One thing about music theory that’s always bothered me is the disconnect between pitch interval and pitch frequency content. Hopefully this will become a little series where I attempt to easily bridge this gap with frequency domain plots.
Chromatic Scale
First things first, let’s take a look at the chromatic scale ranging from middle c (c’) to tenor c (c”), using scientific pitch notation ISO 16, with A 440hz concert pitch (A above middle C having a pitch of 440 hz):
It’s easy to see a few things from this plot:
- The notes of the chromatic scale are A, A#, B, C, C#, D, D#, E, F, F#, G, G#.
- Some people refer to the chromatic scale as the dodecatonic scale as it consists of 12 notes per octave. In this case tonic refers to tones, not the presence of a tonic. In 12-tone equal temperament each tone is separated by a 100 cent wide semitone interval, thereby making the chromatic scale nondiatonic with no tonic note.
- Each interval of the chromatic scale consists of one semitone, or half step, indicated by the letter S.
- Each of the intervals are made up of 100 logarithmic units of measure named cents.
Aside: Like the decibel’s relation to intensity, a cent is a ratio between two close frequencies, therefore, the frequency range encompassed by a cent must be proportional to the two frequencies. The number of cents between two pitches, a and b, can be calculated by: - Mathematically, the pitch frequency increases exponentially as a function of number of intervals, as dictated by the following equation:
whereis the output frequency,
is the starting input frequency, and
is the number of intervals from the input pitch.
Due to the fact that the chromatic scale is exponential, and doubles every 12 intervals, it results in a monotonically increasing linear plot when viewed with semi-log axes.
Intervals
Visualizing the chromatic scale on a semi-log plot makes understanding intervals very straight forward.
minor seconds: 1 semitone wide (100 cents, or 1 interval)
major seconds: 2 semitones wide
minor thirds: 3 semitones wide
major thirds: 4 semitones wide
perfect fourths: 5 semitones wide
diminished fifths: 6 semitones wide (also called an augmented fourth)
perfect fifths: 7 semitones wide
minor sixths: 8 semitones wide (also called an augmented fifth)
major sixths: 9 semitones wide
minor sevenths: 10 semitones wide
major sevenths: 11 semitones wide
perfect octaves: 12 semitones wide
Diatonic Scale
Moving on from the chromatic scale, let’s look at the diatonic major scale (Ionian mode) in C.
Looking at the diatonic major scale in C we observe some readily apparent things:
- The diatonic scale consists of 7 distinct notes per octave, A, B, C, D, E, F, G. Some people may refer to this as being heptatonic as it contains 7 tones.
- The intervals between the pitches of the chromatic scale are structured in a manner to separate the semitone (half step) intervals as much as possible, being separated from each other by a series of two or three tones (full steps).
The notes of the diatonic scale are made up of 8 degrees with the following names:
1) Tonic (key note)
2) Supertonic
3) Mediant
4) Subdominant
5) Dominant
6) Submediant
7) Subtonic (leading tone)
8) Tonic (Octave)
The tonic note is the first scale degree of a diatonic scale. Similar to how a root is the reference note of a chord, a tonic is the reference note of a scale. As the name implies, the diatonic scale consists of two tonic notes, with the second being one octave higher than the first.
Another characteristic of the diatonic scale is the presence of a subtonic note, which is the seventh scale degree. This is a note that resolves to a note one semitone higher or lower, meaning that it moves from dissonance, or an unstable sound, to consonance, a stable sound. The seventh scale degree has a strong affinity for, and leads melodically to, the tonic.
Moving On
Further on in this series I will plot other scales and modes, as well as common chords, chord progressions and songs in terms of frequency vs time.