How to map A3 on the piano

Establishing and Balancing the Prime Octave

Introduction

This section provides an explanation of using an Accu-Tuner and templates to map specific target notes for tuning the Prime octave.  The tuning system begins by establishing the relationship between the prime octave (A3/A4) and the prime 5ths (A3/E4 and D4/A4).

Tuning the Prime Octave

The tuning process begins with the prime octave. Notes A4, A3, D4, and E4 are tuned according to the selected template. The A4 Number provides key information about the prime octave’s width, while the widths of the prime 5ths (A3/E4 and D4/A4) must be measured directly to ensure accuracy.

Measuring and Adjusting Prime 5ths

After tuning A4, A3, D4, and E3 to the template, it is common to find that the prime 5ths have unequal widths—for example, one measuring -2.2 cents and the other -1.2 cents. At this stage, the focus is on achieving a combined width of -3.0 cents for the prime 5ths, rather than their individual equality. If the combined width is greater than -3.0 cents (such as -4.0 cents), it indicates that the prime octave is too narrow and must be widened by approximately 1.0 cent. Conversely, if the combined width is less than -3.0 cents (such as -2.0 cents), the octave is too wide and should be narrowed by 1.0 cent. Adjustments are made by selecting a template with a slightly higher or lower A3 setting, then retuning A4, A3, D4, and E4. After adjustment, the widths of the prime 5ths should be close to the target combined value of -3.0 cents.

At this point, the individual widths of the prime 5ths are not critical; balancing them is addressed later in the process. The primary objective is to reach a combined width of -3.0 cents for the prime 5ths. Most pianos will sound satisfactory with this approach.

Listening to the prime 5ths and the resulting prime 4ths (A3/D4 and E4/A4) may reveal slight differences in the beating pattern of the 5ths. When the prime 5ths are unbalanced, these differences often become apparent in the 4ths, which beat slightly faster than the 5ths. Comparing the upper and lower prime 4ths can help confirm the difference in the widths of the prime 5ths and indicate the direction in which the midpoint of the prime octave should be adjusted to achieve a better balance for both the 5ths and 4ths.

Balancing the Prime 5ths

Once the difference in width between the prime 5ths is known, adjusting them to achieve equality is straightforward. For example, if the lower 5th measures -2.0 cents and the upper 5th measures -1.0 cent, lowering both by 0.5 cent will result in balanced widths of -1.5 cents each. This adjustment is made by lowering the settings for D4 and E4 on the template by 0.5 cent and retuning each note. After retuning, the prime 5ths and the associated prime 4ths should sound considerably improved.

Provided the prime octave 2:1 interval is not wider than about 2.8 cents, the overall sound of the octave should remain pleasing. These adjustments establish accurate settings for A4, A3, D4, and E4, which are used throughout the remainder of the mapping process and for generating the tuning with the Littau-Conrad Spreadsheet. While these guidelines are effective for most pianos, some instruments may require further fine-tuning. Consistent practice and experience are essential for developing the necessary precision.

Considerations for the Prime Octave

For most pianos, keeping the prime octave’s 2:1 width below 3 cents ensures tolerable beating. In some cases, a slightly wider octave may be necessary, but generally, maintaining a width “no wider than 3.0 cents” is a reliable starting point. The prime octave’s width depends largely on the widths of the prime 5ths, making these intervals crucial to the piano’s overall sound quality.

Some subjectivity is involved, as preferences for wider octave widths can vary. The piano itself should inform the process, with the goal being minimal beating. The prime octave should not be wider than necessary, but must accommodate a pair of well-sounding prime 5ths. Since every piano is unique, a tailored approach may be required for each instrument. Nonetheless, the outlined procedure offers a dependable foundation and helps quickly determine whether a piano fits the typical “most pianos” category.

Occasionally, a pure 4:2 (prime octave) may result in a 2:1 that is too wide and fast-beating, while excessively narrowing the octave can cause the 5ths to beat too quickly. For challenging pianos, compromises may be necessary, sometimes resulting in a prime octave’s 2:1 wider than 3.0 cents and prime 5ths narrower than -1.5 cents. The aim is to find the best-sounding—or at least the least bad-sounding—compromise for both intervals.

A balanced pair of prime 5ths, regardless of their absolute width, usually produces a satisfactory overall piano sound. The process involves iterative adjustments to achieve the most harmonious balance between the octave and the 5ths.

With experience, it becomes evident that some pianos, especially those with shorter string scales, require a nuanced approach to sound their best. If the combined width of the 5ths is -3.0 cents and the prime octave’s 2:1 is less than 2.8 cents wide, the Mapping of A3 can be considered complete.

Final Balancing: The Prime Octave and Sub-Prime Octave

To optimize the relationship between the prime octave and the prime 5ths, the prime 5ths must be balanced, which also brings balance to the resulting prime 4ths (A3/D4 and E4/A4). Balancing is achieved by adjusting the midpoint of the prime octave—technically D#4, which is determined by averaging the settings for D4 and E4. The midpoint (D#4) is displayed in the spreadsheet for reference.

The spreadsheet also generates settings for D3, D#3, and E3 in the third octave, allowing for midpoint adjustment in the sub-prime octave. While this adjustment can assist in balancing the sub-prime 5ths, its main function is to shape the curve of the sub-prime octave to better suit the piano’s scale, particularly where transitions occur between string types and bridges.

The same template used for the prime octave is also applied to the sub-prime octave, extending the established relationship. Typically, sub-prime 5ths are not as balanced as those in the prime octave, with the upper sub-prime 5th often being slightly narrower. As tuning progresses into the bass, the 5ths tend to become less narrow.

Midpoint adjustment facilitates good-sounding sub-prime 5ths (A2/E3 and D3/A3), octaves (D3/D4, E3/E4, A2/A3), double octaves (A2/A4), and 12ths (A2/E4, D3/A4). All these settings are incorporated by the Littau-Conrad spreadsheet to generate a tuning that passes through all key target and mini-target notes.

Having the ability to adjust the midpoint in both the prime and sub-prime octaves is a unique feature of this system. For example, if the lower prime 5th is -2.0 cents and the upper is -1.0 cent, raising the midpoint (D#4) by 0.5 cent brings both to -1.5 cents, achieving balance. This small adjustment not only improves the 5ths but also refines the resultant 4ths and other intervals, such as major thirds, within the prime octave.

The capability to shape the tuning curve in both the prime and sub-prime octaves—and to have these adjustments reflected in the tuning generated by the Littau-Conrad spreadsheet—is a distinctive and valuable aspect of this piano tuning method.