It’s just … just … just intonation!

Philosophical Guideposts

Purity, exactness, integrity (integer).

The distinctive buzz of purely tuned harmony, especially in lower limits.

Acoustic instruments. Too much JI music for headphones already. Nothing replicates physical acoustic instruments.

Favor instruments with independently-tunable strings or pipes or bars (no frets, no tone holes): piano, concertina, organ, psaltery, tube chimes, etc.

The harmonic series. I’m not interested in exploring pitch relationships that leave the harmonic series.

The expressive potential of high partials.

Prime-numbered partials: take any set of prime-numbered partials, and all interval relationships will be unique. (Purity=variety)

Nontransposability. It’s more charming.

Gravitational pull toward one powerful tonic pillar.

That tonic shall be C (keyboard topography, not necessarily frequency). It’s in my blood as an organist.

Settle on one tuning. Rather come to know one tuning intimately than dabble in many.

Conceptual elegance.

28-36 pitches.

Octave-replicating in concept….

except that in practice each octave will have some unique pitch content. Why have the same intervallic colors in every single octave?

Equal beating chords.

Tunable without electronic devices. (I suppose I value this for some reason…)

Candidates

Prime

The first seven octaves of the partials of the lowest C on a piano include 31 prime-numbered partials (up to 127) which extend to the top register of the piano. These 31 primes generate 930 different intervals. The one pure low-ratio tonic undeniably shines through as uniquely consonant. The remaining pitches can be heard as consonant in various harmonic contexts. Many equal-beating chords are possible. Additionally, many harmonies approximate major or minor triads and a variety of seventh chords, each with its own distinctive, “detuned” color. Several very small intervals provide a detuned-unison shimmer, much like the celeste on the organ.

The best use of this scale would be to pit the pure consonance of lower-limit ratios against equal-beating chords involving the higher partials. The various “detuned” major triads and other familiar sonorities may be interesting but seem a more arbitrary use of a system designed for maximum integer integrity.

The scale can be rounded out a bit. The largest step, 67:64, at 79 cents, is conveniently filled in with the next prime, 133. This also brings the scale to 32 pitches, which matches the 8×4 or 8×8 orientation of many controllers on the market (though I’m quite uninterested in electronic/electroacoustic JI applications). Additional large gaps can be filled in with 139, 311, 337, and 199, bringing the largest step down to 58 cents and increasing the number of intervals to 1260.

Drawbacks and nothingburgers: You probably must have an electronic tuner to tune this system. There may be ways of approximating the intervals with lower-limit ear-tunable intervals, though that is still approximation. It may also be tunable by fitting pitches into equal-beating chords, but that would take a tremendous ear and the right instrument, and I’m not certain it would cover every pitch. Nothing wrong with using an electronic tuner, perhaps … but it leads me to question the value of an exact tuning for partials so high that we have difficulty perceiving the consonance of the resulting intervals.

With 1200 cents an octave and a human ability to hear intervals within 5 cents’ difference, the 930 or 1260 intervals available is a swindle. We can’t distinguish all those intervals anyway.

The conceptual elegance attracts me, but the practical inelegance prevents me.

Chromatic Harmonic

It would be elegant enough to take the 16 partials of the fifth octave (31-limit, dubbed “chromatic harmonic” by Dante Rosati), or even the 32 partials of the sixth octave (61-limit), and use that as the scale. The evenly-decreasing step sizes would ameliorate tuning issues for the higher primes. Still plenty of color resources and many more small-ratio intervals to capitalize on consonance.

However, the relatively even spacing is unexciting, and the draw toward a single powerful tonic is weakened, due to three purely-tuned 4:5:6 triads. Various compromises could accommodate these deficits. Leaving out some steps would make the scale more uneven, as would adding in some higher partials here and there. Removing partials 15 and 25 from the 16-note scale would leave only one 4:5:6 triad, increasing the gravitational pull toward 1/1.

In the end, the concept promises too little interest.

Fractal

In this concept, a series of intervals is replicated at the level of each of its members. It could be nested and spiral outward at several levels. This concept allows for lower, potentially ear-tunable primes to generate a series of consonant constellations that yield truly exotic intervallic relationships. I explored several different concepts before settling on this one.

Take all primes up to 31 (I quite like the next prime, 37, but I doubt my ability to tune it … 31 is hard enough!): 1, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31. That’s the 11-note initiator set. I’m not crazy about 17 and 19 because they so closely approximate equal-tempered m2 and m3, but they participate in convincing harmonies with other members of the series, so I suppose they must stay.

About that fifth partial… Due to La Monte Young’s influence, there are pockets in which JI fashion dictates the exclusion of the fifth partial. Nothing against those wonderful artists, I get it – their 5 is my 17. I personally can’t live without 11/10, though, or 13/10, and 7/5 is quite the charmer. But rather than sprinkle the fifth partial throughout the tuning system, I put it on a pedestal, as a 5/4, in the initiator set, to draw attention to its unique character and create a gravitational pull toward the 1/1.

Let’s also leave 31 in its place. It’s plenty colorful where it is and will contribute nicely to the distinctiveness of the initiator set.

That leaves 3, 7, 11, 13, 17, 19, 23, and 29 as the generator. Starting with the lowest partial, multiply that partial by each partial of the generator. Incorporate the resulting pitches. Remove any pitches that fall within 15 cents of already-present pitches.

One more arbitrary decision I made was to exclude 3×3=9 from the results. I didn’t want 6:9 competing with 2:3; it’s too circle-of-fiths-y, not prime-like enough for being that low of a partial.

The process yields 32 pitches. I made one more arbitrary decision. The algorithm included 17×11=187, which later excluded 13×29=377. Since I had plenty of 17 (which I dutifully include) and not enough 29 (which I absolutely adore), I switched 187 out to include 377. I was left with the following resulting consonant constellations:

  • Over 3: 7, 11, 13, 17, 19, 23, 29
  • Over 7: 3, 7, 11, 13, 17
  • Over 11: 3, 7, 11, 13, 23
  • Over 13: 3, 7, 11, 17, 23, 29
  • Over 17: 3, 7, 13, 17, 19
  • Over 19: 3, 17, 23
  • Over 23: 3, 11, 13, 19
  • Over 29: 3, 13, 29

The resulting constellations are unified through the initiator intervals, but no two have exactly the same content. Lower partials tend to have a larger constellation. Excluding 3×3 had two unforeseen benefits: it gave the third-partial constellation a unique feature (all the others wound up with 3), and it also made space for 11×13=143 and 17×17=289 – a twofer.

The resulting partials, in order: 1, 33, 17, 69, 143, 289, 299, 19, 77, 39, 5, 323, 21, 87, 11, 91, 23, 377, 3, 49, 51, 13, 841, 437, 221, 7, 57, 29, 119, 121, 31, 253. Despite the simple means of generation, the scale presents us with 760 intervals and, of course, a great many very close approximations to high primes (as long as we’re approximating…).

It just needs a name: 11-Primes Fractal Pillar.

Onward

My attraction to JI is at least as much philosophical as much as musical. I care what it sounds like. And I care what it points to. I’ve designed hundreds of scales over the last year, searching for a tuning that satisfies my underlying philosophy. Perhaps this is the scale I’ve been looking for. Who knows?

Next step: Begin mapping pitches onto instruments and instrument pairs with 12, 14, and 24 pitches to the octave.

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