Measuring the “4th” A4 Number
After determining the A4 numbers, the next step is to locate the “4th” A4 number. To measure the “4th” A4 number, technicians focus on the 2nd partial of A4. This “4th A4” value represents the width of the A3/A4 2:1 interval after A3 has been tuned as a pure 4:2 (from A4).
For example, if the 2nd partial of A4 is found to be at 2.0 cents, the SAT is set so that A5 equals 2.0 cents. Then, A3 is tuned to this setting, resulting in a pure A3/A4 4:2 octave. The width of this 4:2 octave is then measured as a 2:1. The resulting measurement is the “4th” A4 number: the width of the A3/A4 2:1 interval after it has been tuned as a pure 4:2.
Typically, the resulting 2:1 octave width is displayed as a negative value, indicating a wide 2:1 interval. On most pianos the 4th A4 number will have a value between 0 and -3.5 c., or even -4.0 cents, though the extremes are rare. Commonly, pianos register between -..6 and -2.5 cents. A width of 0 is extremely rare.
The “4th” A4 Number’s Significance in Tuning the Prime Octave
The “4th” A4 number is highly useful during the mapping and tuning the Prime octave. Although discussions on octave stretch often revolve around the A3/A4 4:2, the A3/A4 2:1 interval is just as important—if not more—because unwanted beating typically arises in the 2:1 first. While the 4:2 starts as pure, the 2:1 is inherently wide when the 4:2 is pure.
Evaluating Acceptable Prime Octave Widths
Technicians have individual preferences and tolerances regarding octave width in this area of the piano, but an accepted standard for the A3/A4 2:1 interval width is that it should not be much wider than about 3.0 cents. If the interval exceeds approximately 3 cents, it’s beating may be considered objectionable. Widths less than 3.0 cents are generally acceptable, but as the interval grows wider than 3 c., some technicians may begin to find it problematic.
There are, of course, exceptions due to the unique characteristics of each piano. On some instruments, because of 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 such cases, if the 2:1 width is 3 cents wide, the Prime fifths may be too narrow. To achieve the best compromise, the octave may need to be widened beyond the -3.0 c. threshold, resulting in both a wider than usual Prime octave and narrower than usual Prime fifths. However, this compromised adjustment often produces the best sound for that specific piano.
The “4th” A4 Number as an Early Indicator
The “4th” A4 number serves as an early indicator during the mapping process. If this value is above 3 or below 0.5, it suggests significant adjustments and compromises may be necessary to achieve the best tonal result on this specific piano. Being able to recognize this early in the process allows technicians to anticipate potential challenges and address them efficiently, ultimately saving time.
Using the “4th” A4 Number as a Guide
By measuring the “4th” A4 number, technicians can quickly and accurately determine how much they can widen the A3/A4 2:1 octave before reaching the -3.0 cents threshold. What is considered “objectionable beating” will differ from one technician to another. To determine one’s personal threshold, it is helpful to tune the A3/A4 octave as a 2:1 at various widths and listen for the point at which the beating becomes undesirable.
It is best though to maintain the ‘hierarchy’ of the intervals, meaning the octave should beat the least, then the 5ths, then the 4ths, then the major 3rds, then the minor 3rds, and so on. When compromising, it is still best to have the prime 5ths beating faster than the prime octave.