How to Tune Appropriate Stretch

This is often an elusive goal; how much to stretch the treble and bass so that large chords sound good.

Many aural tuners do it by feel; they just tune each octave more or less wider, and hope the end result is pleasing. I think it’s possible to get a feel for how much is too much and how much is not enough, but it takes a very long time; you only know if the final result is appropriate after you’ve tuned a whole piano. Then you have to wait until you tune another whole piano to try something different. And trying to remember the feel or sound of each octave you tuned and repeat or do something just a little different is almost impossible, unless you are gifted with something akin to perfect pitch.

Personally, being a more technical person, I have never liked that approach. I wanted a method that allowed to get the exact same stretch each time, and also allowed me to change the amount of stretch, if I wanted.

I believe I have developed just such a method. It uses Beat Speed Windows; listening to two beat speeds – a slow one and a fast one – and tuning a third interval that beats between these slow and fast beats, that make up the beat speed window.

One step that is required to use this method is to aurally measure the inharmonicity of the piano you are tuning. It is possible. I use the check notes for the 4:2 octave and listen to the size of the 6:3.


Example: A3A4

1. Tune a pure 4:2. To do that, we make the beat speed of F3A3 equal the beat speed of F3A4.

F3A3 = F3A4

F3 is called a Check Note.

Check Note Theory:

The definition of a pure 4:2 is that the 4th partial of A3 must be the same frequency as the 2nd partial of A4.

A3(4) = A4(2)

We use the check note, F3.

Playing F3A3 puts the 5th partial of F3 very close to the 4th partial of A3. This Coincidental Partial is close to the note A5.

F3 is tuned lower than that which would produce F3(5) = A3(4) which would produce no beating at A5. We would call that a pure M3, but we set F3 to be lower than that which would produce a pure F3A3.

Since F3(5) is lower than A3(4), there is beating. And what’s more, that beating is a measure of how high A3(4) is from F3(5); the faster the beating, the higher A3(4) is from F3(5)

Now, when we play F3A4, we also have a coincidental partial at A5 because F3(5) and A4(2) are both close to A5.

And what’s more, the speed of F3A4 is a measure of how high A4(2) is from F3(5).


if F3A3 = F3A4


A3(4) and A4(2) are both high from F3(5) by the same amount,

and therefore, they are the same frequency (or very close to it),

A3(4) = A4(2)

This is the definition of a pure 4:2.

NOTE: The check note is always two octave plus a major third below the coincidental partial. (1)

2. After tuning the pure 4:2, we listen to the different beat speeds produced by the 6:3 check note.

In the case of the A3A4,
the 6:3 partial is where A3(6) and A4(3) are found.
A3(6) and A4(3) are both close to E6.

In case your harmonic series theory is weak, here are the two harmonic series’ above each note:

A3: A3 A4 E5 A5 C#6 E6
A4: A4 A5 E6

From (1), the check note is two octaves plus a M3 below E6, which is C4.

So, we listen to A3C4 and C4A4.

If A3C4 = C4A4, then we have a pure 6:3.

Here are the possible sizes for the 6:3, once we have tuned a pure 4:2, and their respective inharmonicities, which I call Octave Scale. Note that each octave you tune can only have one inharmonicity or Octave Scale.

Aurally Measuring a Piano's Inharmonicity.001

After we know the piano’s inharmonicity, we can use beat speed windows to tune an accurate temperament, and also produce a consistent and appropriate stretch.

For example, on small scale pianos, where the octaves can be tuned as pure 4:2 and pure 6:3, we can have a stretch the produces pure 11ths (octave plus fourth), pure 12ths (octave plus fifth), pure 19ths (two octaves plus a fifth), and pure 22nds (three octaves) all at the same time, and all tuned by using beat speed windows.

I have not written a book on this method but I am open to producing one. Since it would be such a small production, the price would be high. What would you be willing to pay to have me write such a book?

Your other option is to have a personal one on one skype lesson where I teach this method to you live. The cost for that is $100 per hour (2016. Subject to change. CONTACT ME to confirm pricing). I estimate the lesson would take two hours.

I look forward to hearing from you.


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4 Responses to “How to Tune Appropriate Stretch”

  1. Cobrun Sells says:

    I’d pay $40 USD for a 10-20 page booklet covering stretch. Maybe $60 USD for a 20-50 page in depth book discussing stretch to your fullest extent. I still believe that Pure 22ths are larger stretches than Pure 19ths. I dont have extensive evidence, but can gather some if you want me too. Mark, take a look at Mario Igrecs book, Pianos Inside Out! Its pricey, but his section on stretch amplified my knowledge of octave stretch that I learned from you.

    • Hi Cobrun,

      It’s a book I want to write. It’s on my list.

      Re:P19 and P22.

      The analysis I do is strictly from beat speeds. The measurements you use with the Sanderson Accutuner may or may not agree.

      Consider F3F4 and whether it is tuned as a pure 6:3 or a narrow 6:3.
      (See attached image)

      If we tuned F3F4 as a narrow 6:3, (Medium Scale piano) then,

      F3G#3 > G#3F4

      Now let’s tune a pure P19, F3C6,

      F3G#3 = G#3C6

      This produces a wide P12, F4C6. (see image.)

      G#3F4 < G#3C6 The P12 stretch and the P22 stretch are the same. The P12/P22 window is, M3 < M10 < M17 = M6 = m6 below, for our Medium Scale piano where we tune wide 4:2's, So, a pure P19 would also produce a wide P22. But, if we have a Small Scale piano, where we tune pure 6:3's, The P19 and the P22 have the same stretch, IMAGE

  2. Cobrun Sells says:

    Also, I’d like to know the difference between octave stretches (2:1 4:2 6:3 8:4) and non-octave stretches (12ths 15ths 19ths 22nds). I know the differences, but I’d like to know what you have to say. Also, I know you like to tune bass notes to the fundamental of the lower note which leaves the bass notes sharp in comparison to tuning bass octaves by 6:3 or 8:4.

    • Hi Cobrun,

      Sometimes the larger intervals, P12/P22 etc, when tuned pure, may not produce perfectly beatless octaves, in my experience. But the overall sound is better to me. Note that the octaves around the temperament are still beatless because we tuned them appropriate to the inharmonicity there.

      As for Natural Beat bass tuning advocated by Virgil Smith, I don’t think there is any way to show what size they are, unless you were sure you had actually tuned the fundamental clean. Right now, there’s no way to be sure since it’s not a scientific method.

      But one thing to consider: Using a highly accurate and precise stretch method like beat speed windows, to produce pure P12 and P22 as high and low as possible, means that, once you change methods and start using one that is less accurate and precise, the error in the extremes will be less than if you use the less accurate and precise method from the beginning.

      See the NEW POST I just wrote.

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