Table: On Piano Retuning

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.

Retunings


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.

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David Canright -- DCanright@NPS.edu