Scales, tone, pitch and tuning

See also, on music and mathematics

with interactive media

On scales, tone, pitch (and piano tuning)

- with interactive media

Page 1

Musical intervals and ratios

It would be a mistake to regard the relationship between the simple arithmetic ratios and the primary musical intervals in Western music, as a Divine enigma, on the notion that there is no straightforward, mundane explanation. It is indeed psychologically striking that simple whole number ratios in strings and pipes appear to be associated musical intervals, but that does not mean a mundane, rational explanation cannot be provided for the ratios, or that there does not exists a perhaps more complex explanation for the associated pitch perceptions. What is of interest, from the point of view of psychology, is why anyone in the modern world would prefer what is essentially the perpetuation of Pythagoreanism, to the well established rational explanation for the ratios. I hasten to point out that I am not stating this because I wish to promote the views either that there is a meaningful rational explanation to everything, or that the rational view is always appropriate. Nevertheless, to see as magical, unexplained or mystical, what is actually explainable, mundane and rational, is merely to be swayed by psychological effect, and is not to truly see the Divine in music, in any case.

There is, after all, a mundane, rational explanation for why strings (and also pipes) behave in this way. There are indeed presently unanswered questions concerning pitch and musical perception, as there are for all our perceptions, but these do not make the issues connected with music or musical intervals any more salient than the questions that arise in other areas concerning human perception, consciousness, and psychology.

The attachment to the idea that there is a salient connection between simple ratios and musical intervals, as a connection that demands philosophical attention, is a symptom of the legitimate need to connect with the Divine in music, but the idea of a significant connection is itself questionable. On scientific scrutiny, the simple ratios turn out to be not really simple ratios at all, but approximations. it would also be a mistake to think that all musical intervals necessarily even correlate or approximate to these ratios. The view that this is a universal relationship simply ignores cultural variations in music practice and musical instruments. Finally, the idea that a metaphysical explanation is necessary, disregards the corpus of information on the relatively simple origins of the ratios that can be found in most undergraduate texts in acoustics.

The acoustics behind the ratios

The length ratios that are found, on musical strings (and pipes), to correspond to musical intervals, owe their origins to the natural behaviour of waves. Put simply, the sound produced originates in vibrations on the string, which are caused by wave motion. Specifically, there is a special kind of wave called a standing wave, that produces the vibration. There are only certain lengths or shapes of standing wave that can "fit" onto a given string. This gives rise to a set of vibrations that can exist on the string, all other vibrations not being "allowed". This, in turn, produces a "structure" in the sound produced, related to this originating behaviour.

In the case of the string, this structure is what has been called the chord of nature. Briefly, the tone of the string is a recipe of sound ingredients, called partials. These ingredients each have a rather simple sounding tone quality, and occur at different pitches.

As an example, listen to this bass note from a Steinway piano. After the initial noise at the very beginning, around about the time the hammer strikes the strings (which produces a noise known as the transient), the tone of the note then consists of a recipe of distinct partials all sounding together. If we take these partials out of the recipe and listen to them one at a time, putting them in order from the lowest pitched partial to the highest, then we can hear them in order of their pitches. The first partial, the lowest pitched one, is not included here. It is present in this recipe, but for most web listening, for this particular piano note, it is too low to reproduce and be heard properly. It would sound an octave below the 2nd partial, which is where our sequence starts.

Partials 2 - 17, separated out and put in sequence, sound like this. (400KB)

Or, spending a little less time listening to each one, so we can hear the sequence well, partials 2 -17 heard in a faster sequence sound like this.

There are many more partials above the 17th one. Grouping them all together, they sound like this. In amongst these higher partials are some that are more prominent than others. For example, here is the 23rd partial.

It is this recipe of ingredients that largely determines the tone of the note. Notice how some partials, like the 4th partial, die away or decay "smoothly", whilst many others, like the 5th partial or the 16th partial, beat as they decay, meaning the fluctuating in pulsing manner. This kind of beating behaviour of many partials can, as we shall soon see, have a profound influence on the tone of notes and musical intervals.

If you want to study some partials separately, here are the even numbered ones up to and including the 16th:

You may be able to hear some of the partial ingredients within the tone of the note. Or you may find it easier if we turn the volume up as the note decays away. (You will of course hear more hiss and noise though, as the volume increases).

Notice how the even numbered partials in the list above also have a sequence of musical intervals between them. In fact, the sequence of intervals between the even numbered partials is the same sequence of intervals that the whole series (including both odd and even numbered partials) has have starting from partial 1, except that it sounds an octave higher. The musical intervals between the partials appears to proceed as octave, perfect fifth, perfect fourth, major third, minor third, minor third, whole tone (i.e. major second). The intervals get smaller and smaller, the higher the partial, until eventually they form a cluster, such as we heard in the high partials.

Don't worry if some of these pitches do not sound quite right. The description is only a rough one in any case.

The point is, that all musical strings produce partials separated by intervals, many of which, especially in the lower partials, appear as familiar musical intervals. We could in theory construct strings that would behave differently, and would not behave like this, but mostly, musical instrument strings do behave in this way.

The partials of a string have this series of intervals between them, starting from the lowest partial. It is sometimes called the chord of nature, or the harmonic series. If we tune two strings to a musical interval, such as a perfect fifth, then in effect we get two series of intervals, starting on different lowest partials.

Now consider just one specific case of a tuned interval between two strings.

Consider two strings, String G and String D tuned to a perfect fifth. The first six partials of the G would be:

1st partial

2nd partial

3rd partial - compare with D's 2nd partial

4th partial

5th partial

6th partial - compare with D's 4th partial

Whilst the first six partials of the D would be:

1st partial

2nd partial - compare with G's 3rd partial

3rd partial

4th partial - compare with G's 6th partial

5th partial

6th partial

Notice that the 2nd partial of the D is about the same pitch as the 3rd partial of G. Also the 4th partial of the D is about the same pitch as the 6th partial of the G.

The tone of the fifth itself, consists of a sound in its own right, with its own set of partials. Each partial is generated from the G string or the D string, but in the cases where the G and D strings independently would have partials approximately the same in pitch, both strings generate one partial for the fifth's soundscape, together.

It is important to note that on real musical instruments like the piano, two strings generating one partial are not necessarily independent of one another. In the piano, they are, in general, coupled by the bridge and soundboard, and can be capable of behaviour as a string pair, that would otherwise not occur as separate strings. Whether or not bridge and soundboard coupling has significant effects, depends on many factors, including what is regarded as "significant". Generally, bridge and soundboard coupling can have effects that are significant in piano tuning.

In the tone of the perfect fifth, two such partials within the range already listened to, are created by both strings. These occur at around the pitches of the 2nd partial of the D or the 3rd partial of G, and the 4th partial of the D or the 6th partial of the G. Each of these partials is called an adjustable partial, because its behaviour depends on the tuning condition between the two strings.

Adjustable partial 1 is here.

Adjustable partial 2 is here.

See if you can hear them in the tone of the fifth.

The tonal feature of adjustable partials is a critical one in tuning musical instruments, and especially in the fine tuning that is required in the art of piano tuning.

Adjustable partials occur in the tones of all musical intervals between strings. They occur in relation to the ratio 3:2 for the perfect fifth, as can be seen from the example above. For the perfect fourth, the ratio is 4:3, for the major third, it is 5:4. Each interval has its associated ratio, known as its harmonic ratio.

Tone and pitch