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1.1 Oscillations, Period and Frequency

Most musical notes are produced by a mechanical or electronic system which vibrates or oscillates.  If the vibrations repeat regularly the period of the oscillation is the length of time required for one such vibration and the frequency of the oscillation is the number of times the vibration is repeated per second.

Musical notes also have a contour to their loudness or amplitude called an envelope.  This envelope arises from the mechanical characteristics of the system or is influenced by a musician playing an instrument.  In electronic systems an envelope is applied to prevent the note ending with an abrupt click.  All examples on these pages have an envelope applied to them (but you will still hear a click if you select a second note whilst the first one is still playing).

T 1.1.1 A pure tone with a frequency of 110 cycles per second
A musical note
(click me)

The phase of a signal is a measure of how far along a single cycle the oscillation is at a given moment.  For two tones of the same frequency there will be a constant offset between their phases.  If one of the tones has a slightly higher frequency then its phase will advance more rapidly causing the phases to sometimes match and sometimes not match (in and out of phase) over time.  The moving in and out of phase can be heard as a 'beating' sound.  This is one way in which a musician or instrument tuner can tell whether two notes are exactly in tune or not.  In the following example you can click to hear the two tones play one after the other and right click to hear them sound together.  A beating can be heard when they are sounded together.

T 1.1.2 Beating sound between two similar frequencies
Two similar notes
(right click)

1.2 Fundamentals, Pitch, Harmonics and Harmonic Series

All periodic oscillations are either pure tones (sine waves) or are a set of pure tones all sounding together.  The most prominent pure tone is known as the fundamental and determines the fundamental frequency or pitch of the tone.  The others are known as harmonics and have frequencies related to the frequency of the fundamental by integer (whole number) factors.

The process of decomposing a periodic tone into its constituent harmonics is known as harmonic analysis.  A description of the process was given by Jean Baptiste Joseph Fourier (1768 - 1830) in his thesis Analytical Theory of Heat, 1822.  The equations now known as the Fourier Series were first published by Daniel Bernoulli (1700 - 1782) in 1728. [EB62]

An harmonic series contains all the integral multiples of a pitch for as far as the series extends.  For example, a fundamental frequency F has harmonics of frequencies 2F, 3F, 4F, 5F, 6F, etc.  The fundamental, F,  is also referred to as the first harmonic, 2F the second harmonic and so on.  This makes it possible to consider the properties of an harmonic series without having to treat the fundamental separately.  Sometimes harmonics other than the fundamental are referred to as overtones or upper partials: the first overtone is the second harmonic, the second overtone is the third harmonic and so on.

Typically, for a single tone containing harmonics, the strength of the harmonics decreases with harmonic number.  The relative strengths of the harmonics determines the timbre of the tone.

A set of harmonics does not necessarily include all of the harmonics in a series.  For example, the set of odd harmonics includes only odd numbered harmonics.

T 1.2.1 A series of harmonically related tones
First harmonic F (Fundamental) Second harmonic 2F Third harmonic 3F Fourth harmonic 4F Fifth harmonic 5F Sixth harmonic 6F


1.3 Frequency and Intervals

If two notes with a fixed ratio between their pitch are sounded together then they are perceived as being the same distance apart regardless of how high or how low in the audio spectrum they both occur.  This is because human perception of pitch is logarithmic in nature (like most human senses) so a ratio of frequencies is perceived as a fixed size step in the pitch scale.

For example, the lowest note of the orchestral compass, that produced by a 32 foot organ pipe, has a frequency of about 16 cycles per second.  The note an octave higher has a frequency of about 32 cycles per second.  Higher up the audible frequency spectrum the highest note of a piccolo has a frequency of about 4,000 cycles per second and the note an octave lower has a frequency of about 2,000 cycles per second.  So an octave is represented by a doubling of frequency rather than a separation by some number of cycles per second.

T 1.3.1 Perception of pitch intervals is related to ratios between frequencies
16 cycles per second 32 cycles per second 64 cycles per second ... 1024 cycles per second 2048 cycles per second 4096 cycles per second

The ratio of the frequency of one note to the  frequency of another is the physical basis of an interval in music.  For example, an octave is the interval 2 / 1, that is, a doubling of frequency (2 divided by 1 is 2).  Musical intervals are usually measured from the lower note to the higher in which case numerically the interval must be greater than or equal to 1.  Conversely, descending intervals must be numerically less than 1.

T 1.3.2 The interval of an octave
F 2F
(right click for sequence, left click for chord)

1.4 Pitch Class and Octaves

If two notes are sounded together, one having twice the frequency of the other, they are perceived as being 'the same note but different'.  The interval between them is one octave and has the ratio 2 / 1.  All notes having frequencies which are successive doublings of the same fundamental are referred to as being of the same pitch class and are one or more octaves apart.

T 1.4.1 Notes of the same pitch class
F 2F 4F 8F

If one note has a fundamental frequency F then its harmonics will have the frequencies 2F, 3F, 4F, 5F, 6F, etc.  If another note has a fundamental an octave higher, 2F, its harmonics will have frequencies 4F, 6F, 8F, 10F, 12F, etc.  These exactly match every other harmonic of the first series, hence the perceived similarity of notes in the same pitch class and the physical basis of the octave.

1.5 Transposing by an Interval

In numerical terms an interval is the ratio between two pitches.  If the pitch of a tone is multiplied by an interval then the pitch changes.  This is the mathematical basis of transposition in music.  For example to transpose a pitch up by an octave it is multiplied by the interval 2 / 1, i.e. it is doubled.

T 1.5.1 Transposing up an octave
fundamental F octave 2F
interval 2 / 1

To transpose down the interval is inverted or reversed.  For example, to transpose down an octave multiply by 1 / 2, i.e. halve the pitch.

T 1.5.2 Transposing down an octave
octave 2F fundamental F
interval 1 / 2

1.6 Simple and Compound Intervals

Intervals of less than one octave (less than a doubling of pitch or frequency) are referred to as simple intervals.  Intervals greater than one octave can be composed of one or more octaves plus a simple interval and are called compound intervals.

1.7 The 3rd Harmonic and the Interval of a Fifth

The relationship of a frequency, F, and its double, 2F, is an octave, 4F a further octave and so on.  But what of a frequency three times that of the fundamental, 3F?  This is about an octave and a half above the fundamental and is the interval known to musicians as a twelfth, or one octave plus a fifth (because it is that many steps in the conventional musical scale).  This interval is the second most important interval after the octave.

The interval of a twelfth is represented by the ratio 3 / 1.  To obtain the interval of a fifth we must reduce this by an octave, 1 / 2, so the interval of a fifth is represented by the ratio 3 / 2, three times the fundamental but transposed down an octave.  A fifth is the interval which exists between the 2nd and 3rd harmonics of an harmonic series.

T 1.7.1 Derivation of a fifth from the third harmonic
F 2F 3F
twelfth 3 / 1
octave 2 / 1
fifth 3 / 2

1.8 Inversion of an Interval and the Interval of a Fourth

The process of subtracting an interval from an octave is called inversion of the interval.  Because of the logarithmic nature of the scale of pitch the process of 'subtraction' is in fact one of division.  For example, to subtract a fifth, 3 / 2 from an octave, 2 / 1, we divide (cross multiply) 2 / 1 by 3 / 2 giving 4 / 3 (dividing the larger interval by the smaller).

In musical terminology intervals are counted in steps of the scale including both end points.  The octave is 8 steps.  A fifth is 5 steps.  To invert a fifth, i.e. subtract it from an octave, yields a fourth or 4 steps because the end points are counted in each case.

T 1.8.1 Subtracting a fifth from an octave to yield a fourth
octave 2 / 1
fifth 3 / 2 fourth 4 / 3

The interval of a fourth has the ratio 4 / 3 and like the fifth, 3 / 2, is of crucial importance to the construction of musical scales.  It is the interval that exists between the 3rd and 4th harmonics.

T 1.8.2 A fourth exists between the 3rd and 4th harmonics
3F 4F
fourth 4 / 3

Fifths and fourths are called perfect fifths and perfect fourths because they occur in both the major and minor scale.

1.9 The 5th Harmonic and the Intervals of a Third

The 1st, 2nd, 4th and 8th harmonics all represent successive doublings of pitch and are therefore octaves.  The 3rd harmonic gives rise to the intervals of a twelfth and a fifth and has been discussed above.  The 6th harmonic is an octave above the 3rd.

Another harmonic having a direct bearing on the construction of musical tuning systems is the 5th harmonic which is about two and a quarter octaves above the fundamental.  In fact it is two octaves plus the interval known as a major third.  The harmonic major third is thus the interval 5 / 4, the 5th harmonic, 5 / 1, transposed down 2 octaves, 1 / 4.  It is the interval that exists between the 4th and 5th harmonics.

T 1.9.1 A major third exists between the 4th and 5th harmonics
4F 5F
major third 5 / 4

The interval between the 5th and 6th harmonics, 6 / 5 is that known as a minor third.  A minor third, 6 / 5 plus a major third, 5 / 4, is a fifth, 3 / 2 (6 / 5 x 5 / 4 = 30 / 20 = 3 / 2).

T 1.9.2 A minor third exists between the 5th and 6th harmonics
5F 6F
minor third 6 / 5

T 1.9.3 A major third and a minor third constitute a fifth
major third 5 / 4 minor third 6 / 5
fifth 3 / 2

1.10 Consonance, Dissonance and Harmonic Intervals

Most tones are not pure tones and therefore contain higher harmonics.  If two tones are sounded together they will sound consonant or harmonious if there is a good match between the pitches of their upper partials.  They will sound dissonant or inharmonious if there is a poor match between the frequencies of their upper partials.  Consonance and dissonance are relative terms in a spectrum of sound perception influenced by the subjective expectations of the listener.

An harmonic interval is an interval that occurs between two members of an harmonic series.  For example in the series 1F,  2F, 3F, 4F, 5F, 6F, etc. the intervals 2 / 1, 4 / 1, 3 / 2, 4 / 3 and others are all harmonic intervals.  Because the harmonics of harmonics are themselves harmonics any two notes related by an harmonic interval will be at least to some extent consonant.  Because the relative strength of the harmonics typically decreases with harmonic number it is the lower numbered harmonics that have the greatest influence on consonance.

The closest possible consonance is unison, 1 / 1, the consonance between two identical notes.  The octave, 2 / 1, is the next most consonant.  The fifth, 3 / 2, and the fourth, 4 / 3, come next.  These are the only intervals classified as perfect concords.  A musical scale constructed on such consonant intervals should itself be consonant.