Words in italics are subjects you need to know in order to understand this article.

When tuning a piano by ear, one of the first steps is to tune a clean octave. For example, after tuning A4 from the fork or sound source, many methods advocate tuning A3 by tuning the A3A4 octave as clean as possible. Many methods do not really say how to do that other than “make it clean”.

Some methods advocate tuning a pure 4:2. Some say tune a wide 4:2/narrow 6:3. The problem with these methods is that they do not always produce the best sounding octave due to the different inharmonicity of different pianos.

We need a method that allows us to measure the inharmonicity of a specific piano. That would allow us to know which octave size sounds best for that piano.

The following flow chart shows you how to tune the first octaves of a temperament when you do not have any other criteria other than “make it sound as clean and pure as possible with itself.”

Because of the different 6:3 octave sizes associated with each octave, each octave proves a different m3/M3 equality and that equality is listed below the octave scale size.