has given rise to some exquisite music and many lively discussions.  I’ve appreciated and enjoyed a lot of this music greatly, and the surrounding discourse as well, but I’ve never been active in the field myself. (For one of the reasons why, see here.)

Never, that is, until recently. There has long been one scale – a complementary pair of scales, actually – that intrigues me and that I’ve wanted to explore on my own. But I had put off the idea simply because of the work and time that would be required. For people who work in computer music it’s easy to check out and explore new scales, but it’s much more of an undertaking for people like me who are mostly into making acoustic instruments. Excuses, excuses.  Anyway, the point is, I finally did it.  I built an instrument to play the relevant tunings pair. Further, I’ve managed to finagle things so that the instrument will not be entirely alone; there will eventually be at least a couple of other instruments around that can play in the same tuning. You can see and hear my instrument here.

The tuning pair can be described as follows: one set is an octave-repeating scale consisting of harmonics 8 through 16. The other is subharmonics 16 through 8.

If you’re into tuning theory, that description may be sufficient. But to explain a little further: These tunings are built upon the harmonic series and its inversion, the subharmonic series.  In the harmonic series, it happens that the pitches are widely spaced in the lower part of the series – for instance, the interval between the first and second harmonics is an octave – but the pitches get closer together as you go up the series. When you get up to the 7th or 8th harmonic, the tones are about the right distance apart to feel scale-like – that is, the series of tones comes across like the steps of a musical scale.  This remains true up through somewhere around harmonic 16. Beyond that point, as the intervals between the tones get smaller, the series starts to feel more microtonal and less scale-like. Thus the 8-through-16 scale sits in the sweet spot where the tones of the harmonic series can readily function as a scale, in keeping with people’s ingrained sense of about how far apart the notes of a scale should be. The 16th harmonic is an octave above the 8th, so this segment of the harmonic series can comprise a one-octave scale. To continue up further, the scale is octave-repeating, meaning that you can simply repeat octave duplicates of the same eight tones up through this next octave, and the next and the next as far as the compass of your instrument allows.

The subharmonic series is less familiar than the harmonic series, and for good reason: while the harmonic series appears frequently in nature and is integral to many musical sounds, the subharmonic series appears much more rarely.  (It’s sometimes said that it does not occur in nature. I can think of an instance or two in which it does, but that’s a topic for another discussion.)  The subharmonic series is the inversion of the harmonic series: while the harmonic series consists of integral multiples of a base frequency (f, 2f, 3f, etc.), the subharmonic series consists of submultiples: f, f/2, f/3, etc. While the harmonic series is conceptualized as an upward progression of frequencies, the subharmonic series is usually thought of as a downward progression. And while the harmonic series starts with large intervals that get smaller as you ascend, the subharmonic series starts with large intervals which get smaller as you go down.  As with the harmonics, with the subharmonics once again we get a sweet spot of scale-sized intervals between 8 and 16. This, then, is the second of the two complementary scales: subharmonics 16 through 8, repeating at the octave.

Of course, I am not the first to suggest these two scales. I first ran into the idea in an essay and accompanying cassette called Sounds of Just Intonation written by Ralph David Hill in 1985. Prior to that, Harry Partch, in his seminal work Genesis of a Music, discussed the harmonic series and subharmonic series. He coined the terms otonality (from overtone) and utonality (from undertone) to refer to tonalities based in the two series respectively. I’ve been using the term over-under scales to refer to scale systems which bring the two tonal approaches together as complements to one another. By extension, I’m also using  over-under a noun to refer to instruments built to play in such scales (of which, admittedly, there are very few at present). Example of the term used in a sentence: “Nice over-under you got there.”

One of the appealing things about the harmonics 8-16 scale is that it’s fresh and different, having several exotic intervals in it – exotic in that we don’t usually get to hear them because they’re quite unlike anything available in the standard western scale – yet there’s a certain naturalness to it; it seems sort of strangely familiar, like some forgotten dream. Perhaps this “natural” quality is because the intervals of the scale are integral to a lot of natural sounds including many musical sounds. Also, they line up nicely with what seems to be the ears’ innate way of interpreting the frequency relationships that are the basis of musical intervals. Or, who knows, perhaps this sense of naturalness is just fanciful on my part; a trick I play on myself aesthetically in response to an intellectual predisposition.  But the point is, for my ears at least, this overtone scale has an appealing quality of seeming at the same time exotic yet natural.

And what about the subharmonic scale? Perhaps less familiar, because the subharmonic series isn’t widely present in nature? No, to my ears the subharmonics 16-8 scale has that strange-yet-familiar feeling to an even greater degree. Don’t ask me why; it’s just my ears. But I take great pleasure in it.

So I built the instrument! It consists of two sets of rectangular aluminum tubes tuned by length and played by percussion.  This is a suitable material for such a project because the hollow aluminum bars can be tuned to pitch quite accurately and, once tuned, retain the tuning well (although they are subject to some variation with temperature). With percussion bars like this, there’s often a problem with unwanted inharmonic overtones within the bar tone coming through too strongly and dominating the fundamental, especially in the lower bars. In this instrument the problem is mitigated in those lower bars by air resonance which strengthens the fundamental but not the overtones. The air resonance is that of the air column contained within the rectangular tubes themselves: by placing flute-like toneholes in the wall of the tube, the resonance pitch of the air column can be tuned to match the fundamental of the percussion tone. Strike the bar, and this internal air resonance comes into play and reinforces the tone of the metal bar itself. This is like the idea of placing tuned air resonator tubes below a marimba bar, but in this case the resonator is not a separate tube, but rather is the hollow bar itself – a much more compact arrangement. The choice of rectangular bars rather than round tubes has to do with a certain detuning problem that arises in connection with the drilling of the toneholes. Without going into details, I can say that this problem is easier to manage in hollow rectangular bars than in cylindrical tubes.

The higher notes of the two scale sets are arranged like a flat keyboard in two parallel rows, while the tones of the lower octave are suspended vertically in rows above.  This arrangement is mostly a matter of practicality: how to keep the instrument as compact as possible while making sure that all tones are easily accessible and the whole thing is reasonably ergonomic. Inevitably though, it’s a fairly large instrument, as the three-octave range for both scales requires fifty tubes for altogether, the longest of which are around two feet.

The instrument is called tangular arc – tangular for the use of rectangular bars, and arc as a shorthand for air resonated chimes. To be really proper, the full name would be tangular arc over-under.  The idea is that one could make additional over-unders – other instruments designed to play in an ensemble of over-under-tuned instruments. This seems an improbable fantasy, but not entirely hopeless, since as mentioned at the start of this article, a couple of other instruments capable of over-under-ing have been made or are in the pipeline.

Before closing, a couple more observations about the over-under scale pair:

The harmonic and the subharmonic 8 -16 scales have a sort of natural integrity, but they each lack some basic intervals that one might like to have available. Most notable among these are the fourth for the harmonic scale, and the fifth for the subharmonic scale. But with the two scales paired and positioned adjacent to one another, these intervals become available as the player can borrow tones as needed between the two.  In addition to occasional borrowing for essential intervals, less obvious tones borrowed between the scales can provide color and interest and complexity within the musical context.  For that matter, one could create entirely new scales from various combinations of tones from the two.

With these over-under scales, it can be interesting to explore different modes. The word modes is used to mean many different things: the relevant meaning here is the idea that, given a particular set of notes comprising a scale, one can in effect create different tonalities within the set by treating different notes as the tonal center or home tone. As a familiar example, you can get a C major scale or an A natural minor scale from the same set of white-notes on the piano, depending on the melodic and chordal patterns you play and how you resolve them.  In the harmonics and subharmonics scales described here, it would seem natural to treat the tones at degrees 8 and 16 as the tonal center of the music, as they are octave displacements of the fundamental tone on which the harmonic or subharmonic series is based. But for a different tonal feeling, you can structure whatever you play so that some other pitch within the scale stands out as the home tone and defines the key of the music. In fooling around with these scales, I’ve found that particularly lovely tonalities can be found in the modes based on the 12th harmonic or subharmonic. (It’s an interesting coincidence that the mode based on the twelve turns out to be especially appealing for both scales.)

To determine the actual frequencies for the tones of any harmonic or subharmonic scale, the starting point is the frequency of fundamental over or under which the series is to be built. The choice of this frequency is an arbitrary one; you could use any frequency as a starting point from which to build the scales. Different actual pitches will result from different choices for the fundamental, but the relationships between the pitches and the resulting musical structure will be the same whatever the fundamental. As the fundamental for my over-under scales I chose 62.4Hz, which corresponds to the note B natural plus 18.5 cents. The reason I chose this particular frequency as the root of all my subsequent labors was: another local instrument inventor/builder, in the person of David Samas, has recently been building and/or commissioning  a set of instruments to play in a harmonics-based scale, consisting in this case of harmonics one through thirty-two, with 62.4Hz as his base frequency. This is conceptually rather different from my over-under scales, but the two pitch sets have enough in common to be on speaking terms, allowing some exchange of instruments between the two. Should anyone reading this get the bug and decide to build over-under instruments of their own, I encourage you to use this same fundamental.  That way, should your instruments and mine ever in some imaginary future find themselves in the same room, they could play together. (Technical note: Building a subharmonic series downward from a starting frequency as low as 62.4Hz would take you into ridiculously low musical territory, so for the subharmonic side it’s easier to conceptualize the scale as being built downward from a higher octave displacement of the same fundamental. Since the interval of the octave corresponds to a doubling of the frequency, you can find a suitably higher octave version of the fundamental by multiplying the original frequency by two a few times.)


Share This