Relative pitch

We’ve seen how to specify a single pitch in absolute terms (see Absolute pitch). But of course most music uses a number of different pitches, and our notation needs to be able to represent that too.

It is perfectly possible to specify multiple pitches in absolute terms, in the same way as we did for a single pitch. But it is almost always more musically meaningful to specify the various pitches in a piece of music in relation to each other.

This is because music’s effects depend far more on relative pitch than they do on absolute pitch. That can be proved by performing the same piece of music in different keys: although the pitch of every note is changed, the effect of the music is much the same because the relationships between the pitches are unchanged.

Consequently, it is useful to be able to specify relative pitch independently of absolute pitch. In global notation, this is done by choosing one pitch as the “reference pitch” and specifying all of the other pitches in relation to that. If it is desired to specify absolute pitch, this is done for the reference pitch only.

A relationship between two pitches is called an “interval.” Intervals can be measured by the difference in pitch: a large difference between a higher and a lower pitch means a wide interval, a small difference a narrow interval.

However, a difference in pitch is not measured by the difference between the two frequencies, but rather by the ratio between them. The simplest ratio, 2:1, gives the interval that is most basic to most forms of music, the “octave.” (To know what an octave sounds like, think of the first two notes of the song “[Somewhere] Over the Rainbow.”) Because of the simple vibration ratio, our ears interpret the two pitches of an octave as somehow “the same,” even though we can hear that one is higher than the other, and they are given the same name, as with the various “Cs” on the piano. With rare exceptions, this holds true around the world.

Of course, music uses lots of other intervals too, and unlike the octave, these vary widely between different forms of music. The challenge for a global notation system is to be able to specify them all equally accurately. To do this, we need a fairly precise way of measuring intervals.

Since the octave is so basic to most music, it seems logical to measure intervals in fractions of an octave. In effect, the tuning of modern keyboard instruments does this by dividing the octave into 12 equal parts. Hence there are 12 keys to each octave, counting black as well as white keys. The interval between any two adjacent keys, whether white or black, is called a “semitone” or “half-step.”

However, as mentioned before, the pitches used in the world’s music are not always the ones on the piano keyboard, and the intervals between them are not always equal to an exact number of semitones. In other words, one-twelfth of an octave is not a fine enough unit of measurement for dealing with all the world’s music. To provide greater precision, music theorists have devised a system of dividing the semitone into 100 equal parts, called “cents.” This allows them to speak of intervals like 150 cents or 240 cents which couldn’t be played on a piano. A cent is one 1200th part of an octave, which is precise enough for most purposes, since a difference in pitch of one cent is practically impossible to hear.

Global notation adopts the cents system as a means of specifying relative pitch. That is, the interval between the reference pitch and any other pitch is stated as a figure in cents.

For an example that uses only two different pitches, and hence one interval, take the sports crowd jeer “What a load of rubbish!”

As there are two recurrent pitches, there are two horizontal pitch lines, with the bar and beat lines unbroken between them since the two pitch lines constitute a single “layer.” Absolute pitch is specified only for the “reference pitch,” the lower one. (Regarding which pitch to choose as the reference pitch, see Scales and melody.) The relative pitch of the upper pitch line is specified by the number 300, meaning 300 cents above the reference pitch: an interval of three semitones. (In practice this figure is likely to be an approximation; the exact interval used in a particular performance can be measured more precisely if required, using sound analysis software.

Note that the cents system can also be used to specify absolute pitch more precisely. In this case, the lower pitch sung is actually 30 cents lower than where E3 would be in concert pitch, and is written as “E3-30.” (30 cents higher than E3 would be “E3+30”; and similarly for other pitches and numbers of cents.)

The jeer can of course be chanted in any key, so to notate a different performance of it, we might want to change the absolute pitch (without changing the relationship between the two pitches). This is done by simply changing the specification of absolute pitch for the lower pitch line by writing a different pitch name, such as G3 or C4+22. Nothing else need be changed: the interval between the two pitch lines will still be 300 cents. (This is in contrast to staff notation, where changing the key requires rewriting every note of the piece.)

Now that we are beginning to notate melodies that move between different pitches, there is a new point to note about the shape of the symbols as well. Remember that sounds of specified duration are represented by “extendable” symbols (as opposed to “discrete symbols of fixed shape” for sounds of unspecified duration; see Unspecified duration). Just as the horizontal part of the rotated T symbol can be extended to indicate duration, so can the vertical part be extended to connect with the previous symbol and so indicate that there is no “break” in the sound—that the melody is performed “legato.”

In this way, a legato melody appears in the score as a continuous, bending line. The “tops” of the T shapes are still visible as vertical extensions beyond that line, indicating where a new “note” is articulated, for instance by a new syllable in the lyrics.

We now know how to specify an interval between two recurrent pitches. Very often, however, there will be more than two such pitches, and further pitch lines will need to be added, with their relative pitch also specified. The result will be a set of pitch lines representing a “scale” in the musical sense.

Next: Scales and melody

4 comments on Relative pitch

1. Han says:

Your use of cents seems a bit biased towards western music. This way western intervals will look a lot ’rounder’ than non-western ones. I suggest using thousandths of an octave or something similar using octaves as the base interval.

I was also wondering if it is possible to set multiple absolute pitches as a way of marking that the music is atonal. Some music simply doesn’t have a root note. And even if music is tonal, can you just change the root note as an indication of a modulation?

1. Yes, the cents system is of course based on Western equal temperament, so using it in global notation does incorporate a cultural bias. It seemed a compromise that was worth making since the idea of an octave divided into 12 semitones is so well known to musicians around the world and the fact that each of those semitones is a “round” number of cents will help convey a sense of the size of an interval quite readily. We could just as well divide the octave into 1000 (let’s say) “millioctaves” but that might seem to incorporate a different cultural bias in that it would privilege scales like the Javanese Slendro which divide the octave into five (roughly) equal intervals. Each of these intervals would come out as (roughly) 200 millioctaves in comparison to, say, an equal-tempered whole step of 167 millioctaves. For purposes of pitch measurement, so long as the octave is divided into equal parts, it doesn’t really matter how many equal parts, and 1200 has an advantage over 1000 in being divisible by more factors.
There’s nothing to prevent you from marking the absolute pitch of each pitch line if you want to, though I think the advantages of basing the notation on relative pitch still apply with atonal music, hence my recommendation of using the lowest pitch as the reference pitch if there isn’t a clear tonic. Modulation in tonal music is something I haven’t yet covered in the website, but essentially my solution would be to specify the new tonic by thickening and lengthening the relevant pitch line (from the point where the modulation occurs) and to adjust the position of any pitch lines that need to be different in the new key (e.g. F-sharp instead of F).

2. Jackson says:

Maybe let people label lines with pure ratios like “2/3” (two thirds of an octave aka a perfect fifth)? Irrational cent values aren’t very intuitive.

1. Sure, no reason why you shouldn’t label the pitch lines with frequency ratios if that is more intuitive for you and your intended readers. What is intuitive for anyone probably depends what they are used to. Personally I find the cents values more intuitive because I can relate them to the notes on a keyboard (any two adjacent keys are 100 cents apart) and thus get a clear sense of how big the interval is. With ratios I would have to think for a minute to figure out whether, say, 19/16 is larger or smaller than 6/5 and by how much.