Piano Midrange Mapping: Tuning Templates and Procedures

Piano Midrange Mapping: Tuning Templates and Procedures

Overview

This section details the process of selecting and utilizing tuning templates for correctly mapping the midrange of pianos, with a special focus on scenarios where the 4th A4 Number is less than 2.0 cents. The steps include choosing the appropriate template based on A4 and A3 values, tuning key notes, measuring and adjusting the widths of prime fifths, and refining the mapping to ensure the prime octave is sized within ideal parameters. The objective is to achieve a prime octave that supports optimal piano tuning.

Procedure 1: Standard Process for Most Pianos

This procedure applies to pianos with a 4th A4 Number generally less than 2.0 cents. In such cases, the prime octave’s 2:1 can be either expanded or contracted, as dictated by the prime fifths, while still remaining under 3.0 cents in width. Typically, the prime 4:2 octave will be wide or pure, indicating a workable mapping for A3 and the prime octave.

Templates are employed to map the midrange (A3/A4). Selection of a template (loaded into the SAT) is guided by the A4 Numbers and the 4th A4 Number, which initiate the mapping process for A3. Starting with a template that yields a pure 4:2 prime octave is recommended.

For example, if the 4th P of A4 is 10 and the 2nd P of A4 is 1.5, a template featuring A4 at 10 and A3 at 1.5 is a good starting point. Tune A4, A3, D4, and E4 to this template. Then, measure the lower (A3/E4) and upper (D4/A4) fifths and sum their widths to assess proximity to -3.0. If the sum is -3.0, the prime octave width is likely appropriate.

If the combined width measures -2.6 cents (not as narrow as -3.0), the octave is probably too wide. In this case, select a template with an A3 value 0.4 cents higher, re-tune A3, D4, and E4, and re-measure the fifths (e.g., transition from an A3 of 1.5 to 1.9 cents). The sum should approach -3.0.

When the sum of the prime fifths is -3.0 cents and the prime octave 2:1 is less than 2.8 cents wide, A3 is likely mapped correctly. If the sum is -3.6 cents (narrower than -3.0), the octave is too narrow; in this situation, select a template with an A3 setting 0.6 cents lower (e.g., from 1.5 to 0.9 cents). Again, check for the -3.0 sum and a prime octave 2:1 width below 2.8 cents to confirm proper mapping.

The process involves trying several templates to locate A3 such that the prime octave width remains under 2.8 cents and the sum of the prime fifths is -3.0.

Balancing the Prime Fifths

Once the above conditions are met, the next step is to balance the prime fifths by adjusting the template locations for both D4 and E4. If the lower fifth’s width is -1.9 cents and the upper is -1.1 cents, the lower is 0.4 cents too narrow compared to -1.5, and the upper is 0.4 cents not narrow enough. Adding 0.4 to both D4 and E4 settings will balance the fifths.

Refer to the template settings for D4 and E4, add 0.4 to each, and re-tune these notes. The result should be a well-tuned prime octave less than 2.8 cents wide, two balanced prime fifths at -1.5 cents each, and a pair of good-sounding resultant prime fourths.

Document these D4 and E4 settings as part of the mapping notes for the prime octave. Balancing the fifths ensures the tuning curve aligns with the piano’s scaling in this range.

Procedure 2: Handling Exceptions

Exceptions arise with pianos exhibiting a 4th A4 Number greater than 3.0 cents. Such a high number signals that a pure 4:2 yields an overly wide 2:1. In these cases, the priority shifts to controlling the octave, preventing excessive beating. A relationship exists between the prime octave and prime fifths—sometimes, the fifths must be narrower than -1.5 cents, yielding a sum narrower than -3.0. After balancing, each fifth may be -1.7, -1.9, or even -2.0, depending on the instrument.

A 4th A4 Number over 3.0 cents suggests the best result may be a narrow prime 4:2 octave and prime fifths narrower than -1.5 cents. These exceptions may require exceeding the ‘no wider than 2.8 cents’ guideline for the prime 2:1 octave. Double Octave Beat (DOB) adjustments are useful, allowing fine-tuning of template widths during mapping, particularly if templates with sufficiently low A3 numbers are unavailable.

Although the general procedure remains the same, high 4th A4 Numbers mark the piano as an exception. For instance, if the 4th P of A4 is 9.5, the 2nd P of A4 is 1.0, and the 4th A4 Number is 3.6, then tuning A3 as a pure 4:2 results in a 2:1 between A3 and A4 that is 3.6 cents wide. Select a template with A3 at 1.0 and A4 at 9.5, tune the relevant notes, and measure the fifths. The sum may approximate -3.0, but the octave might beat excessively.

If the prime octave’s 2:1 is too wide (e.g., 3.6 cents), raise A3 by 0.6 cents (to 1.6) to narrow the octave toward 3.0 cents width. Re-tune and re-measure; the fifths may now total -3.6 cents, and when balanced, each will be about -1.8 cents. This is an acceptable result.

Recall the theoretical fifth width of -2.0 cents. Further narrowing the octave (e.g., to -2.6) will yield fifths near this value, which typically sound fine. Since changes to the octave split between two fifths, a change of 0.8 cents in the octave affects each fifth by only 0.6 cents.

Using Double Octave Beat (DOB)

DOB allows subtle adjustments without referencing new templates. It can widen or contract the tuning, keeping A4 unchanged, and spreading changes outward in both directions. For instance, a 0.1 DOB adjustment shifts A3 by 0.2 cents. To try a slightly higher A3, use a -0.1 DOB, re-tune A3, D4, and E4, and re-measure the fifths.

At this stage, focus solely on the sum of the fifths, regardless of their individual widths. Once the sum is acceptable, balance the fifths as the final step. For example, with a lower fifth of -2.3 and an upper of -1.5 (sum: 3.8), balancing will bring each to -1.9 cents. Add 0.4 to both D4 and E4 template settings, re-tune, and check the results. If executed correctly, the outcome should be satisfactory.

Document the settings for A4, A3, D4, and E4 for use in the LC spreadsheet during tuning creation. This completes the mapping of A3 for the exception piano.