Balancing and Tuning the Prime Octave and 5ths
Establishing the A3/A4 Octave
Once the A4 numbers have been determined, begin by tuning a pure A3/A4 4:2 octave. To accomplish this, select a template where both A3 and A4 figures match the 2nd P and 4th P of A4 that were identified during A4 measurement. Tune the notes A3, D4, E4, and A4 according to the template.
Measuring and Adjusting the Widths of Prime 5ths
Begin by measuring the width of each of the prime 5ths and then summing their widths. It is common to find that the upper and lower 5ths have different widths. If the total width of the prime 5ths adds up to -3.0 cents, the octave is likely at a workable starting point width. When the sum is -4.0 cents, this indicates the octave is probably too narrow by about 1.0 cent. Conversely, if the sum is -2.5 cents, the octave is too wide by approximately 0.5 cent.
If the sum of the prime 5ths is not -3.0 cents, select a different template that adjusts the A3 value so that the total width of the prime 5ths falls within 0.1 cent of -3.0 cents.
If the sum of the prime 5ths is about -3.0 cents, check that the width of the prime 2:1 remains less than about 2.8 cents wide. An effective setup at this stage will have a pair of prime 5ths, each about -1.5 cents narrow (for a total of -3.0 cents), and the prime octave less than 3.0 cents wide.
However, if the prime octave is too wide (greater than 3.0 cents) when the sum of the prime 5ths is -3.0 cents, the octave may produce excessive beating. In such cases, some compromises may be needed to achieve the best sounding result. These compromises will involve adjusting both the prime octave width and the width of the prime 5ths.
Selecting the Best Sounding Template
After selecting a template that yields the best sounding octave, tune and measure the upper and lower prime 5ths (D4 and E4) to evaluate whether the width of the prime octave is optimal. A recommended width for the prime 5ths is -1.5 cents, though this value may vary depending on the piano. For further details, refer to resources about The Prime 5ths.
Balancing the Prime 5ths and Octave
While A3 and A4 remain unchanged, to balance the prime 5ths, the mid-point of the prime octave is raised or lowered. This adjustment, made possible by using templates and the LC Spreadsheet, is a unique and valuable feature of this tuning system.
Balancing the Prime 5ths Through Prime Octave Curve Adjustment
Balancing the Prime 5ths requires careful adjustment of the prime octave curve. This process involves making slight changes to the mid-point of theping A3 and A4 unchanged. By doing so, the overall balance between the upper and lower prime 5ths can be fine-tuned for the best possible sound.
For instance, consider a chart where A3 is set at 2.5 cents and A4 at 10 cents. In this scenario, a straight blue line visually connects A3 and A4, representing the linear relationship between these two notes. The note D#4 can then be positioned at different points along the curve, as indicated by the values 5.8, 5.3, and 4.8 cents on the red, green, and black lines, respectively. These positions reflect how the mid-point adjustment of the prime octave curve affects the placement and tuning of D#4 relative to A3 and A4.
Adjusting the Prime Octave Curve
The LC spreadsheet allows for this curve refinement. If the lower 5th is not as narrow as the upper 5th, balance can be achieved by lowering the D# in LC and adjusting the prime octave curve to make the lower 5th narrower and the upper 5th less narrow. This process results in evenly beating prime 5ths and 4ths, and a smoother progression of the 3rds.
Raising or lowering the center of the prime octave tuning curve deepens or shallows the curve. When both prime 5ths are tuned to about -1.5 cent width, the optimal tuning curve for the prime octave is achieved.
Since D#4 is impractical as a mapping note, D4 and E4 are measured and used instead. D#4 can be easily interpolated between the settings of D4 and E4. The LC spreadsheet uses D#4 to adjust midpoint of the curve.
For instance, if the lower 5th A3/E4) is -1.0 (not narrow enough) and the upper 5th (D4/A4) is -2.0 (too narrow), lowering both D4 and E4 by 0.5 cent will narrow the lower 5th and widen the upper 5th by 0.5 cent, resulting in both being balanced at -1.5 cent.
Lowering the center of the curve (D#4) increases the ‘dip’ in the prime octave tuning curve and raising D#4 will reduce the dip in the prime octave curve. This curve adjustment customizes the tuning curve to the particular piano being tuned. This adjustment can be checked by using the 6th/10th aural test to determine if the prime 5ths are beating similarly, or of course the intervals can be measured with a SAT.
Understanding the Prime Octave Tuning Curve
The shape and smoothness of the Prime Octave tuning curve are directly influenced by the piano’s scaling and the string lengths within that specific region. Because these factors can vary significantly from one instrument to another, the resulting curve may differ greatly between pianos.
The Role of the LC Spreadsheet in Tuning Adjustments
The LC spreadsheet, with its mapping routine, provides a practical method for identifying and applying a tuning adjustment to the middle of the Prime octave’s curve. This tool allows tuners to make precise modifications where needed. While an experienced aural tuner might make these adjustments intuitively, electronic tuning devices (ETDs) typically fail to recognize or accommodate such nuances in the tuning process.
Challenges with the D# Adjustment
Certain pianos may require a greater adjustment to the D# note to achieve a pair of balanced Prime fifths. In some cases, it becomes necessary to lower D#4 by one or one and a half cents, which creates a noticeable dip in the Prime octave’s tuning curve. However, omitting this correction can lead to even more pronounced issues with the progression of thirds or other intervals within the prime octave. Once again, ETDs do not detect or correct for this, but a skilled aural tuner is likely to notice and address it to some degree.
Identifying Issues with Some Pianos and Saving Time
Fortunately, pianos that require these exceptional adjustments are relatively rare. The method of discovering the ‘A numbers,’ the ‘4th A4 number,’ and using templates to map the prime octave and the prime fifths makes it possible to identify these exceptions early in the tuning process. This not only serves as a valuable time saver but also helps in achieving the best possible compromises for sound quality right from the start.
Outcomes of Balanced Tuning
When both prime 5ths are tuned to -1.5 cent width, the 5ths, the octave, the resulting prime 4ths (A3/D4 and E4/A4), and the major 3rds will all progress smoothly and sound good. It is important to note that even though both prime 5ths are set to the same width, the upper 5th will still beat ever so slightly faster than the lower one due to inharmonicity. The same phenomenon occurs with other intervals, such as the F3/A3, A3/C#4, and C#4/F4 major 3rds, When those 3rds are tuned to an equal width of 13.0 cents—their beat rates gradually increase because of inharmonicity.
Prime 4ths as Aural Check Indicators
The prime 4ths (A3/D4 & E4/A4) serve as excellent aural check indicators, as they beat a little faster than the 5ths, making them easier to hear. The upper prime 4th (E4/A4) should be the fastest beating interval in this 4th and 5th routine, but its beating should not be objectionable.
Al Sanderson’s Two Octave Temperament, when tuned using Direct Interval tuning, begins by tuning the five lower major 3rds to the same width. The increasing beat rates are a result of inharmonicity, not a continually widening interval in terms of cents.