On scales, tone, pitch (and piano tuning)

See also, on music and mathematics

with interactive media

On scales, tone, pitch and piano tuning

- with interactive media

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Turn speaker or headphone volume right down first, and listen quietly.

If using headphones, listen very quietly.

Scales

A musical scale (or mode) can be constructed in any number of ways. It is a just set of pitches that are arranged in ascending or descending order. An example of the familiar modern Western chromatic scale, with 12 degrees (pitch steps) to the octave, played in simple tones, and ascending, sounds like this. Or the modern Western diatonic scale, with 7 degrees to the octave, ascending sounds like this. Descending versions are here and here.

These scales are said to preserve the octave. That is, there are a certain number of degrees or pitch steps in the complete scale, and when all the steps have been completed, the bottom note and the top note are an octave apart.

More odd sounding scales that preserve the octave are here and here. A scale that does not preserve the octave, but preserves a double octave, can be heard here.

A scale that had 100 degrees to the octave might sound like this.

Scales can even be constructed so that after all the rising steps of the scale, the last note is still the same as the first note (Shepard's scale).

Clearly there is no universal principle preventing us from creating whatever kind of scale we choose. The above examples, it is true to say, are electronically generated, but it would be perfectly possible to reproduce most of them using suitably designed musical instruments.

The issues caused by natural acoustical principles only begin to arise when two or more notes are played together. As discussed on the previous two pages, the question then arises, of the tone of the interval in relation to the perceived pitches of the notes, in addition to consideration of the tones of the individual notes.

The structure of the recipe of partials in the tones of strings (see the previous pages), leads to a situation in which the tone qualities of the intervals can vary radically, depending on the size of the interval. A small change in the size of this interval, can lead to a radical change in its tone quality, so that it sounds like this. This change is in addition to, but distinct from any change of perceived pitch interval. Nevertheless, perceived pitch is not necessarily independent of perceived tone. Now there is no intrinsic reason why this tone should not be culturally acceptable for music making, if you happen to like that kind of tone, and pitch interval. Strings tuned together, however, also provide intervals with tones like this, which certainly in Western culture is the preferred form. Ultimately, what you find beautiful or ugly in sound is a matter of culture, philosophy, conditioning, an training, as well as of acoustical principles. In some situations, for some musical genres, such as "honky tonk" piano, this tone for a piano note, would be preferable to this tone. In most situations, it would be the other way around, because the first tone would be regarded as "out of tune". The term "out of tune" ultimately just means the interval or note has not got the kind of tone and/or pitch intonation you want or expect. Neither of these tones are "in tune" by the normal expectations of piano tuning on the concert platform. If we are sensitive to this as being more beautiful than this, then we will naturally choose to include the former in our musical art, and to exclude the latter.

On the previous pages we looked at the partial recipe. Now if we are using similar strings at the same tension, to create our musical intervals, we will find, due to the natural partial recipe, that the intervals whose tones contain the fewest and slowest beating partials, are those intervals created by strings whose sounding lengths are in (what at least appear to be) simple, whole number ratios. From this comes the idea that the musical intervals are defined by whole number ratios.

In fact, as a generality, this idea is untrue. In the case of string lengths, it is true to a good practical degree of accuracy, but even if it were 100% accurate, the generalised statement derived from this observation, that musical intervals are defined by whole number ratios would still be false, and highly misleading.

As we have seen, musical intervals are not just things that happen between strings. And even if they were, complex and interacting issues of both tone and pitch interval are present in determining the nature of the interval. They determine the qualities of an interval that are important as it is perceived either in isolation, or in a musical context. Neither of these can be defined as a simple ratio, so an interval itself cannot be defined by a simple ratio..

In this, occurs the bifurcation between tuning theory and practice, and temperament theory. Temperament theory is a continuation of the "science" allegedly started by Pythagoras, originating from simple observations of string lengths and musical pitches, and perhaps even musical tones. It starts from the premiss that musical intervals are defined by simple ratios, most likely to have been originally observed in strings, and then proceeds, over time, to conclusions that are out of all proportion to the legitimacy or supposed generality of its original premisses.

The important sonic aspects of musical intervals, considered in the context of the music in which they are used, are of course pitch interval and tone quality. The doctrine of simple ratios simply does not address the complexity of these aspects and their complex relationship to each other, but it is precisely this that is so important in the proper musical context, and becomes the true core of the matter in the production of good tone and intonation, in activities like singing, strings playing, and piano tuning.

It may arguably be that the human predilection for the intervals associated with simple ratios in strings, arises from the human voice itself. The acoustical evidence does support this view, in the sense that the partial components of the human voice can emulate those found in the tones of strings. In particular, the effort to effect a communication at a distance, by "raising" the voice, may also involve these intervals. For those who see things in terms of evolution through the principle of survival advantage, this has obvious implications in speculating the development of music. Nevertheless, it is difficult to see how this could be seen as linked to the simple arithmetic ratios, in early civilisations. For the link between intervals and ratios, we need strings or pipes, or a more advanced understanding of wave function principles that was not possessed by earlier civilisations, and did not appear until Fourier presented his theorem on periodic functions, in the early 19th century.

Non Western intervals

If we are not using strings for our musical notes, then the same situation does not necessarily arise. The Javanese gamelan uses two "scales", the slendro and the pelog, which have 5 and 7 notes to the octave, in unequal intervals. Listen to this slendro scale played on a Javanese gender, a bronze bar metallophone with resonator tubes. Each note is struck with a disc-like beater, and damped with the other hand.

Here, again, are the 2nd note and the 5th note. They are 653 cents apart (a cent is 1/100th of an equally tempered semitone). In terms of normal Western intervals as found on a piano, that is about six and a half semitones. Sounding together the slendro notes sound like this. This may not be a Western pitch interval we are used to hearing, but the tone is no less beautiful than the tones of either of the two notes alone. The tone of this interval can be appreciated, even by the western ear, as an entirely beautiful sound, because the tone does not contain a multitude of past beating partials that we would tend to find ugly. Of course, perceived tone, like pitch, also has a subjective aspect, so some listeners can be expected to say that they dislike the slendro tones in any case.

If, however, we used such intervals in the piano, it would involve notes like this and this, which sounding together, would sound like this. These two piano notes are tuned to the same interval. Here, it is not just the pitch interval that we find unusual, but the tone of the interval is very different to the tone of either note, being much "coarser" or "rougher" due to the presence of fast beating partials, like this one and this one. Partials like this do not occur in the individual piano notes.

In the case of the gender notes, the individual tones are practically pure tones, the characteristic tone quality being provide mainly by the transient and the decay pattern. When both sound together, no adjustable partials are created. The pitch interval of this slendro interval could vary considerably, therefore, without creating beating partials in the tone of the interval.

Difference tones

Sometimes, intervals between such tones can produce audible difference tones, which can also cause a perceived "roughness" in the tone. This is not necessarily an undesirable consequence of tones produced by metallophones.