|(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
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:l
Octave plus a fifth
Further information on intervals and octaves can be found in Baldassin's book "On Pitch".