The Undertone Instruments Collection is an ensemble of musical instruments tuned to a subharmonic scale. This article describes the scale and the thinking behind it. The scale consists of selected notes from the subharmonic series, also known as the undertone series, which is the inversion of the more familiar harmonic overtone series. While the harmonic series arises frequently in nature and is familiar to both physicists and musicians, the subharmonic series is less common. Although there are some isolated instances where it arises naturally in the physical world, it may be seen more as an intellectual construct than a natural phenomenon.
To get the familiar harmonic overtone series, you start with some chosen frequency, call it f, and multiply to get the frequencies of the series: f, 2f, 3f, etc. For a subharmonic series, instead of multiplying you divide: f, f/2, f/3 etc. In music it’s natural to think of the harmonic series as starting from a low fundamental frequency and progressing upward, but with subharmonics the progression from the starting point is downward, so it sometimes calls for what feels like upside-down thinking.
The subharmonic scale used in the Undertone Instruments Collection consists of subharmonics eight through sixteen – that is, f/8, f/9, f/10 …. f/16. That portion of the series covers the range of an octave (f/16 is an octave below f/8). To continue the scale up or down from there, we simply cycle through the same eight notes in the higher and lower octaves. Thus, in speaking to cognoscenti you could adequately describe this scale simply by saying “subharmonics 8-16, repeating at the octave” to which they would nod knowingly and reply, “groovy.”
The reasoning behind the range selection of eight to sixteen is as follows: The pitches in the subharmonic series get closer together (the intervals between them get smaller) as you go down through the series. This is analogous to what happens with the harmonic series as you go up. At the beginning of the descending subharmonic series, the intervals are too large to seem scale-like, starting with a descending octave, followed by a fifth, then a fourth, and so on. By the time you get to subharmonics eight through sixteen, the intervals between the steps are of a size that the ear recognizes as typically scale-like – major and minor seconds, roughly speaking. Continuing down the series beyond sixteen, the intervals become smaller than typical musical scale intervals. Thus, eight through sixteen is a sweet spot for a ready-made scale, and that’s why this range was chosen.
As mentioned above, a subharmonic scale could not really be considered “natural” in the way that an overtone scale might be. Then again, the subharmonic musical intervals involved are just intervals (in the sense of just intonation), and so we can expect them to sit nicely with the ear. In fact, as the numbers play out, some of the intervals are very basic and familiar just intervals, while others are technically just but are pretty far out (in Partchian terminology, this scale could be described as 13-limit). In short, the scale intervals are a nice blend of familiar and exotic. So we could ask, does this scale come across as an artificial intellectual construct, or does it feel in some sense naturally musical? The answer is subjective, but to my ear it seems pretty natural and musical, yet with enough spicy intervals thrown in to make things interesting.
So far we’ve only been talking about intervals – relationships between notes – without reference to specific frequencies or pitches. But a subharmonic series, like a harmonic series, has to have a starting point – a specific base frequency on which you can do the arithmetic to arrive at frequencies for the ensuing notes of the series. For the Undertone Instruments Collection I used a base frequency of 62.4HZ. Why did I choose that frequency? It was really just a matter of convenience in re-utilizing some work I had done a few years earlier in another context. Generally speaking, the choice of base frequency is arbitrary. In building such a scale, you can make the choice based on whatever practical, historical, philosophical, spiritual, cosmological or entomological considerations you wish.
What about names for the degrees of the scale? Nomenclature of this kind can be helpful in talking about the scale, and perhaps even developing notation. You can identify the notes of this subharmonic 8-16 scale by numbers corresponding to their place in the subharmonic the series. The ascending scale would then be 16, 15, 14, through to the octave above at 8. Since 16 and 8 are octave equivalents of the same note, we can refer to this first/last degree as 16/8.
When playing in this scale, it’s natural to think of 16/8 as the tonal center, but you can also play or compose in other modes – that is, you can play in such a way as to establish some other note of the scale as tonal center, in a manner analogous to the old church modes in European music. Each mode has its own characteristic mood and flavor based on the interval relationships between the chosen tonal center and the surrounding notes. As it happens, the mode based on 16/8 actually isn’t the most ear-friendly of the lot. It has a just fourth – that’s nice – but it has nothing approximating a perfect fifth (which, in traditional tonal music, is important in establishing tonality), and it has a lot of seemingly off-kilter intervals. The most familiar sounding and immediately appealing is mode is the one built on note 12, which has a just fourth, a just fifth, and a just minor third, along with several other interesting and appealing intervals. It’s really lovely. Other interesting modes include those built on 9, 10 and 15, all with mixes of familiar just intervals, and intervals that to varying degrees farther out. For something more exotic, try modes 13 or 14, which, other than the octave, have no intervals that relate closely to familiar intervals of western music. Mode 11 is an odd one; it has a sort of eccentrically cheery feeling, because it has several intervals that come across as slightly detuned versions of intervals from a major scale.
When I first toyed with the idea of building an instrument set in this scale I thought of it as one of a pair or complementary scales. The other scale was to have been – no surprise – the more familiar inversion: harmonics eight through sixteen. The idea that scales can come in pairs isn’t new; it has a couple of well known cultural manifestations including the apposition of major and minor scales in western music, and the pelog and slendro in Indonesian gamelan. Perhaps there are other examples as well. A pair of scales can work together musically in several very nice ways, both for contrast and as a source of augmented tonal resources (think of “borrowing from the minor” in western tonal music). The harmonic and subharmonic pair of scales could have complemented each other nicely in both of these respects. And someday it may happen! But for now I have focused on creating the subharmonic half. (The reason was nothing more than practicality: there are only 24 hours in a day, plus I’m out of room for storage of more instruments.)
In deciding what instrument types to include in the ensemble, I focused on practical considerations. Mainly, I wanted instruments with clear and consistent pitch, and instruments not inclined to go out of tune. As it stands now, eight instruments are included. Some of these are newly built versions of instrument types that I had previously built in standard tuning. Others are instruments that were already on hand around here, retuned for this collection. For two of them, both of the pair of scales mentioned above are included. For the remainder only the subharmonic scale is available. A couple of them are string instruments, and that is not ideal due to the need for regular retuning. The other instruments are all nicely tuning-stable. The list of instruments follows here. (Note that the links below will take you to information focused on the original instruments, not the instruments as remade for the Undertone Instruments Collection.)
Tangular Arc: An aluminum bar-percussion set. Some of the bars are hollow, with interior air resonance tuned to reinforce the tone. For this instrument, both the subharmonics 8-16 scale and the harmonics 8-16 scale are included.
P-Boos: A set of boos made of ABS plastic. Boos are like tubular tongue drums in which the enclosed air resonance of each tube is tuned to reinforce its tongue tone.
Rumba Box: A bass kalimba, in the style of the Caribbean folk instrument of the same name.
Bowed T-Rods: Tuned metal rods, fixed at one end, played by bowing.
Dansal 3: A tuned set of aluminum rods mounted for playing by longitudinal friction.
Pliffers: A tuned of plastic rods mounted for playing by longitudinal friction (similar in concept but very different in sound from Dansal 3).
Mahogany: A kalimba-like instrument, overtone-tuned so that the most prominent overtone in each tine is an octave above ethe fundamental. The tines have light rattles to impart a glittery sound and bring out the octave overtone.
Movable Fret Guitars: Two guitar-like instruments with adjustable fret positions. In one of them the frets have been positioned for the subharmonoics 8-16 scale; in the other they’ve been positioned for the harmonics 8-16 scale.
Wine Barrel Zither: A fairly conventional 15-string zither. (The soundboard is cut from one of the wide bottom slats of a very old, decommissioned, industrial-sized wine barrel made from old-growth redwood.)