The Well-Tempered Tune

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7D2_3539-5The Well-Tempered Tune
Stretch and Stretch Points

The Verituner adjusts the stretch (Style) of the octaves across the keyboard by optimizing the coincidence of the partials which are most likely to be heard (or lead to noticeable beating if not matched). The table below shows the overtone frequencies for selected notes on a Baldwin F piano (tuned with Verituner on an iPad using a stretch defined by Ron Koval which he labelled 4.6). The predominant overtones are indicated in bold type. For example, for A0 there is little f1 and f2 present, but then a significant contribution from f3, f4, f5, and f6, diminished f7, f8, and f9, and then a stronger contribution from f10 (see the plots on the Partials page). As we progress upward to C8, the major contributions shift increasingly towards the fundamental, with decreased contributions from overtones. For A5 and A6 the dominant partials are the fundamental and the first overtone, while for A7 only the fundamental is present to any significant extent.
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|(High resolution version of the above table (tiff).

This is to be compared to the partials which are sampled by the Verituner:
A0 6 - 12
A1 3 - 10
A2 2 - 8
A3 1 - 8
A4 1 - 8
A5 1 - 4
A6 1 - 2
A7 1

To optimize the coincidence of the partials between A0 and A1, a logical choices would be the 6th partial of A0 overlapping with the 3rd partial of A1, referred to as a 6:3 partial for A0. Another would be 10:5 (4:2 and 2:1 would not work due to lack of sampling of the 2nd and 4th partials in A0).

Since the inharmonicity of a piano is normally negligible in the temperament octave (A3-A4), the stretch is adjusted outwards from the center (typically A4). The notes below the temperament (the tenor and bass) are stretched down (flat) so that their partials will coincide with the fundamentals closer to the center of the keyboard; and the notes above the temperament are stretched upwards (sharp) so that the overtones of the notes in the middle range (alto) will coincide with the fundamentals of the higher (treble) notes. Thus the Railsback curve (which has nothing to do with temperament). See the Optimization page for an example of the Railsback curve.

For example, a 6:3 stretch point for A1 defines the 6th partial of A1 relative to the third of A2 (high lighted in yellow above), while a 4:2 partial of A5 defines the position of the second partial of A5 relative to the 4th of A4 (light blue above).

Color coding of the other partial pairs (stretch points) indicate the location of stretch points used in Ron Koval's 4.6 style (described in his “Cornucopia” on the Verituner discussion forum for users):

A0 6:3 45% / 10:5 55%
E1 6:3 100%
C2 6:3 100%
A3-4 4:2 70% / 6:3 30%
A5 4:2 50% / 4:1 50%
A6 4:2 40% / 4:1 60%
A7 4:1 100%
C8 4:1 60% / 8:1 40%

In this case, the position of A0 is pushed downwards to optimize the overlap of the 6th and 10th partials with the 3rd and 5th of A1. Using E1 and C2 allows for better definition of a change in inharmonicity in the bass. On this Baldwin, the inharmonicity was nearly flat below A3 (see the B values page).

Note: some stretch points define the positions of overtones in octaves, while others define double octaves and higher:
Double octave
Triple octave
Octave plus a fifth
Further information on intervals and octaves can be found in Baldassin's book "On Pitch".

Some more stretch examples with figures showing all the overtones:

The SK style (from Bill Fritz) is as follows:

A0 6:3 0.0 60% 10:5 0.0 40%
G1 6:3 0.0 65% 8:4 0.0 35%
G2 6:3 0.0 100%
A3-4 4:2 0.0 72% 6:3 0.0 28%
F5 4:2 0.0 50% 2:1 0.0 50%
F6 4:1 0.0 100%
C8 4:1 0.0 70% 8:1 0.0 30%

The distribution of partials for all 88 notes using this stretch on a Steinway B (with equal temperament) is shown here.
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(A high resolution image of this chart is available here.)

What seems to distinguish this style is the high level of agreement in the partial pairs used for the set points. The level of agreement is on the order of 1 Hz, usually much less, especially from G1 to G5. For example, note the 6:3 pairs for G1 (293.4 to 294.5 Hz), G2 (588.0 for both), A3 (1324.1 for both), the 4:2 pair for A3-A4 (880.9 to 881.1), and the 2:1 pairs for F4 (698.7 for both), and F5 (1400.2 to 1401.4). It creates a very nice full sound on some pianos, e.g. a Baldwin grand.

Another way to look at this stretch is with the Railsback curve, which shows the deviation of the fundamental for each note from an ideal equal tempered scale (with A4 set to 440 Hz). The curve below shows the deviations (in cents) for each note (from A0 to C8) with the "temperament octave" (A3-A4) shown in red. For comparison, the solid curve from A3 to C8 is the curve obtained from the positions of the f1 overtone indicated by Veritune in the Fine mode display. (The offsets for the fundamental are not provided by the Veritune software for notes below A3.)
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(A high resolution image of the Railsback curve is here.)
Ron Koval’s Universal 2.0 Style
This style has a significant increase in the number of stretch points in the upper treble and creates a pronounced clear bell-like quality. Note the use of only octaves with no 3:1 intervals. The resulting tune is very open, rich, full. I like it very much. It works very nicely on some pianos with temperaments other than equal temperament and I suspect that this is due to the use of only pure octaves. I especially like it with the 1/7 comma meantone temperament of Romieu (1750).

A0 6:3 35% / 8:4 30% / 8:1 35%
A1 6:3 40% / 8:4 30% / 4:1 30%
D3 6:3 40% / 4:1 30% / 4:2 30%
A3-4 6:3 30% / 4:2 40% / 2:1 30%
E5 4:1 30% / 4:2 40% / 2:1 30%
C6 4:2 30% / 4:1 40% / 2:1 30%
A#6 8:1 40% / 4:1 40% / 2:1 20%
C8 4:1 10% / 2:1 5% / 8:1 85%

A frequency mapping of the partials resulting from Koval’s Universal 2.0 with equal temperament on a Steinway B is shown below. The color coded entries indicate the positions of the stretch points. The frequencies outlined in bold boxes show the range of the most pronounced partials.
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A high resolution image of Kovals Universal 2.0 on a Steinway B with equal temperament can be found here.