Error in Stretch Methods

I have developed what I believe to be a very accurate method of producing consistent stretch that creates pure P11/P12/P22 and even P19 if we are tuning a Small Scale piano. (Small Scale pianos are pianos whose F3F43 and A3A4 octaves can be tuned to a pure 4:2 and a pure 6:3 simultaneously.)

But this beat speed window method can’t be used in the extremes because the beats are too fast (treble) or too slow (bass), so I use a different, less accurate/precise method.

For the treble I just listen to the P8 (octave), P12 (octave plus fifth) and P22 (triple octave) below and try to get them to sound as clean as possible.

For the bass, I listen to what Virgil Smith called, “The Natural Beat”. There’s no good science to prove that when playing a bass octave, there is a beat produced at the fundamental of the bottom note if the octave is not clean, but that’s what Virgil said he did, and that’s what I do.

If this method is not as accurate or precise as the beat speed window method, I still think it is appropriate because the error is so small.

For example, consider the Railsback curve, shown below, that shows the sharp treble and flat bass of a piano tuned by a “fine tuner”.


Now, let’s consider that the green line represents the “perfect” stretch, or at least the P12/P22 stretch produced by my beat speed windows.

Pure 12:22 stretch

Now let’s consider that my stretch method in the extremes, is not as accurate. Once I begin using that method, there will be error.

Minor error using inaccurate method in extremes only

However, if I was to use an inaccurate method from the beginning, like just listening to octaves, the error would start closer to the center, and the accumulated error would be huge once we would get to the extremes.

Major error using inaccurate stretch everywhere

Looking at the above curve reminds me of what my tunings used to sound like before I started using beat speed windows for stretch. I could never get the treble and bass to sound clean enough for my ear.

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5 Responses to “Error in Stretch Methods”

  1. George maalouf says:

    Mr. Mark, In the name of all piano tuners and technicians around the world, and in my name, I thank you for your great work and efforts in contributing for the science of piano tuning in such an altruistic spirit… In the name of human knowledge, music and science, I thank you also.

  2. Cobrun Sells says:

    I really hope that book on Stretch you plan on writing makes its debut, because I still have some confusion. Why do you say that P12s and P22s are the same? I have heard on my piano that a P22 is much larger of a stretch than P12. Is there something I’m doing wrong? My piano needs to be tuned again so I’ll re-verify my observations. I’ve also found that P22s are larger than P19s in contrary to what you have mentioned to me.
    Also, Virgil Smith’s “Natural Beat” does have good science behind it. It has very good science behind it. Because A0 is theoretically 27.5Hz and A1 is 55Hz when I play A0A1 the resultant tone is 27.5Hz. On pipe organs there is a stop called the “Resultant tone” stop. It plays the P5 above the note being played so that the two pitches (lowest note and P5) will subtract frequencies and produce an octave lower than the lowest note already being played.
    The same thing happens with the piano. If A1 is 55Hz and A0 is 25 Hz then the resultant tone is 30Hz which beats against 25Hz at 5beats. A0 would therefore be 5Hz flat if I was tuning it to the “Natural Beat” of A1.
    But, as I mentioned in a comment a few days ago, tuning a lower note to the natural beat of the octave above leaves that lower note a tad sharp in comparison to tuning the lower note as an 8:4 octave or a 6:3 octave.

    • Cobrun Sells says:

      I’ve been pretty busy the past few days and forget that I already mentioned this information and personal observations on another recent blog comment post. Please disregard the comments above on my part.

      But, Mark, I now see what you mean about the “natural beat.” The natural beat may be narrower than a 6:3 or 8:4 octave, it may be the same as a 6:3 or 8:4 octave, or it may be wider than a 6:3 or 8:4 octave depending on the notes’ inharmonic coefficient. Mario Igrec in Pianos Inside Out book explains that the tenor to bass bridge break sometimes has horrible interval beats speed progressions because of the break and poorly scaled string diameters, tensions, and lengths. This poor break may leave upper bass notes less inharmonic than lower tenor notes. This makes upper bass notes sharper than one would expect (when tuned with the octave above) and may actually make upper bass notes narrow octaves with the their corresponding notes an octave higher than them if tuned at the 4:2 or 6:3 level. Albeit, the 2:1 level would still be wide no matter what. I don’t doubt poor scaling may also have an effect in the deep bass obscuring our measurements of natural beat octaves versus 6:3 or 8:4 octaves.

    • Hi Cobrun,

      Just saw this post late.

      The P12 and the P22 are pure if all the notes have been tuned that way.

      The specific P12 and P22 that I test are relative to the top note in the downward direction.

      Example: Tuning C6
      P12 down is F4C6
      P22 down is C3C6

      Test for the P12 is M6 = M17
      G#3F4 = G#3C6

      Test for the P22 is m6down = M17
      C3G#3 = G#3C6

      Notice that the P12 and the P22 will be pure if C3F4 is tuned right.

      If you look closely at the two tests, you can see that C3G#3 =G#3F4
      That is m6down = M6, the test for the pure P11!

      So, it is possible to tune P12, P22 AND P11!

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