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.

How Good is an Electronic Piano’s Tuning?

There are so many good things to say about electronic pianos.

  • They’re easy to carry.
  • They have many sounds.
  • They can be hooked up to a computer.
  • They can be used as a midi controller.
  • They can be used with headphones so you can practice at night.

But, “They don’t have to be tuned” is not one of them.

A piano that does not have to be tuned, implies that the piano is already in tune.

In all my years as a piano tuner, I have been looking for an electronic piano that is in tune. Almost every electronic piano I check, is so badly out of tune, that I can use any electronic piano after I give a 20 hour course in aural piano tuning, as an example of how badly electronic pianos are tuned, and the beginners who take my classes, with their barely learned aural skills,  can tell.

We shouldn’t say, “They don’t have to be tuned.” We should say “They can’t be tuned, but they should be tuned!”

Many aural piano tuners tune the octave F3 to F4 first, setting each interval size so that they fit with each other. One criteria for equal temperament is that the major thirds increase in beat speed smoothly and evenly.

Listen to this recording of a Roland electronic piano. The first recording is of the major thirds from F3A3 to C#4F4. The second recording is filtered around the beating partial of each major third. You can easily hear how bad the tuning is. The major thirds vary wildly from progressive.

I have added a graph of the beat speeds so you can see, as well as hear the changing beat speeds.

I offer to score aural piano tunings that people send to me, and this one scored 55%. The minimum to pass is %80.

Roland Electronic Piano. Major Thirds. Unfiltered.

Roland Electronic Piano. Major Thirds. Filtered.

Graph of Beat Speeds.
The beat speeds should be within the green lines.


Also, listen to the first beating major third. The beat speed actually starts at 7.8 beats per second, but by the end of the interval, it has risen to 10 beats per second!

Electronic pianos have many benefits. But being in tune is not one of them.

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.




1) Learn how to use the extremely accurate Go APE system. 

2) Use bisecting beat speeds windows to produce accurate and precise ET and stretch. 

3) Understand Stability. 

4) Develop your “Tuner’s Ear”; the ability to filter out all unwanted noise and hear beats clearly. 

This method is appropriate for beginners and experienced tuners. 


Two Sessions:

8am to 12pm, $350USD.($440CAD)

1pm to 5pm, $350USD.($440CAD)

Both sessions, $595USD.($740CAD)

These workshops will be given by Mark Cerisano, RPT, who has been teaching piano tuning and repair to hundreds of people since 2006. Read more and get free video lessons and resources at HOWTOTUNEPIANOS.COM

Electronic Piano On Trial

As some of you may know, I provide a temperament analysis service. (For more information, CLICK HERE) The first one is free.

One of my subscribers sent me a recording of the temperament on his electronic keyboard. You can read and hear the interesting results HERE.

FREE Meeting – Piano Tuning

You are invited to a free meeting where I will be presenting some of the elements of my aural piano tuning method I call the Go APE method. It was developed to provide an Accurate, Precise, and Efficient way for beginners and experienced tuners to tune pianos aurally. It is easy to follow but demanding on the specific skills needed to tune a piano to a high standard.

Here are the details:

WHERE: Ottawa Pianos, 1412 Bank Street, Ottawa, Ontario.
WHEN: Wednesday, April 20th. 7pm – 9pm. (TOMORROW!)

Just show up.

This is a regular meeting of the Ottawa Chapter of the Piano Technicians Guild.

The Piano Technicians Guild is an invaluable resource for anyone wishing to advance their skills, no matter their level. I strongly encourage all of you to attend if you are within driving distance.

Hope to see you there,

P.S. You do NOT need to be a member of the PTG to attend!

New Lesson on Measuring Equal Temperament

Use this method to super train your piano tuner’s ear!

CLICK HERE for the free lesson.

New Lesson on Unisons

This lesson gives you insight into how clean a unison should sound and how to listen to tell if a unison can be cleaner.

It also is an introduction to the resolution possible with Double String Unison, a technique where the tuner mutes one string of a trichord, and moves the resulting double string up or down as needed.

CLICK HERE to view the lesson.

Piano Tuning and Repair Workshops in Montreal, Quebec. April 11 – 15, 2016

I’m looking for people who are interested in a live course with me, on the topics of aural piano tuning and basic piano repair and regulation.

The course will be given at Westend Pianos in Montreal Ouest.

Each course is 20 hours long, and while you will not be a piano tuner after 40 hours, you will have a great headstart and be able to start practicing using some excellent methods.

Get more information below:
Piano Tuning Workshop
Piano Tuning Tools
Piano Repair Workshop
Piano Repair Tools

If you are interested, please contact me as soon as possible because I need to prepare your tools.


What You NEED to Know to Tune Pianos Aurally.

Screen Shot 2016-03-19 at 5.45.49 PM

I have been teaching piano tuning since 2006 and in that time I have constantly struggled with this question: What do people really need to know in order to tune pianos aurally.

I have boiled it down to three skills:

1. The ability to set pitches and have them stay. (Stability)

2. The ability to make fine adjustments to pitch and have them stay.

3. The ability to hear beats clearly. (Beat Speed Clarity)

(Tuners must also identify small differences in beat speeds, but my research has shown that if we can hear beat speeds clearly, this is actually easy. So, it is the ability to hear these beats clearly that is the bottleneck; get to the point where you can hear beats clearly, and tuning aurally becomes much easier.)

#1 and #2 can be learned through study but #3 is the only skill that can not be rushed; the brain and ear must change and that takes time and practice. (See the study that measured the brains of aural tuners.)

Many piano tuners today use electronic aids to help them tune pianos. Many use them as a tool, but many more use them as a crutch. I believe the struggle to hear beats clearly is what has caused many of these tuners to give up on the idea of learning to tune a piano aurally.

It doesn’t have to be that way. Historically, learning to tune a piano has been a very difficult process. But now, with the help of technology, we can speed up the process. We can learn what the beats sound like and measure them.


In order to help people improve their Beat Speed Clarity, I have designed a tool that shows you what your beat speeds sound like and helps you hear them more clearly.

You simply send me a recording of each note in your temperament, and I put them together and measure the beat speeds of the M3, P4, etc, and show them to you in a graph.

I then ask you to confirm what I’ve measured.

I also analyse the tuning to find notes that could be improved; notes that if they were changed slightly, would smooth out the beat speed progressions, and move the tuning closer to ideal Equal Temperament. The logic of these analyses is explained clearly.

Some of my subscribers have already taken advantage of this free offer and you can listen to their tunings HERE.

Click the link above and you will see a list of different tunings that I have analysed for others. See if you can confirm my findings.

I am very excited that this method may be a tool that can improve your beat speed clarity and therefore reduce the time needed for people to learn to tune pianos.

I wish this resource was available to me when I started learning how to tune pianos aurally.

Your comments are welcomed.


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