90% of this book is standard theory that all technicians learn when they begin their studies, but some of the theory has been developed by data that the author has collected.
(The following example may be of interest to tuners who enjoy using beat speeds to check octave sizes. The author is aware that many technicians do not favour octave checks and prefer to tune octaves directly. This discussion, therefore, may not be of interest to them.)
Example: Tuning pure 4:2 or wide 4:2/narrow 6:3 octaves.
This has been a question that has been debated and there are advocates for both.
The author performed a study that may help one determine when to tune a pure 4:2 and when to tune a wide 4:2/narrow 6:3.
The study used the concept of Octave Spread.
Octave Spread has been defined by the author, as how "out of tune" the 6:3 partials are, when an octave is tuned as a pure 4:2. On some pianos, the 6:3 partials are very close. On other pianos, they are very far apart.
The author has defined the following categories:
Small Octave Spread: 0.0 - 0.5 cents
Medium Octave Spread: 0.5 - 1.0 cents
Large Octave Spread: 1.0+ cents
(The cents value indicates how far apart the 6:3 partials are in an octave when it is tuned as a pure 4:2.)
When using the standard octave checks, there is a window where unequal beat speeds will "sound" equal to the tuner. This is defined by the author as The Human Limitation in Beat Speed Difference Recognition.
From tests with subjects, this limitation has been found to be about 3%. It does not depend on tuning experience.
This 3% window is very close to the 0.0 - 0.5 cent bin for the Small Octave Spread.
So, even if an octave is a wide 4:2/narrow 6:3, but also has a Small Octave Spread, it will "sound" like a pure 4:2/pure 6:3, when using the standard octave tests.
When tuning wide 4:2/narrow 6:3 (when the spread is wide enough to be heard using the standard octave tests) there is a limit when subjects indicate that the octave does not sound "clean". This limit is equal to a spread of about 1.0 cents.
If an octave has a spread of approximately more than 1.0 cents, it sounds better as a pure 4:2, which has a very narrow and wildly beating 6:3, according to subjects' responses.
The attached flow chart may be used to aurally assess the spread (which is an indication of the inharmonicity of the piano) and possibly determine the best octave size.
(The author uses "Octave Spread" and "Octave Scale" interchangeably.)
*In some cases, the octave will not tune as a pure 4:2/pure 6:3, will not sound good as a wide 4:2/narrow 6:3, and will not sound good as a pure 4:2. Sometimes, tuning it as a pure 2:1 will work. Sometimes, it may sound good as a narrow 4:2/wide 6:3, if it can be tuned as such. Rarely, the octave may have to be tuned directly, without check notes.
