Topic 5 Representation of intervals

Let us now return to our relationship between geometry and music and represent some intervals. Let’s remember that building, for example, a fourth interval means starting from a note and counting the following 4 notes. If we take C as the first note, its fourth will be F, since: C (1) D (2) E (3) F (4). But how many semitones is a fourth interval made of?

Simply join the C note (at 12 o’clock) with the F notewith the rubber band and count, starting from C and going clockwise, the number of semitones up to the F note. You will find out that the value of a fourth interval corresponds to 5 semitones (or 2 T and 1 sT). More precisely, this interval is called a perfect fourth. How much is a fifth interval?

Again, we just need to count the number of semitones starting from C to will discover that a fifth interval is 7 sT (or 3 T and 1 sT).

If we consider the chromatic scale (represented by all 12 nails) we can obtain 12 intervals. Let’s list them:

  • C-C = unison (0 sT)
  • C-C#/Db = 1 st, called a minor second (Db) or augmented unison (C#)
  • C-D = 2 sT, called a major second
  • C-D#/Eb = 3 sT called a minor third (Eb) or augmented second (D#)
  • C-E = 4 sT, called a major third
  • C-F = 5 sT, called a perfect fourth
  • C-F#/Gb = 6 sT, called an augmented fourth (F#) or diminished fifth (Gb)
  • C-G = 7 sT, called a perfect fifth
  • C-G#/Ab = 8 sT, called an augmented fifth (G#) or minor sixth (Ab)
  • C-A = 9 sT, called a major sixth
  • C-A#/Bb = 10 sT, called an augmented sixth (A#) or minor seventh (Bb)
  • C-B = 11 sT, called a major seventh
  • C-C (higher) = 12 sT called an octave

Now let’s see what the graphical representation of all the intervals together looks like. What do you think?

As the picture shows, the distance of the intervals is represented by a straight line joining two points. Since each interval is specific, then it can be represented by one and only one straight line. It is appropriate to recall the analogy with Euclidean geometry, in which, by virtue of the first axiom, “one and only one straight line passes through two distinct points on a plane”. Furthermore, if we were to position our regular dodecagon in a Cartesian plane, where the center of the circumference in which it is inscribed coincides with the origin of the plane itself, it would be possible to determine the equation of each of the lines mentioned.

At that point we could then assign to each line a specific musical meaning corresponding to the various musical intervals. In other words, if every musical interval has a specific sound and this can berepresented by one and only one straight line, then, conversely, starting from the specific equation of a straight line we could say that, musically speaking, it represents one and only one musical interval.

As you can see, we have always started from C, but we would get the same values ​​if we started from any other note. Now try to say what the following intervals are called (second, third, fourth….) and how many semitones they correspond to, as in the proposed example:

  • G-B = third (4 sT)
  • D-G ​​=
  • F#-C =
  • Ab-Eb =

At this point sit down at the piano or take a virtual piano and try to play these intervals trying to describe the sensations they arouse in you.

Which notes and how many semitones correspond to the following intervals:

  • F + minor sixth =
  • D + augmented fifth =