The Well-Tempered Tune

What stretch will give the perfect tune? Where is the "sweet spot" for a piano? How many stretch points are needed and where should they be placed? Does the style need to be tailored to the piano, or can one style fit all, or nearly all, pianos? These questions have been discussed by Kent Swafford in a series of articles in the Piano Technicians Journal, beginning with the July 2017 issue.

Kent Swafford has created what he calls a Twenty-First Century Tuning Style based on the work of Bernhard Stopper (see Europiano 3//1988). Swafford argues very convincingly that the stretch should not be limited by focusing on pure octaves (i.e. the traditional approach used by most tuners), but pure perfect 5ths, 12ths, 19ths or 26ths should be considered as well. The focus should not be on any one note or interval, but more on the unity of sound. Some have argued that a pure 12th style may be universal. Swafford seems to prefer the pure 12th, but has provided the other styles that might be more appropriate for pianos with low or high inharmonicity, or for those who find the pure 12th style too much.

Swafford's tuning styles adjust the stretch of a piano to give beat rates that would be heard if the piano were tuned to give pure perfect intervals with no inharmonicity.

One can simulate this by defining with a scale from A0 to C8 that is equally tempered but has pure intervals. For pure P12's this can be achieved by defining A4 to be 440 Hz, and then the frequencies of the notes above (and below) A4 are obtained by multiplying (or dividing) sequentially each frequency by 1.05953 (i.e. 3^(1/19) or the 19th root of 3). The validity of this can be seen by multiplying this number by itself 19 times to obtain 3, the factor defining the ratio of the upper and lower frequencies of a pure 12th (with 19 steps).

Next, the overtones (or partials) of each note are defined by integral multiples of the fundamentals since we are assuming there is no inharmonicity. A table (shown below for pure 12ths) can be used to define the stretch and as well as the sizes of the perfect 5ths, octaves, and perfect 12ths.

The table of fundamentals and partials can now be used to specify the partial intervals in the tuning style (e.g. 2:1, 3:1, 4:1, 4:2, 6:3, 6:2, 8:1, 8:2, and 9:3), and they are consistent with those given by Swafford. These numbers are outlined in the columns on the right, with color coding to aid in locating the positions of the frequencies used for each octave. In this example, the boxes outlined with a bold border indicate the pure 12th intervals (with 0.00 beat rates), the others indicate the octave intervals with the beat rates imposed by the pure 12th tuning.

A high resolution image of the above pure 12th table can be found here.

Note that the stretch relative to an octave based equal temperament is not constant, but varies linearly. For the pure 12ths it varies from -5 to 4 cents.

In order to see what is achieved by Swafford's tuning, I have measured the positions of the fundamentals and partials for a Steinway B that was fine-tuned using Swafford's Pure 12 style. For comparison, I have obtained similar results on a Steinway D fine-tuned with the Pure 5 style (results are shown at the bottom of this page). Both were tuned with the Veritune software. The style parameters for the Pure12 were those on the Veritune website, i.e. Pure12, Standard (8-1-17):

A0 / 9:3 0.00 33% / 8:1 0.47 33% / 6:3 0.12 33%
A1 / 6:2 0.00 33% / 8:2 0.62 33% / 6:3 0.24 33%
A2 / 4:1 0.62 33% / 6:2 0.00 33% / 6:3 0.47 33%
D3 / 4:1 0.83 33% / 3:1 0.00 33% / 6:3 0.62 33%
A3-4 / 4:2 0.63 100%
A5 / 3:1 0.00 33% / 4:2 1.25 33% / 4:1 1.25 33%
A6 / 3:1 0.00 33% / 8:1 3.77 33% / 4:1 2.50 33%
C8 / 8:1 8.97 33% / 4:1 5.96 33% / 2:1 2.98 33%

and those for the Pure 5 were from the Pure 5th 8-1-17 file:

A0 / 9:3 0.48 33% / 8:1 1.27 33% / 6:3 0.32 33%
A1 / 6:2 0.64 33% / 8:2 1.95 33% / 6:3 0.64 33%
A2 / 4:1 1.95 33% / 6:2 1.28 33% / 6:3 1.28 33%
D3 / 4:1 2.61 33% / 3:1 0.85 33% / 6:3 1.70 33%
A3-4 / 4:2 1.70 100%
A5 / 3:1 1.70 33% / 4:2 3.42 33% / 4:1 3.91 33%
A6 / 3:1 3.42 33% / 8:1 10.25 33% / 4:1 7.85 33%
C8 / 8:1 24.38 33% / 4:1 16.27 33% / 2:1 9.14 33%

The parameters in the P5 tuning can be obtained from a table similar to that shown above by constructing a scale from A440 using a multiplier of 1.059526 (i.e. 1.5^(1/7)). A table of fundamentals and partials is shown below for pure 5th tuning and no inharmonicity.

A high resolution image of the above pure 5th table can be found here.
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For those who are interested, the parameters in the P19th tuning can be obtained by constructing a scale from A440 using a multiplier of 1.059502 (i.e. 6^(1/31)). A table of fundamentals and partials is shown below for pure 19th tuning and no inharmonicity.

A high resolution image of the above pure 19th table can be found here.
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And for the sake of completeness, the parameters in the P26th tuning can be obtained by constructing a scale from A440 using a multiplier of 1.059491 (i.e. 12^(1/43)). A table of fundamentals and partials is shown below for pure 26th tuning and no inharmonicity.

A high resolution image of the above pure 26th table can be found here.
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The fundamentals, partials, and inharmonicity values were measured using single strings for each of the 88 notes of the piano (see the bottom of the Partials page for details on how this is done) and the results are shown in the tables and figures below. The tuning of each string was adjusted to within 0.1 cents as indicated by the Verituner prior to recording a complete transient following a mf to f strike. Fourier transforms of the transients provided all the partials, from which the inharmonicity was calculated. As on the Stretch Points page, only those partials that are utilized by the Veritune software are shown in the table. The strongest partials observed in the analysis are outlined by a bold border. Color coding of the partials indicates the stretch points and intervals as defined in the files. For example, the dark blue background for the f8 partial of A0 and the f1 partial for A3 indicate the 8:1 interval used in the first stretch point. Also shown in the nine columns on the right is the measured stretch for each note (the difference in cents between the ideal frequency (without stretch) and the actual measured values), the octave widths along with the widths of the P5ths and P12ths.

A high resolution image of the above table can be seen here.

A high resolution image of the above table can be found here.

It is interesting to compare the inharmonicities of the Steinway B and D. The difference is negligible in the treble and becomes noticeable in the bass where the D (blue) is consistently about half that of the B (red).

The stretch resulting from the Swafford P12 Standard tuning (red) can be most easily compared to that observed for the Pure 5 (blue) using a Railsbach curve, which shows the distribution of the stretch for each note across the piano. The Veritune software uses the measured inharmonicity of each string to calculate the stretch, as done in these calculations. The Railsbach curve is therefore a result of the tuning, and it is not used by the Veritune software to achieve the stretch. The slope of the curves in the midsection is near to that observed above for the ideal model with no inharmonicity. The tails at high and low frequencies result for the stretch caused by the real inharmonicity pulling the stretch out to achieve the beat rates defined by the Pure P12 model above.

The curves below compare the widths of the octaves resulting from the Pure 12 tuning (red) and Pure 5 tuning (blue). A pure octave (in the absence of inharmonicity) would be 1201.23 cents wide. Thus, all octaves are stretched, with the largest stretch occurring in the low bass and most of the treble.

Next, the widths of 12th's resulting from the Swafford Pure 12 Standard tuning (red) compared to those measured following a P5 tuning (blue). For reference, a pure 12th would be 1901.955 cents wide (not 1900, since 1200*Log2(3) is 1901.955. The cent is defined using a pure octave scale.).

Next, the widths of 5th's resulting from the Swafford Pure 12 Standard tuning (red) compared to those measured following a P5 tuning (blue). A pure 5th would be 701.995 cents wide (1200 Log2(1.5)).