Why do we need two measures of distance in pitch?

The mysteries of musical interval can be explored on a single guitar string.

There are two measures of interval and we need both. The measure that is most evident on guitar is the semitone, equivalent to one fret (as counted in guitar tablature). The semitone is useful when we want precision. The semitone count is shown in the middle of the fret.

There are 12 semitones in an octave. However this is too dense and featureless for  people to remember when learning melodies. Most popular songs use scales that have between 5 and 7 notes per octave. This smaller number makes the patterns more distinctive. 

The intervals of the European music tradition are based  7 note scales with unequal distances between steps. If the interval spans two scale steps it is called a 2nd. If it spans three scale steps it is called a 3rd and so forth. But we do not yet know the exact size of the interval because of the unequal steps along the scale.

Sometimes we need to know the exact size and sometimes not. For example if we hear Autumn Leaves  

                                                        scales steps    size to the nearest semitone

                                        a b c f       2nd 2nd 4th   M2 m2 P4

                                        g a b e      2nd 2nd 4th    M2 M2 P4

                                        f g a d      2nd 2nd 4th    M2 M2 P4

                                        e f# g# c   2nd 2nd 4th   M2 M2 dim4

 -from one point of view the repeating interval pattern - up a 2nd, up a 2nd, up a 4th - is the design principle. However, the fact that the precise size of the intervals can change between one phrase and the next gives the tune its character. Particularly striking is the diminished 4th between the g# and c.

Note the the scale steps are counted 1st, 2nd, 3rd... while the semitones are counted 0,1 2...  This is traditional and the methods could be swapped without having too serious effect. The semitone is a more recent concept than scale step and was adopted when poeple had got used to zero in their calculations. Counting from zero produces more elegant maths. You can add and subtract semitones. When you add scale steps counted from 1st you get strange results eg a 3rd + a 3rd = 5th . But subtract 1 from each number and order is restored.

Intervals can be considered at four levels according to how general or specific we need to be

Contour : does the note repeat, go up or down

Scale steps: how many steps does the melody go up or down

More precise size of scale step: what was the size of the step in semitones? During the course of a piece of music the semitones are like a line of bowls lined edge to edge collecting pitches that fall within their reach, in other words we are quite generous with our listening so long as the note is near the target. We perceive pitches to be within familiar categories.

Intonation: exactly what was the tuning of the note, was there any vibrato or glides between notes? Exact tuning is measured in cents , 1/100 of a semitone.

 

This table has further mysteries for me. The first is why the perfect intervals are less elastic than the imperfect intervals (all the rest). The perfect intervals are Pythagorean (see the Ratio Engine page).

The second mystery is who gave the interval names this classic form, and when.