The Well-Tempered Tune

Devoted to perfecting the sound of pianos and harpsichords
7D2_3530-1The Well-Tempered Tune
Stretch points and the distribution of partials

To begin, one needs to understand the concepts of overtones and harmonics. A vibrating string vibrates not only at its fundamental frequency (f1, the first partial) but also at multiples of the fundamental (the overtones, or partials f2 and above). The table below shows the frequencies (in Hz) of the partials (the fundamental and overtones, f1-f10) for all eight A's (A0-A7) on a Steinway B piano. The stronger overtones are shown in bolder type. (f0 is the fundamental used to calculate f1 through f12 in the table.) The second partial for an ideal string would be twice the fundamental, or an octave above the fundamental, the third would be three times the fundamental or an octave plus a fifth, the fourth 4 times the fundamental or two octaves above f1, etc. The overtones of an ideal vibrating string are the harmonics. The overtones of a real piano string do not occur exactly at the harmonics. The deviation between the harmonics and the real overtones is characterized by the inharmonicity.

The positions of the notes above and below A4 are adjusted so that the partials which lead to unpleasant beating are brought to the same frequency.

Stretch points define which partial of one note is to be compared to the partial of another note above it. Examples are color coded in the table below to help indicate the partial pairs specified by stretch points in a particular tuning style (Ron Koval's 5.1). For example, the olive green highlighting at 163.8 Hz and 164.5 Hz in the top two rows of the table indicates the A0 6:3 stretch point where the 6th overtone of A0 is to match the 3rd overtone of A1.

The success of the Verituner is indicated by the degree to which the frequencies of the pairs are able to be matched. Where multiple pairs overlap (e.g. A0 6:3 and 10:5), a compromise must be made, and the individual matches may not be exact.

The frequencies of the partials were obtained as described on the Partials page, and the inharmonicity coefficients (B values) for each A were obtained as described on the Inharmonicity page.

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High resolution version of the table (tiff).

This piano was tuned using Ron Koval's 5.1 style and the iPad version of Verituner with the following stretch points:

A0 6:3 45% / 10:5 55%
E1 6:3 100%
C2 6:3 100%
A3-4 4:2 60% / 6:3 40%
A5 4:2 40% / 2:1 60%
A6 2:1 70% / 4:1 30%
A7 2:1 100%
C8 4:1 100%

The meaning of these stretch points is described in the Stretch Points page. The results of applying various styles to different pianos are given there also.