Introduction to the “4th” A4 Number
The “4th” A4 number is an essential measurement in this routine. While the first three A4 numbers identify the locations of A4’s 2nd, 4th, and 8th partials, the “4th” A4 number refers to the resultant width of the A3/A4 2:1 after it’s been tuned as a pure 4:2. This measurement is particularly valuable when mapping the Prime octave, as it helps establish the best sounding relationship between A3 and A4.
Measuring the “4th” A4 Number
To measure the “4th” A4 number, the 2nd Partial of A4 is used. After the A4 numbers are determined, the next step is to locate the “4th” A4 number. The “4th” A4 number represents the width of the A3/A4 2:1 after A3 has been tuned as a pure 4:2 from A4.
For example, if the 2nd partial of A4 is measured at 2.0 cents, the SAT is set to( A5 @ 2.0). A3 is then tuned to that setting resulting in a pure A3/A4 4:2 octave. The width of this 4:2 octave is then measured as a 2:1. That’s the “4th” A4 Number: the width of the A3/A4 2:1 after it’s been tuned as a pure 4:2.
The width of the resulting 2:1 octave is typically a negative value—indicating a wide 2:1 octave. This number usually falls between 0 and -3.5 or even -4.0 cents, depending on the piano. While the extremes are uncommon, most pianos will register between -1.0 and -2.5 cents. A width of 0 is very very rare.
Significance in Tuning the Prime Octave
The “4th” A4 number is extremely useful for mapping the Prime octave during the tuning process. Although discussions surrounding octave stretch often focus on the A3/A4 4:2, the A3/4 2:1 is equally important—perhaps even more so—because objectionable beating tends to occur there first. While the 4:2 starts pure, the 2:1 is inherently wide as a result of the 4:2.
Evaluating Acceptable Widths
Technicians have individual aural preferences and tolerances for octave width, but a fairly common acceptable width for the A3/A4 2:1 is, it should not be much wider than about 3.0 c.. More than about 3. c. wide and the 2:1 octave may be considered objectionable. Less than 3.0 c. wide is fine, but when it starts to get more than 3 c. wide some may start to find it objectionable.
There are exceptions of course. After all, it’s a piano! On some pianos, due to inharmonicity in the scaling, the A3/A4 2:1 may need to be wider to accommodate the widths of a pair of Prime fifths (A3/E4 and D4/A4). In these cases, when the 2:1 is 3 cents wide, or wider, the Prime fifths can be too narrow. In order to widen the prime 5ths, good sounding compromise requires widening the octave further, resulting in both a wider than normal Prime octave and narrower than usual Prime fifths. However this adjustment will probably yield the best possible compromise and sound for that particular piano.
An Early Indicator
The “4th” A4 number can act as an early indicator during the mapping and tuning process. When the “4th” A4 number is above 3 or below 0.5, it signals unusual adjustments and compromises will be necessary for the best sounding result. Recognizing this early is very advantageous, as it allows technicians to anticipate and address potential issues efficiently, saving valuable time.
Using the “4th” A4 Number as a Guide
By measuring the “4th” A4 number, technicians can quickly and easily determine how much they can widen the A3/A4 2:1 octave before approaching the -3.0 cents threshold. “Objectionable beating” occurs varies from one technician to another. To learn one’s threshold with this, it’s helpful to tune the A3/A4 octave as a 2:1 at various widths and listen for when the beating becomes objectionable.