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Tempered perfect fifth
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These are demonstration sound samples to accompany the book Theory and practice of piano tuning (Brian Capleton PhD).
These samples are made available independently by the author as a free educational resource, and do not form part of any contract or sale.
Licensing information
These samples are licensed for single user non-commercial educational use only and may not be redistributed.
Beating in fifths
Owing to the "traditional" theoretical model's limitations, even the concept of the beat rate itself, can often be only an approximate representation of true partial behaviour.
All partials in piano tone decay (die away), and in the case of a beating partial, the beat itself may decay at a rate different from the overall partial decay rate. Like all adjustable beat patterns in piano tone, both the beat rate and the beat amplitude in the tempered fifth can vary throughout the decay time.
All tempered intervals have more than one adjustable partial (partial whose beat rate can be adjusted), and hence more than one simultaneous beat rate. In the cases of the thirds, sixths and fourths, the lowest adjustable partial is invariably the most important, because of its relative strength.
The tempered fifths are "special case" in that the second adjustable partial (whose pitch is an octave above the first adjustable partial), is very often comparable in strength to the first adjustable partial, or even stronger than the first adjustable partial. The "traditionally" cited rates for the fifths in the central compass scale area, of less than 1 Hz, in the theory itself, apply only to the first adjustable partial.
Because adjustable beat rates in the tempered intervals can vary by 0.5 Hz or more over the course of the decay time, the precision citations of beat rates by "traditional" theory can be misleading, and can belie the true nature of tuning by beat rates.
Example tempered fifth
Below is an example tuning of the fifth G-D in the scale area on the Steinway model M, illustrating the relative strengths of the adjustable partials. The scale for each envelope is the same. Despite starting at a greater amplitude, the first adjustable partial rapidly decays so that very soon (after less than 1 second) the second adjustable partial is more prominent than the first.
The partials can be studied in detail, visually and aurally, simultaneously. If you are not already familiar with this kind of ear-training, you should find you become aware of your aural perception improving notably, as a result of "reading" the visual representation at the same time as listening.
Clicking on a graphic will launch the MP3 audio of that partial.
Click here first, to hear the fifth G-D
First adjustable partial - click picture to hear this partial
Second adjustable partial - click picture to hear this partial
In this particular example (every fifth has its own set of features), the following features can be observed visually and aurally:
The first adjustable partial has a "null" (the little "kink") at about 0.4 seconds into the decay, which gives the initial impression of a beat rate developing at more than 2 per second, but the null turns out to be isolated. The first actual beat after this establishes itself only at 2 seconds, indicating a "beat rate" after the null of 0.5 beats per second. The pattern then settles into a more regular beat with a rate of around 1 per second, but the beat amplitude is very small, so the beat itself is not very prominent.
The second adjustable partial has a more regular beat pattern, as far as the beat rate is concerned. The beat amplitude, however, is very variable. In particular, the section between 1 and 2 seconds into the decay, has a very small beat amplitude, whilst the partial amplitude is still relatively large, briefly giving the impression of a zero beat rate.
The beat patterns themselves and their effect on the tone quality of the interval, are only roughly described by the "traditional" theory's concept of a beat rate. In the art of aural tuning, the reality is that they are not used as a "tool" for achieving pitches or frequencies, but can be adjusted as a tonal feature in their own right.
We do not, however, have the freedom to just adjust each fifth in isolation, however we choose. The fifths and their beat patterns still "fit together" in a complex network, the beat patterns in the other tempered intervals also forming part of the same network. We still need to apply the knowledge of beat patterns and how they affect each other, for which the "standard" theory of beat rates, is a "rough guide".
Inheritance of beats
The apparent beat "rate" in either adjustable partial of the fifth is also subject to influence from adjustable beating inherited from the unisons, and false beating inherited from single strings. The latter may be sufficiently slow that if it is inherited by a third or sixth (which is normally fast beating), it manifests only as a slow modulation in the third's or sixth's beat amplitude. In the case of the fifth (which is normally slow beating), the effect on the perceived beat rate can be much more significant.
The art versus the theory
Attempting to apply the "rough guide" of "traditional" theory literally, will tend to produce arbitrary results over the whole, rather than results that bring out the best in the individual instrument, not least in the fifths. In the learning process, however, the "rough guide" is an essential "map" for finding one's way through acoustical territory that is otherwise unfamiliar.
