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

Page 2

Tone and pitch

On the previous page we saw how musical intervals tuned between two strings have tones that include adjustable partials in their tone recipe. Adjustable partials are responsible for the obvious difference in tone between this pair of strings and this pair. The first pair of strings are tuned for one note, or unison. The second pair have a micro-interval called a comma tuned between them.

The characteristic of the comma interval is not just one of a slight pitch difference between the two strings. Perhaps the most obvious additional feature of the comma interval's tone, is beating. This is beating in the adjustable partials of this (micro) interval. Unlike the perfect fifth illustrated on the previous page, in this (micro) interval every partial in the interval's soundscape is adjustable. The first four adjustable partials are:

You may be able to hear these partials if you listen again, in the soundscape of the string pair.

The comma's soundscape contains many partials, all of which are beating. If we use two simple tones, each tone having only a single ingredient or partial, rather than the tones of two piano strings, to create the comma interval, then the resulting interval will have only one adjustable partial. Two such tones are here and here. Can you hear the pitch difference? The tone of the interval when they are both sounding together sounds like this. It is very similar to just the first partial of the comma interval tuned between the two piano strings.

Any interval or micro-interval tuned between any two tones has at least these two qualities - a pitch interval (pitch difference) quality (which may of course be zero), and a tone quality. The perceived pitch interval can be affected by many factors, including physiological and psychological ones. The tone of the interval can also depend on psychological factors, but beating in its partials are important properties affecting the tone. However, this itself depends on the partial recipe of the interval's tone, and this is largely dependent on the partial recipes of the notes making the interval.

Let's consider some examples. Here is a piano note C. And here is a piano note E, a major third above. Together, they create the tone of this major third.

Now here is another major third, same two notes, but the E is now raised a little. Can you hear the difference? Which one of these two tunings for the major third is right? Does the difference matter? (I'm not saying it doesn't matter - but I'm not telling you it does, either). What is the pitch interval difference? What is the tone difference?

If you have done all the previous listening attentively, or if you are a piano tuner, you may be aware that as far as the tone is concerned the first major third contains this adjustable partial. The second major third contains this adjustable partial. You should be able to hear a difference.

Piano tuners would refer to a difference of beat rate here. The second beating is faster. But if you are aurally astute, you will hear that there is more to it than this. If you are very aurally astute you will hear that the beat rate of the partial is not fixed. You will also hear that it fades and increases at various points.

The two intervals differ in both tone and pitch interval. The tone of the interval is not the same thing as the beat rate in the adjustable partial. The beat rate is just one factor in the tone of the interval. It is a factor that can be changed in the tuning process. The pitch interval itself contributes to the tone of the interval, because musical perception does not necessarily just separate out pitch and tone quality in musical sounds, even though we can hear them as separate aspects of a musical sound.

Allow me to elaborate a little further on this point, using an illustration from piano tuning, because it is very important to pitch, tone and tuning generally. Students of piano tuning are taught, out of necessity - because they do not begin with an intimate knowledge of piano tone - that the interval we just heard will be "right" when the beat rate is at a particular value. There is a vague notion that "right" means that the notes will be at the right pitch, or the strings at the right frequency. So the beat rate gets treated as a kind of "tool", in abstracto, rather than as a genuine feature of musical tone, in its proper musical context. It is a necessary starting point.

However, many master tuners begin tuning with this interval, and "know" when it is right, for a given piano, even though the chosen beat rate is deliberately varied from piano to piano very slightly. The variation occurs because the beat rate itself is not really an isolated thing. It is contextual, and occurs in the full context of the tone (including the pitch interval), that I have just discussed. The tuner "knows" what is going to be necessary on that particular piano, precisely because beat rates are a part of tone in the context of tone and pitch interval, and not simply an isolated "tool" to which some number value should be applied.

These differences are real. You may be thinking they are small, and in isolation they are. But in a process like tuning the piano or performing music, they collectively "add up" to much more than the sum of the parts, making a large difference to the final result. Master musicians pay attention to tone and pitch as a matter of course, often without thinking about it rationally or analytically in the way we have just been doing, but nonetheless, the attention to both tone and pitch, is there.

There are differences between this major third, and this major third and this major third and this major third. They are differences of pitch interval and/or tone. (If you can't hear them, the file names will give you a clue). Although these each take just one factor and make it into an extreme version they illustrate the generic kind of difference that matters in producing tone through the tuning process.

The complex network in tone

Some tuners still regard beat rates merely as a "tool", with no tonal or musical context, even after years of experience. The vague notion that the beat rate is some kind of pitch or frequency control tool still persists. The reason is simple. The features of tone that occur in all intervals on a keyboard instrument, are interdependent and connected. No one can be changed without affecting others. The intervals, and hence their tones, form a complex network. As a result, the task of tuning the intervals is rather like trying to solve a Rubik's cube, and the wrong change of beat rate in the wrong interval, can have undesirable consequences further down the line, preventing a result in which all beat rates are satisfactory, and in which the whole tuning "works out". We could tune some intervals so that they sound fine, only to find the one that we didn't tune, but is inevitably made as a result of the ones we did tune, sounds like this, when it was meant to end up sounding like this. An interval sounding like this would be bad enough. Even if you liked the pitch interval here, the tone is not acceptable because it contains adjustable partials like this and this and this. In some intervals, such as major thirds, such beat rates might be acceptable, but in the context of what is supposed to be a perfect fifth, they are not. Tone and pitch interval are interconnected, not just because one affects the other through acoustical mechanisms, but also because of psychological and cultural factors.

Our current preference for this major third over this major third, which is a much smaller pitch interval, is a cultural one. In the West, in the Middle Ages this smaller major third, whose pitch interval is from this C to this E, would generally have been the preferable one, except that, of course, it would not have been heard in piano tone. It would have been heard in organ tone or the tones of plucked strings. The acceptability of its size as an interval depends not only on its cultural context, but also on all the factors determining the tone, not just the beat rate (which would be about the same for the piano, plucked strings, or organ).

Once you start trying to tune all the intervals, rather than one or two in isolation, you are in effect engaged in a puzzle solving activity of tone and pitch. You may think you could sort it out, but in fact, it is a lot more difficult than you might think. So to help students of piano tuning solve the puzzle, a "solution" is provided, in the form of "prescribed" beat rates. Of course, this "solution" is a tool, in effect. But it is not perfect, because piano tone is very complex, and does not generally oblige by conforming to the "solution" with perfection. In fact, the biggest drawback of the "solution" (derived from a somewhat idealised and simplified 19th century acoustical model) is not so much that it fails as an accurate, universal solution, but that it does not really say very much at all about either tone or pitch, the things which really matter, musically. It is, in fact, just a prescription for beat rates, as a tool, for supposedly achieving what the theory assumes would be a good set of frequencies for the strings. True, it is better than ending up with this, which you may well end up with unless you follow some kind of instructions, but really, despite all the reiterated "instructions" for tuning that still abound, and are even being sold to unsuspecting buyers, we all really ought to be re-educated on it.

Next:

Scales