Time Signatures and Staves

Part 2 of Mathematics and Music

In my post of June 29th 2021 titled Equal Temperament Tuning, we saw that ratios were at the heart of heptatonic scales like the major scale (see my post of July 2nd 2021 titled Concrete, Laplace Resonance and Heptatonic Musical Scales). Figure 1 is taken from the former post.

Figure 1

As can be seen from Figure 1, the ratios 9:8, 5:4, 4:3, 3:2, 5:3 and 15:8 are most prominent. It might seem that ratio/fractions appear in musical time signatures. To quote from Wikipedia:

Simple time signatures consist of two numerals, one stacked above the other. The lower numeral indicates the note value that represents one beat (the beat unit). This number is typically a power of 2. The upper numeral indicates how many such beats constitute a bar.

For instance:

• \( \frac{2}{4} \) means two quarter-note (crotchet) beats per bar
• \( \frac{3}{4} \) means three quarter-note (crotchet) beats per bar
• \( \frac{4}{4} \) means four quarter-note (crotchet) beats per bar
• \( \frac{3}{8} \) means three eighth-notes (quavers) per bar

Figure 2 gives an example of the use 3/4 time:

Figure 2

However, ratios compare two quantities of the same type, and here that's not the case. Time signatures are really musical shorthand combining two numbers representing different types of things (beat unit or length of beat on the bottom versus beats to the bar on the top). It looks like a fraction but it's not really.

The stave (a set of five parallel lines on any one or between any adjacent two of which a note is written to indicate its pitch) shown in Figure 2 has a Cartesian coordinate aspect to it where the vertical axis represents pitch and the horizontal axis represents time. See Figure 3.

Figure 3

Figure 4 shows the precise changes in pitch as we move along the time line in whole notes or semibreves:

Figure 4

The notes of course encode additional time-related information, namely how long the note is sounded. The position of the note on the time line simply denotes its position in the sequence of notes but tells us nothing about its duration. This is equivalent to marking a point on the 2-D Cartesian or (\(x-y\) plane with a specific shape that represents its position on the \(z\)-axis. Figure 5 shows the different types of notes:

Figure 5: source

Thus the ratio of semibreve to minim to crotchet to quaver to semiquaver is 16 : 8 : 4 : 2 : 1 or \(2^4 : 2^3 : 2^2 : 2^1 : 2^0\). As with the differences in frequencies between notes, the powers of 2 predominate here also.

Figure 6 shows a table of simple time signatures while Figure 7 shows a table of compound time signatures.

Figure 6: source

Figure 7: source

There's great complexity to be found as we delve deeper into musical theory but that's probably sufficient detail for this post.