Appendix A – Definitions
Accidentals are the sharp/flat/natural (♯/♭/♮) symbols in a note’s name. A note with either no accidentals or with a natural symbol in its name is natural. Each sharp or flat symbol in a note’s name indicates that the note has been sharpened or flattened respectively from its natural form.
When a note is flattened, its pitch is lowered by one semitone, thereby shifting it counterclockwise on the twelve-tone clock. Flattening a sharpened note moves it back clockwise towards the natural note.
A term meaning rate of oscillation. An oscillator completes a particular number of oscillations, or cycles, per second. This rate is physical in that it can be measured with instruments or calculated from other properties. It is called the frequency of the vibration and its unit of measurement is cycles per second, also known as Hertz (Hz for short). A sound with a smaller number of Hz is said to possess a lower frequency than one with a larger number of Hz, which is said to have a higher frequency.
An oscillator can produce a single frequency, or it can produce many. A pendulum has just one frequency. A harmonic oscillation such as a single note played on a guitar-string produces a fundamental frequency and, at the same time, a series of other more subtle frequency components known as overtones. Whether a single or multiple frequencies are produced by a musical oscillator, they maintain their rate even as the amplitude diminishes.
Motion characterized by a regular change in position and direction. A pendulum oscillates; its behavior is also called (simple) harmonic motion. Each complete back-and-forth motion is one oscillation, or cycle. Similarly, waves and vibrations are oscillations. A musical instrument has parts which vibrate: a plucked guitar-string, for instance.
Harmonic oscillation is regular in that the number of oscillations per second – the frequency, or rate of oscillation – does not vary over time (except whilst deliberately being modified, e.g. bending a guitar-string). Even when a pendulum or a vibrating guitar-string loses its energy, it continues to oscillate at the same rate until it stops. This is a quality of oscillation crucial for tonal music to work. If notes arbitrarily changed their pitch whilst being played, it would be impossible to predict the nature of their interaction, and thus impossible to make tonal music with them.
(Harmonic) overtone series
The series of frequencies which vibrate in harmony with a fundamental frequency. Overtone frequencies are simple integer multiples of the fundamental frequency and, being components of sounds produced in the natural world, are ratios the ear finds natural. Overtone ratios early in the series are the most important to the theory of tonal harmony and to the construction of the frequency ratios used to make music.
An interval is the distance between two notes. Notes are complex, consisting as they do of a named tone and an alteration. And the distance between them is also complex because intervals consist of both a quantity and a quality.
An interval’s quantity is counted from the first note in the sequence and named for the last. So C up to D is called a second because D is the second of the sequence. And G up to B is a third. If you know an interval’s quantity then you know the general interval. If you also know the interval’s quality then you know the specific interval. By way of analogy you could say that knowing a note’s alphabetic name is knowing the general note; and also knowing its alteration is knowing the specific note. Any C to any D is a general second. If we can be specific about the C and the D then we can be specific about the second. For example the interval between C♮ and D♮ is a major (which means larger, or greater) second.
You can think of the interval between two natural notes as the natural interval between them. The natural intervals above C♮ are special and their sizes are the default sizes for intervals – all qualities of intervals are defined relative to these default sizes. Seconds, thirds, sixths and sevenths can be diminished, minor, major (the default) or augmented; unisons, fourths, fifths and octaves can be diminished, perfect (the default) or augmented.
Intervals larger than an octave are compound intervals. Every interval has a direction: either ascending (positive) or descending (negative). Therefore F up to A and F down to A differ in both magnitude and direction; whereas F up to A and A down to F differ only in direction.
The operations which are defined for intervals are: addition of two intervals, giving an interval; and addition of an interval to a note, giving a note. Subtraction is defined by the addition operator, after reversing the direction of the second operand.
The term note is overloaded with meaning. Sometimes the physical divisions on an instrument (piano-key, fretted guitar-string) are referred to as notes. But also, like tone, the specific pitches produced by a tuned musical instrument are known as notes.
Each instrument tone corresponds to several possible notes, distinguished by their different note names and existing for different musical circumstances. So, although a note uniquely identifies a tone on the twelve-tone clock, the converse is not true. For example, C♯ and D♭ are different notes which identify the same tone in common tunings. In cases such as this, the choice of name – the tone’s spelling – depends on harmonic context. The semantic of note is more complex than that of a pitch or a tone. Tones are measured simply by their twelve-tone clock position, whereas a note’s clock position and name are found by adding an alteration to a named tone. So, a note’s name gives directions for how the clock position it occupies is reached.
A note takes the alphabetic part of its name from its named tone. The alteration, which can be zero, indicates the direction and distance in semitones the note’s clock position is relative to the named tone. Moving counterclockwise on the twelve-tone clock takes you lower in pitch – descending – and moving clockwise takes you higher in pitch – ascending. In music, lowering pitch is known as flattening, and raising pitch is known as sharpening. A note’s alteration, if it is not zero, is written as a series of either sharp (♯) or flat (♭) symbols known as accidentals.
A simple example is the note C whose named tone is C and which has no alteration. It is therefore at the same position as tone C at 12 o’clock. The path to the note C♯ also begins at tone C but is sharpened (raised a semitone in pitch, clockwise) once, so it occupies the 1 o’clock position. The note B and the flattened note C♭ both occupy position 11 and they identify the same tone but they are not the same note because they give different names, or spellings to the tone.
When its alteration is zero, you may optionally write a natural (♮) accidental in a note’s name to indicate that the note shares the natural name and position of its tone. For example, the note C can be called C♮ to make it clear you haven’t missed a symbol off a C♭ or a C♯. Another accidental is the double-sharp (x), which makes, for example, Cx a concise form of C♯♯.
It should be clear that the natural name, if any, of the tone identified by a note needn’t bear any relation to the note’s own name. For example, the note Cx occupies position 2, the same position as tone D, but D does not appear in the spelling of this note. What’s important is the name of the tone we begin from and the magnitude and direction of the alteration.
The operations which are defined for intervals are: subtraction of one note from another, giving an interval; and addition of an interval to a note, giving a note. Subtraction of an interval from a note is defined by the addition operator, after reversing the direction of the interval.
Consecutive tones are grouped together into sets of twelve, called octaves. In common tunings, the tones in an octave are spaced approximately equally in pitch even though their actual frequencies vary logarithmically.
Octave is also used to describe the musical interval between same-named notes in two neighboring octaves. The frequency ratio of these equivalent notes in octave n and octave n-1 is 2:1. This straightforward relationship causes a similarly straightforward aural effect. When played either in succession or together, the two tones are perceived to have an unmistakably similar quality.
Frequency is a physical property of an oscillator. Pitch is the subjective, mental sensation of frequency and, since the mind is the measuring instrument, pitch can’t generally be measured or described with great accuracy. Pitches are perceived as low and high in proportion to the frequency of the sound being heard.
A general interval; the class of intervals between two notes whose named tones appear in consecutive order on the twelve-tone clock. A major second is two semitones in size and can be natural (e.g. C♮ to D♮) or altered (e.g. B♮ to C♯). A minor second is one semitone in size and can also be natural (e.g. B♮ to C♮) or altered (e.g. C♮ to D♭).
Of the seven natural seconds on the twelve-tone clock, five are major and two (E-F and B-C) are minor. Therefore the varieties of second which occur naturally are major and minor.
The interval in pitch between any two adjacent tones. The ear perceives all semitones in common tunings as approximately equal intervals even though the frequency delta between adjacent notes varies logarithmically. From other sources you will hear the term half-step used to mean a semitone; I will not use that term.
When a note is sharpened, its pitch is raised by one semitone, thereby shifting it clockwise on the twelve-tone clock. Sharpening a flattened note moves it back counterclockwise towards the natural note.
Longitudinal waves which propagate through a medium such as air from an originating oscillator such as the vibrating parts of a musical instrument. Hearing is the subjective experience of sound and the human ear is sensitive to sound in a range of frequencies from about 20 Hz to about 20,000 Hz.
A general interval; the class of intervals formed by two consecutive seconds. A major third is formed by two major seconds and is four semitones in size. Major thirds can be natural (e.g. C♮ to E♮) or altered (e.g. B♮ to D♯). A minor third is formed by a major second and a minor second in any order and is three semitones in size. A minor third can also be natural (e.g. B♮ to C♮) or altered (e.g. C♮ to D♭).
Of the seven natural thirds on the twelve-tone clock, three (C-E, F-A and G-B) are major and four are minor. Therefore the varieties of third which occur naturally are major and minor.
The term tone is overloaded with meaning. The Greek word tonos means tightening; that is, a taut string. This suggests that tone should mean the sound produced by such a string, so a reasonable definition for the word is as a synonym for pitch. But better than this is to limit tone to mean those pitches produced by a tuned musical instrument, and that is the sense in which I will use the word.
Tones are named by grouping them into sets of twelve (i.e. octaves) and repeating the names for successive sets of twelve. Seven of the twelve tones in an octave are named A-G and the remainder are unnamed. However, all tones have equal status whether named or not. It’s worth stressing that these are tone names and not note names. Tones are given names so that notes may be named in relation to them.
From some sources you will hear tone used in the sense of the combined interval of two adjacent semitones. You will also hear the term whole-step used to mean the same thing. Instead, for the sake of clarity, I will simply say two semitones. Neither will I use tone to mean timbre.
The twelve-tone clock is a visualization of the tones in a single octave. The pattern of the octave is repeated on musical instruments (tones of the same class in different octaves serve the same musical function) and between musical instruments, so this illustration focuses on the octave-in-general in an instrument-agnostic form.
Figure 1 – The Twelve-tone clock
Because the tones Figure 1 map right onto the hour positions of a clock, you can equate each of the tone positions with an hour position. The position of the C tone can be thought of as either 12 or 0 (midnight in the 24-hour system) as you prefer. You can also think of the angular distance between any two tones as a ‘time’ in five-minute units. Use the clock analogy as a tool to help you picture the twelve-tone clock as a mental image.
There is no particular starting-point on the twelve-tone clock, and nor does it need to be oriented with C at the 0 (or 12) position. You should aim, in time, to be able to visualize the circle rotated to any orientation.
Copyright (c) Steve White 2005