Understanding Unusually High “4th” A4 Numbers

Understanding Unusually High “4th” A4 Numbers

Introduction

This section explores the rare phenomenon of extremely low A4 numbers in Baldwin Hamilton pianos. The discussion focuses on a specific example featuring a 4th Partial (P) of A4 at 6.4 cents and an exceptionally low 2nd Partial of A4 at just 0.6 cents sharp. These values are notably uncommon, as demonstrated by a collection of over 800 midrange mapping templates, most of which exhibit higher A4 numbers.

Significance of Low Inharmonicity (iH) Values

Low inharmonicity values can be desirable, but their advantages are not absolute. In this context, the 2nd Partial number is particularly important because it is used in tuning the pure A3/A4 4:2 octave. Although low iH is generally considered beneficial, it may not always yield optimal results for every tuning scenario.

Case Study: Baldwin Hamilton S# 193416

On one Baldwin Hamilton piano (S# 193416), the 4th A4 Number was measured at 4.2 cents. This means that when A3 is tuned as a pure 4:2, the resulting A3/A4 2:1 octave is 4.2 cents wide. Such a wide octave produces excessive beating, which is usually considered undesirable in piano tuning.

The measured A4 numbers for this piano are as follows:

• 4th Partial of A4: 6.4 cents
• 8th Partial of A4: 29.2 cents
• 2nd Partial of A4: 0.6 cents
• 4th A4 Number: 4.2 cents (when A3 is tuned pure 4:2, measured as a 2:1, the width is 4.2 cents)

All of these A4 numbers are relatively low compared to most other pianos, especially the 2nd Partial number.

Rarity in Template Collection

To put these figures into perspective, among over 800 templates used for mapping the midrange, the lowest A4 number found is 4.2. Templates with such low numbers are rarely used because pianos with these characteristics are extremely uncommon. Specifically, only 10 templates fall within the 4.0–4.9 range, about 19 in the 5.0–5.9 range, and approximately 30 in the 6.0–6.9 range, with 20 of these being 6.5 or above. In total, about 740 templates exceed the Baldwin Hamilton’s 4th Partial number of 6.4.

Implications for Tuning: The 2nd Partial Number

The 2nd Partial number for this piano, at 0.6 cents sharp, reflects exceptionally low inharmonicity. This value is critical because it is directly used in tuning the pure A3/A4 4:2 octave. Although low inharmonicity is often valued, it does not guarantee optimal results in every tuning circumstance.

Octave Widths and Their Relationships

In tuning, there are few absolutes. One guiding principle is that both the A3/A4 2:1 and the A4/A5 2:1 octaves should be tuned wide; however, the exact width of these octaves can vary. A well-tuned A3/A4 2:1 octave should accommodate a pair of balanced prime fifths, which in turn produce good-sounding resultant prime fourths (A3/D4 and E4/A4).

A 4th A4 number of 4.2 cents immediately indicates that a pure A3/A4 4:2 will result in excessive beating at the 2:1. Contracting the 2:1 by raising A3 will narrow the A3/A4 4:2. To achieve an octave width closer to 3.0 cents—a reasonable maximum—an adjustment of about 1.2 cents is necessary.

Balancing Octaves and Fifths

There is a useful relationship between the width of the prime octave and the widths of the prime fifths. Contracting the A3/A4 octave affects all intervals within it, including the prime fifths. Most pianos yield satisfactory results with a prime 2:1 octave less than 3.0 cents wide, which allows for a pair of balanced prime fifths at approximately –1.5 cents narrow.

Pianos like this Baldwin Hamilton require a different approach. Achieving balance between the fifths is particularly important in these cases. After adjusting the octave to be near 3.0 cents wide, one fifth measured –3.0 cents and the other –1.0 cent, resulting in an unbalanced pair. The narrower fifth can be problematic for most listeners.

When the prime octave width is set at 3.0 cents, the combined width of both prime fifths totals –4.0 cents. In pursuit of balance, adjusting both fifths to –2.0 cents narrow allows them to beat equally and may produce an acceptable sound. However, if both fifths still beat too rapidly, further compromise of the 3.0-cent octave width may be necessary.

Finding the Best Compromise

In some cases, contracting the octave to 3.0 cents causes the fifths to sound excessively narrow and unacceptable. The relationship between the octave and the fifths must be considered holistically, seeking the best overall compromise. For example, one successful compromise resulted in a prime 2:1 octave width of 3.4 or 3.5 cents and a pair of fifths each at –2.2 cents narrow, which produced a pleasing result.

The tuning process begins with establishing the temperament. Only by tailoring the temperament to the piano’s unique scale can a pleasing overall tuning be achieved. Ultimately, the final decision is made by ear, aiming for the best possible compromise. As Al Sanderson once remarked, the goal is to find “the least bad sounding compromise.” Being mindful of the specific trade-offs involved greatly aids in making such a piano sound its best.