There are an infinity of numbers, and an infinity of prime numbers. There are an infinity of fractions. There are, correspondingly, an infinity of pitches within any octave. Studies have suggested that, depending on circumstances of timbre, register, volume, and so on, the human ear and brain can distinguish about 250 pitches per octave. I myself have found, as a composer, that two pitches only 5 or 6 cents apart turn out to be impractically close, and I can’t meaningfully distinguish them within the context of a piece of music. Other composers, with other, more acoustically pure musical aims, may well find a vast gulf of difference between 200 and 205 cents. To be comfortable, I need at least 16 to 20 cents between the pitches in my scales, but Partch‘s scale contains pitches only 14.4 cents apart. There are, refreshingly, no rules here, and no limitations besides the composer’s own personal idiosyncrasies.

The number 13 opens up still further territories. The 13th harmonic is 840.53 cents above an octave of its fundamental, and some simple 13-based intervals include:

13/12 = 139 cents

13/11 = 289 cents

16/13 = 359 cents

13/10 = 454 cents

18/13 = 563 cents

13/9 = 637 cents

20/13 = 746 cents

13/8 = 841 cents

13/7 = 1072 cents

Note a preponderance of pitches about 40 cents away from our equal-tempered pitches.

The only composer I know of working consistently in 13-limit tuning is Mayumi Reinhard, and she swears that’s where the action is.

The 17th and 19th harmonics come too close to equal temperament to sound very exotic in most contexts; 17/16 is 105 cents, and 19/16 is 297.5 cents, both nearly divisible by 100. Ben Johnston‘s Suite for Microtonal Piano (1977), though, is a fine example of a piece in 19-limit tuning. The 12 pitches of the piano are tuned to the 16th, 17th, 18th, 19th, 20th, 21st, 22nd, 24th, 26th, 27th, 28th, and 30th harmonics of C. (This is still only 19-limit because all the other numbers factor down to prime numbers smaller than 19: 21 = 3 x 7, 22 = 2 x 11, and so on.) I don’t know of a better work for demonstrating the fascinating possibilities just intonation holds for modulation. The first and fifth movements are in the key of C, the second movement is in D, and the fourth is in E, meaning you get some pretty strange scales over D and E. The third movement is dodecaphonic. Johnston is possibly the only major composer who’s written dodecaphonic music in just tunings. The Suite for Microtonal Piano is recorded by Philip Bush on the Koch label, along with Johnston’s Sonata for Microtonal Piano (1964), a complexly prickly work in a highly extended five-limit tuning.

An equally fascinating work in 31-limit tuning is Johnston’s String Quartet No. 9, recorded by the Stanford Quartet (now renamed the Ives Quartet). Johnston has his intrepid string players play in a harmonic series scale from the 16th to the 32nd harmonic, including transpositions and inversions of the scale. The 31st harmonic is basically a quartertone between the 30th and 32nd, and when the strings cadence from dominant to tonic with that 31st harmonic, that cadence sounds nailed down for good. The work is mostly sweetly neoclassical, providing an unexpectedly normal context for odd harmonic events, and it’s very well performed. (The original 1964 tuning of La Monte Young‘s *The Well-Tuned Piano* was also in 31-limit tuning, not seven-limit as it eventually ended up.)

Johnston has more recently gone up to the 43rd harmonic (43-limit tuning) in recent works that aren’t recorded yet. The only person I know of to go higher than that in just intonation is La Monte Young in his sine-tone installations. For 20 years, Young has explored in his scintillating sound sculptures the harmonics between the higher octaves of the 7th and 9th harmonics. His current installation at the Mela Foundation is The Base 9:7:4 Symmetry in Prime Time When Centered Above and Below the Lowest Term Primes in the Range 288 to 224 with the Addition of 279 and 261…. The complete title is many times longer), which you can visit at 275 Church Street in New York City (call 212-925-5098 for times). It includes the 1072nd, 1096th, 2096th, and 2224th harmonics over its base drone, as well as other, lower tone complexes. All of these are octaves of prime-numbered harmonics: 1072 = 67 x 16, 1096 = 137 x 8, and so on. In more recent works Young has gone up above the 5000th harmonic. My sketchy introduction here can’t begin to do his sine-tone installations justice, but you can read more about them in my article “The Outer Edge of Consonance: The Development of La Monte Young’s Tuning Systems,” in *Sound and Light: La Monte Young and Marian Zazeela* (Lewisburg, PA: Bucknell University Press, 1996, pp. 152-190).

Until some madman surpasses Young, this takes us as far as we can go in discussion of new just-intonation dimensions.