Acoustics

The Physics Of Music

Sound is a longitudinal (/ compression / pressure) wave in air or in solid material such as wood. It can be created by the transverse motion of objects. Waves are measured by their cycles from high to low pressure or position and back again. The frequency of a wave is the number of high-low cycles completed per second. The wave-length is the distance occupied by one high-low cycle. This depends on the medium in which the wave is travelling.

Longitudinal Wave

Transverse Wave

Noise can be defined as sound at many different unrelated frequencies simultaneously. A musical note is coherent sound - the auditory equivalent of a laser. A laser produces coherent light by setting up a standing wave in a crystal. A musical instrument produces music by setting up a standing wave in a string or pipe. Standing waves appear not to travel like ripples but just to vibrate in place. It is interference between waves travelling in both directions that produces this effect. At certain points along the length of the string or pipe, called nodes, there will be no resultant movement. Between these nodes will be the points of maximum oscillation.

More than one wavelength and therefore frequency of standing wave will fit a given string or pipe. The distinctive sounds of different instruments result partly from the combination of frequencies that can co-exist. However, the biggest distinguishing factor is the messy noise at the start of producing a note. These are called transient tones. Remove the start of a note from a recording and it becomes very hard to recognise the source.

The lowest possible frequency of standing wave that an instrument can produce is called its fundamental. It depends on the length of the string or pipe (for a given material and shape). The fundamental may not be a note that is actually usable by the player (eg in the horn). It is also very unusual to get just one frequency of vibration at a time. In addition to the intended note, there will be higher notes at related pitches. They are called harmonics or overtones. The intensities and frequencies of these depend on the mechanical properties of the instrument.

In a stringed instrument, the string is held under tension and necessarily fixed at both ends. This means that the fundamental is a single cycle with no movement at the ends and maximum motion in the centre. The first harmonic overtone is a double cycle with no movement at the ends or centre and maximum motion in the middle of each half. The next harmonic is a triple cycle and so on. The intensity of vibration reduces with each successive harmonic to the point at which they can no longer be discerned. Normally a string is plucked or bowed (equivalent to repeated tiny pluckings) partway along its length. However, bowing near the bridge of a violin or similar instrument can bypass the fundamental and force just the harmonics.

Two-ended String Fundamental & Next Harmonics

The equivalent of a one-ended string is seen in the metal prong of a "Jew's Harp" or the multi-pronged African version. In this case the fundamental is a half cycle with no movement at the fixed end and maximum motion at the open end. The first harmonic overtone is one and a half cycles. There is no movement at the fixed end or one third of the way from the open end. There is maximum motion at the open end and one third of the way from the fixed end. The next harmonic is two and a half cycles and so on. The intensity of vibration reduces with each successive harmonic to the point at which they can no longer be discerned. This gives a different pattern of harmonics than the two-ended string.

One-ended Pipe Fundamental & Next Harmonic

In a pipe instrument, any reed merely sets up the vibration rather than defining its frequency. Where both ends of the pipe are closed, the pattern of harmonics is the same as for two-ended strings. More commonly one end of the pipe is open, so the pattern of harmonics matches that of a one-ended prong. Note that if both ends of a the pipe were open, this would be the same as both being closed. However the positions of maximum and minimum motion would be reversed. Early wind instruments used straight pipes but then it was discovered that the tube still worked when bent. This allowed long pipes with lower frequencies to be made more portable by being coiled. Compare a modern French horn with an alphorn or didgeridoo!

From the harmonics it should be obvious that halving the length of a given string or pipe will double the frequency produced. Any change in length changes the frequency by a proportionate amount. However, increasing the tension in a string will also increase the frequency produced. This is how fixed string instruments are tuned. A peg or screw at one or both ends is turned to adjust the tension. Increasing the density used for a string will decrease the frequency. This is why some strings are wire-wrapped. They produce lower notes without getting too long.

The piano and harp have a lot of pre-tuned strings and a player just selects the correct one for each note. The organ and pan-pipes have a lot of pre-tuned pipes to select. Percussion instruments usually have pre-tuned slabs of metal, blocks of wood or membranes under tension. Some instruments of each type allow players to access more than the pre-tuned notes in order to play a melody. Usually this is by "fingering".

Most stringed instruments allow a player to decrease the effective length of each string by pressing a finger against it. Guitars have frets (metal rods) at pre-calculated positions to make this easier. Most woodwind instruments have holes which decrease the effective length of the pipe when uncovered. For long pipes, lever-operated pads cover the holes that fingers alone can't reach. Calculating the necessary sizes, positions and combinations of holes is much more complicated than calculating fret positions. Most brass instruments have valves which can include additional pieces of pipe (called crooks) to increase its effective length. A player has to select the correct valves as well as adjusting the air pressure used. Some drums have levers that allow a player to adjust the tension and therefore the pitch.

The relationship between frequencies of notes is more important in music than the actual number of cycles per second. So the actual pitch attributed to named notes has varied over time. Modern orchestral convention puts an A at 440 Hz. Scientific or philosophical pitch (also used in computers) puts a C at every power of 2, eg 256 Hz is middle C. These two definitions differ by more than 2% on any note and are not the only options. Before starting to play together, the members of an orchestra always make sure that they are in tune with each other. One instrument (or a tuning fork) is selected to provided the absolute pitch and everyone else tunes to this.

Note that the sound produced by plucking a string or blowing on a reed is not very loud. Instruments evolved to use a sound-board, hollow shell or bell to magnify and disperse the sound. Electric instruments use microphones and amplifiers while electronic ones generate notes entirely artificially! A sound-board also transmits vibrations to other parts of an instrument. On a piano, the strings corresponding to the harmonics of a note can be made to resonate when just one key is struck. First depress the sustain pedal or quietly hold down the relevant keys. This will release the dampers that normally prevent unwanted vibrations (such as cross-resonance) in the strings.

The Mathematics Of Music

Ignoring the actual physical pitch of a string or pipe and calling its fundamental frequency "1" gives its sequence of harmonics. The two-ended sequence becomes 1, 2, 3, 4, 5, 6 etc. The one-ended sequence becomes 1, 3, 5, 7, 9, 11 etc. Note that bypassing the fundamental starts a sequence part way through. For example, starting the two-ended sequence at 2 and treating this as the new 1 gives 1, 1.5, 2, 2.5 etc.

Two-ended Harmonic Sequence (G) 1 2 3 4 5 6 7 8 9

One-ended Harmonic Sequence (G) 1 3 5 7 9

The ear can recognise simple mathematical relationships between frequencies. When one frequency is twice another, it is regarded as a pleasant combination. This ratio of 2:1 (= 2) is the interval now called an octave. The term octave or eighth is because the interval encompasses 8 notes on a 7-note scale - including both end notes. It occurs at the beginning of the two-ended sequence of harmonics. The next most significant intervals follow it. They are 3:2 (= 1.5) and 4:3 (= 1.333) and are now called a fifth and a fourth respectively. Note that the jump from 2 to 4 is the same ratio as 1 to 2 and is also an octave - the second one. Therefore a fifth and a fourth make an octave (adding the intervals means multiplying the ratios).

The ratios of an octave, fifth and fourth are so simple that some people cannot distinguish them from the original note. This meant that people with different voice pitches naturally sang at these intervals from each other. So unison or plainsong melody singing was in fact nothing of the sort! The tenors sang a fourth or fifth above the basses and the trebles sang an octave above. Even the majority of people who can distinguish these intervals regard them as harmonious.

The next intervals are 5:4 (= 1.25) and 6:5 (= 1.2) and are now called the major and minor thirds respectively. Note that the jump from 4 to 6 is the same ratio as 2 to 3 and is also a fifth. Therefore a major and minor third make a fifth (multiplying the ratios again). Although thirds are now regarded as very harmonious intervals, for a long time they were regarded as dissonances. It was only as musical taste became more sophisticated that they were accepted. This marked the beginning of what is now regarded as harmony in music.

The remainder of this third octave is another fourth which itself consists of two intervals. 7:6 (= 1.167) is somewhat less than a minor third and 8:7 (= 1.143) is even smaller but still more than a major second. The harmonics in the third octave together form a major triad with a flat 7th note added. The intervals from minor third onwards are rather similar and were not separately defined.

Comparing the early intervals gives interesting results. If the difference between a fifth and a fourth is to be a second then dividing these ratios gives 9:8 (= 1.125). This actually is the next harmonic interval and starts the fourth octave. It was the original "tone" used in modes. It is also the arithmetic half of a major third (the geometric half or square root is irrational). Taking the arithmetic half again gives 17:16 (= 1.0625) for a semitone. This starts the fifth octave. Note that a fourth could be considered as a minor third plus a major second or a major third plus a minor second. This would give tone and semitone ratios of 10:9 (= 1.111) and 16:15 (= 1.066) which are not consistent with the previous calculation.

No individual ratio can be repeated to build an octave. Three major thirds (5:4) would be 125:64 which is 5% too small. Four minor thirds (6:5) would be 1296:625 which is 7% too large. Six tones (9:8) would be 531441:262144 which is 3% too large, but the alternative (10:9) ends up 12% too small. There is a "reductio ad absurdum" proof that the irrational number made from the root of an integer cannot be represented by a fraction. In this case the original integer is 2 (the octave) and the roots just attempted are the 3rd, 4th and 6th.

The eight intervals in the fourth octave are all major or minor seconds. They are generally thought of as dissonant unless merely heard in passing. Hence the term "passing notes" for transitions from one accepted harmony to another. The closer frequencies become in terms of ratio, the more dissonant the combination seems. The actual difference in Hz does play some part though, so low note pairs sound worse than high ones. When approaching a ratio of 1 between frequencies, "beats" are heard. This is because the sum of two similar sine-waves is a mean sine-wave with amplitude modulation at the frequency of the difference. The phenomenon is used when tuning instruments (when the "beats" have reduced to nothing, the two notes are the same). Some composers have used microtones (small intervals) in music, but not many people like the result! The larger intervals form the basis of melody and harmony in music.

The pentatonic (5-note) scale has been invented by many cultures. This is because its intervals are derived from the simplest mathematical ratios. In Europe it was extended to a system of modes which were 7-note scales. Each mode consisted of 2 tetrachords separated by 1 tone. Each tetrachord was an interval of a fourth sub-divided into 3 unequal smaller intervals - 2 tones and 1 semitone. The order of the intervals within the tetrachord and the choice of tetrachords defined the mode. Each mode had a distinctive style but there was no flexibility in moving from one to another. The tuning of an instrument defined which note had to start each scale. A change of key within a piece is called modulation and comes from the word "mode".

The modal system was later abandoned in favour of 12 equalized intervals between notes. Scales still only used 7 notes at a time but now the starting note made no difference. Modulation became easy. The scales in common use were reduced to just major (happy) and a couple of variations on minor (sad). A starting note of a scale now defined its key rather than its mode. This innovation was celebrated in Bach's "Well-Tempered Klavier" pieces. Of course the original precise mathematical ratios are lost, but the resultant intervals are close enough approximations for most people.

Dividing an octave ratio of 2:1 into 12 equal parts means taking the twelfth root of 2. This gives an interval of 1.059463 per semitone which is about 0.3% smaller from the 17:16 ratio. Twice this or the sixth root of 2 gives an interval of 1.122462 per tone. This value lies between the 9:8 and 10:9 ratios and is about 0.3% smaller than the former. The semitone and tone are the minor and major seconds respectively.

A minor third is three semitones or a tone plus a semitone or the fourth root of 2. This gives an interval of 1.189207 which is about 0.9% smaller than the 6:5 ratio but about 2% larger than the 7:6 ratio. A major third is four semitones or two tones or the third root of 2. This gives an interval of 1.259921 which is about 0.8% larger than the 5:4 ratio. A diminished chord consists of four notes at intervals of minor thirds. An augmented chord consists of three notes at intervals of major thirds. Both chords sound a little dissonant and, when used in music, should be "resolved" into a more normal harmony.

A perfect fourth is 5 semitones. This gives an interval of 1.33484 which is about 0.1% larger than the 4:3 ratio. A perfect fifth is 7 semitones. This gives an interval of 1.498307 which is about 0.1% smaller than the 3:2 ratio. Between these two, half an octave or six semitones is called an augmented fourth or a diminished fifth. The interval is 1.414214 and is dissonant unless part of a chord of the 7th. The closest natural equivalent is between harmonics 5 and 7 in the third octave. The second or square root of 2 is about 1% larger than this ratio. Given that a semitone is about a 6% difference between notes, the equalised fourth and fifth sound quite accurate. Also remember that none of the exact ratios can be repeated to build an octave on its own.

The perfect fifth and perfect fourth are complementary intervals in that together they make an octave. Similarly, the complement of a major third is a minor sixth and that of a minor third is a major sixth. Also the complement of a major second is a minor seventh and that of a minor second is a major seventh. The complement of an octave is a unison! Although technically both the same interval, for notation purposes the complement of an augmented fourth is a diminished fifth. Intervals such as a ninth or tenth are merely an octave plus a second or third respectively. A twelfth (octave plus fifth) occurs at the beginning of the one-ended sequence of harmonics. It has a ratio of 3:1. The next interval in this sequence has a ratio of 5:3 and is a major sixth. The next is 7:5 which is quite close to the geometric half of an octave (6 semitones). The clarinet is a one-ended pipe that has its overblown upper register a twelfth above its lower register.

Note that since frequency is inversely proportional to length, the ratios are inverted when calculating distances. So 1/2 a string or pipe gives the octave 2:1 frequency ratio. 2/3 of it gives the perfect fifth 3:2. 3/4 of it gives the perfect fourth 4:3 and so on. This can also work for an equalised system using the 12th roots of 2. Semitone N is at (2) ^-N/12 of the original length. In practical terms for a guitar-like instrument: put the octave fret at the central point along the length of its strings. Then divide the finger board back from this into 12 calculated spaces using 11 more frets.

Natural Intervals As Length Ratios

Equalised Intervals For All Semitones