This article first appeared in 1/1, the Journal of the Just Intonation Network, Volume 1, Number 4, p.10 (1985).
To some, the idea of a justly tuned piano is an oxymoron. A piano, after all, is the very symbol of equal temperament. They are not easy to retune. The effort required demands that one consider very carefully just which tuning to use out of the infinity of possibilities. Here I'll talk about some of the more versatile possibilities. I'll only consider 12-tone tunings where all "semitones" are at least twice as big as a comma, so each tone is distinct. (This excludes the trick of including two enharmonic versions of a note, which could be very useful in some circumstances.) The four tunings (see Table at end) are organized by increasing prime limit, i.e., by exploring intervals progressively further up the harmonic series. At the end I'll mention some of the problems inherent in retuning a piano. Of course, these tunings could be used on any retunable keyboard instrument. In this article ratios used to represent note names are indicated by colons, while ratios representing intervals are slashed.
Historically, our tonal system derives from a "circle of fifths." Indeed, the closing of the circle results in equal temperament, with all the "fifths" slightly squashed in the process. The payoff, of course, is that you can modulate indefinitely and still retain the same harmonic resources, such as they are. If extensive modulation is important for your music, but you still want some sort of pure intonation, then Pythagorean tuning (see Table at end) is your best bet. That is to say, don't close the circle, and all the 3:2's stay perfect, except where the circle doesn't close you get a 262144:177147, which is 23.5 cents flatter than a 3:2. Then eight of the "major thirds" will be a nice bright 81:64, and the other four will be 8192:6561, which is only 2 cents flat of 5:4. (Unfortunately, one of these falls in the major triad with the bad "fifth.") And you can modulate by 3:2 through six identical Pythagorean diatonic scales. The other scales are useful too, just not identical, varying in where they include the flat "fifth" and thus some number of the more restful "thirds." This gives a great deal of harmonic mobility, but if you want some pure thirds you'll have to transcend the 3-limit.
Pure major and minor thirds (5:4 and 6:5) involve the prime 5, so necessitate at least a 5-limit scale. And this brings up the comma of Didymus, a problem discussed in this journal (1/1 Vol.1, No.3). in other words, to get pure thirds you must give up some perfect fifths The 5-limit scale given below (see Table at end) gives the best combination that will fit in the twelve tones. In a 3x5 harmonic grid representation it forms a rectangle. (Terry Riley uses a variant of this, with a 27/16 instead of 5/3, which effectively moves the tonal center to 3/2.) The just major scale 1:1 9:8 5:4 4:3 3:2 5:3 15:8, occurs twice, on the 1/1 and the 8/5, but the major scales based on the 3/2 and the 6/5 differ from the above only in having a smaller second (10:9 instead of 9:8). Just minor scales of the form 1:1 9:8 6:5 4:3 3:2 8:5 9:5 occur on the 1/1 and the 5/4. And of course there are lots of other usable scales, and six each major and minor triads. (A complete harmonic analysis is available for the asking. Send a SASE to the network requesting Supplementary Paper #2.) However, modulation through identical scales is very limited (to one step at best).
For my purposes and maybe for yours, it is necessary to go to a 7- limit, because that provides intervals that are radically different from traditional harmony, constituting new territory for exploration. The most versatile 7-limit twelve-tone scale that still contains standard major and minor scales is given below (see Table at end). This offers the chance to contrast, for example, a standard minor scale (on 5/4) with a septimal minor on 1/1: 1:1 9:8 7:6 4:3 3:2 14:9 7:4. This is the tuning I use in my piano canon for seven hands, which employs all twelve tones, as well as in a work in progress. A variant of this tuning, used by Other Music, is to replace the 45/32 with a 7/5. This variant tuning, while lacking the standard minor scale, does include a pentatonic undertone scale (Partch's "utonality") on the 7/4.
The final, intriguing tuning goes all the way to a 13-limit (see Table at end). This includes a complete eight-tone harmonic scale on the 4/3 and a seven-tone one on 1/1. Such scales sound fascinating in their variety, coherence, and newness. However, the other tonalities (except 3/2 minor) are pretty strange. Due to the inharmonicity problem (discussed below) this tuning would have to be classified experimental for pianos although it would work fine for other keyboard instruments.
Pianos, by their very nature, have some problems that make tuning them in Just Intonation particularly trying. The biggest problem is that the strings are stiff enough to make the overtones sharper than the harmonic series, progressively more so as you go up the series. This effect varies throughout the range of the piano. The best solution to this problem is to use the largest piano possible (e.g., a Bosendorfer Imperial), since increasing the string length raises the tension and thus relatively lessens the effect of the stiffness, making the overtones closer to harmonic. In any case, the inharmonicity means you cannot get both the fundamentals and the overtones in tune. For a 2:1 or a 3:2 the problem is slight, but for a 7:4 or higher harmonics the difference is obvious. So you can have fundamentals in tune and beating overtones, or coincident overtones with out-of-tune fundamentals, or some compromise in between, chosen by ear. As the ear must be the final judge, the latter choice is recommended.
Another problem is that when a piano is retuned, it immediately begins to go out of tune, so it should be tuned either soon before performance or repeatedly over an extended period of time (a week or so). Also, there almost inevitably is some beating between paired and tripled strings, which partly accounts for a piano's characteristic sound. If this is annoying, or to avoid the work of tuning all the duplicate strings, you can leave in a tempering strip, so that only one string per note sounds. (Terry Riley often uses this approach.) In addition, room acoustics and "false beating" can further confuse the issue. Only experience can be the guide. For further information, I suggest you direct your questions to the well-experienced Shabda Owens, from whom these tips come.
This table lists the four piano tunings for the four prime limits, each in a double column of intervals (ratios) followed by the deviation, in cents, from equal temperament. For example, to get a Bb of 7/4, tune it 31.2 cents flatter than a tempered Bb, i.e. 968.8 cents instead of 1000.0 cents.
Limit: 3 5 7 13
C 1/1 (0) 1/1 (0) 1/1 (0) 1/1 (0)
Db 256/243 (-9.8) 16/15 (+11.7) 28/27 (-37.0) 13/12 (+38.6)
D 9/8 (+3.9) 9/8 (+3.9) 9/8 (+3.9) 9/8 (+3.9)
Eb 32/27 (-5.9) 6/5 (+15.6) 7/6 (-33.1) 7/6 (-33.1)
E 81/64 (+7.8) 5/4 (-13.7) 5/4 (-13.7) 5/4 (-13.7)
F 4/3 (-2.0) 4/3 (-2.0) 4/3 (-2.0) 4/3 (-2.0)
F# 729/512 (+11.7) 45/32 (-9.8) 45/32 (-9.8) 11/8 (-48.7)
G 3/2 (+2.0) 3/2 (+2.0) 3/2 (+2.0) 3/2 (+2.0)
Ab 128/81 (-7.8) 8/5 (+13.7) 14/9 (-35.1) 13/8 (+40.5)
A 27/16 (+5.9) 5/3 (-15.6) 5/3 (-15.6) 5/3 (-15.6)
Bb 16/9 (-3.9) 9/5 (+17.6) 7/4 (-31.2) 7/4 (-31.2)
B 243/128 (+9.8) 15/8 (-11.7) 15/8 (-11.7) 11/6 (-50.6)
Note: Since I wrote this article, a piano technician friend, Tom Winter, suggested that one should try to keep the overall tension of the strings the same as for equal temperament. To do this, add to each column of cents deviations the negative average for that column (-0.15, -1.0, +15.3, +9.2 cents, respectively). This is really only an issue for the last two tunings.