Pitch

We reproduce below Ellis' famous table entitled History of Musical Pitch which demonstrates the various pitches used at different times in different places.

Why did pitch vary so much even during the same period in history?

One obvious answer is that there was no universal pitch standard. Before the widespread use of keyboard instruments, most serious music in the Middle Ages, both sacred and secular, was sung. The monochord, used to check intervals, was too rudimentary a device to be of use as a pitch reference.

Even into the sixteenth century, the pitch for a cappella performance was set not by the notated parts but rather, as Ludovico Zacconi writes in his Prattica di Musica, pub. Venice (1596), "to have regard for those who are to sing, that they be at ease with the pitch, neither too high nor too low." Something similar continues to this day, for example, in Sacred Harp singing where tunes are sung in relative pitch, rather than at an absolute pitch derived from A=440Hz.; referred to as " Pitch of Convenience", a long standing tradition as can be seen from directions for setting the first note from the Bay Psalm Book*.

(* For more on this topic see here)

Once we reach an era where pitch and tuning were anchored to that of a pre-tuned keyboard instrument, any freedom all but disappeared. Where musicians performed in a band, an orchestra, at court, in the opera house or in a church they would have to cope with several different working pitches. For stringed and keyboard instruments the solution was to retune the instrument. It is said that Isabella d'Este (1474-1539) considered stringed instruments, such as the lute, superior to winds, which were associated with vice and strife. Maybe for wind instruments this association reflected their inability to cope with changes in the pitch, a problem that could be solved only by purchasing completely new instruments when moving from place to place, venue to venue, or by working from parts specifically transposed to take account of the difference in pitch.

Once the Hotteterre family had redesigned woodwind instruments to be made in sections rather than in a single piece, transverse flutes could be made with extra sections which, if longer, lowered or, if shorter, raised the pitch of the instrument. An adjustable plug in the head section was used to correct the tuning and speaking properties of the flute as the middle sections were exchanged. Brass instruments were also made with extra crooks, small lengths of tubing called corps de réchange, which could be fitted to the instruments to change their pitch.

Quite apart from the problems of starting at the same pitch, there was also the reality of playing together as the ambient temperature changed. If the ambient temperature rises, the pitch of stringed instruments, like harpsichords, lutes and violins, drops, while that of wind and brass instruments rises. Played together, the two groups move in opposite directions and what might start out well enough would soon become increasingly strained particularly if the instruments were being played in small concert halls, theatres or opera houses. Churches were less of a problem because they tended to remain cool whatever the weather outside.

Sir John Hawkins, writing in 1776, tells us that the tuning fork, originally called the 'pitch-fork', was invented in 1711, by John Shore, a trumpeter in the band of Queen Anne. It provided the first and, until the advent of electronic meters, the most trustworthy pitch-carrier, and was in every way superior to the 'pitch-pipe' about which the French philosopher Jean-Jacques Rousseau (1712-1778), writing in 1764, noted "the impossibility of being certain of the same sound in two places at the same time".

As an interesting aside, in Korea, pitch was set using resonant stones, called kyong-sok, which whatever the temperature or the humidity would, when struck, produce a reliable pitch reference.

Until comparatively recently, most musicians and scientists, set the note C rather than A. Today, we tune our instruments to internationally agreed pitch standards set for A (actually a' or la₄ ) although it should be pointed out that in a world where equal temperament is widely used, setting A also uniquely sets every other note of the chromatic scale, including C.

The most widely used standard, first proposed at the Stuttgart conference of 1838, but not properly established until 1938 in Britain and in 1939 by the International Organization for Standardization (ISO), is a'=440 Hz. Hz is an abbreviation of the name of the German physicist, Heinrich Hertz (1857-1894), and is a unit of frequency equivalent to one cycle per second. Neither the Stuttgart (1838), the ISO conference (1939) nor its successor held in London in October 1953, was completely successful in setting an internationally agreed pitch. These points are discussed in more detail in A Brief History of Musical Tuning by Jonathan Tennenbaum

Sound is a wave associated with the transmission of mechanical energy through a supporting medium. It can be shown experimentally that sound cannot travel through a vacuum. The energy available in a sound wave disturbs the medium in a periodic manner. Periodicity is important if a sound wave is to carry information. In air, the disturbance propagates as the successive compression and decompression (the latter sometimes called rarefaction) of small regions in the medium. If we generate a pure note and place a detector (our ear, for example) at a point in the surrounding medium, a distance from the source, the number of compression-decompression sequences arriving at the detector during a chosen time interval is called the frequency. The time interval between successive maximal compressions is called the period. The product of the frequency and the wavelength is the velocity.

You are probably aware that the speed of sound is far lower than the speed of light (the speed of light is 299,792,458 metres per second). When, in the middle of a thunder storm, the flash of lightning is followed, noticeably later, by a clap of thunder, we take ever greater comfort the longer the delay. At ground level and at 0° C. the speed of sound is approximately 331.5 metres per second (c. 1,194 km or 760 miles per hour). This is approximately equivalent to 1 mile every 5 seconds (or very roughly 1 km every 3 seconds). The wavelength of the note we call a'=440 Hz proves to be about 753 mm (approximately 30 inches). Bats, who use a signal of about 35,000 Hz (usually written 3 kHz) to search for food, switch to higher frequencies (40-90 KHz) in order to more accurately locate their prey. At these higher frequencies the wavelength is in the range 8-4 mm.

It has long been established, and was described thus by Rayleigh, "that within certain wide limits the velocity of sound is independent, or at least very nearly independent, of its intensity, and also of its pitch (that is, its rate of vibration)". In general terms this must be the case otherwise how could music remain coherent even when it has travelled some considerable distance from performer to listener. However, high frequency sounds do lose more energy than do low. This is one of the reasons why we can tell if a known sound is distant: it has lost more high frequency energy, and this contributes to the 'muffled' sound (see FAQ in music acoustics).

The credit for the first correct published account of the vibration of strings is usually given to Marin Mersenne (1588-1648) although Galileo Galilei (1564-1642) published a remarkable discussion of the vibration of bodies in 1638, derived from his study of the pendulum and of the relationship between pendulum length and frequency of vibration. Although this appeared two years after Mersenne published his Harmonicorum Liber, Galilei's discoveries pre-date those of Mersenne. Wallis (1616-1703) and Joseph Sauveur (1653-1716) noticed that along a vibrating string there are points where there is no motion and others where the movement is particularly violent. Sauveur coined the term 'node' for the former and 'loop' for the latter, although, today, we use the term 'antinode' instead of 'loop' and also suggested the terms 'fundamental' and 'harmonic', applied to frequencies that are integer multiples of a particular frequency. In the discussions that follow, we have adopted the convention that the fundamental is the first harmonic although, in some books, the first harmonic is the name given to the second, not the first, note in the harmonic series. By the sixteenth century, it was clear that the interval relationships between notes, applied to the frequencies of those notes, was identical to the ratios discovered by the Greek from their study of the sounding length of vibrating strings.

We have prepared an article entitled the Physics of Musical Instruments - A Brief History to which you may wish to refer for further details on this topic.

Our appreciation of pitch stability has changed as some instruments notorious for their pitch and tuning instability have been replaced with instruments that are much more stable. For example, modern electronic instruments are almost entirely insensitive to changes in ambient temperature, while even the humble modern piano, with its full metal frame, is a much more stable platform than the half metal half wood framed pianos made three quarters of a century ago, or than the harpsichords, clavichords and spinets made three centuries earlier. Similarly, the relative uniformity of pitch standards around the world, makes it much easier for the modern musician to travel and perform abroad.

However, one must not ignore that fact that 'being' sharp or flat may have its cause outside the immediate mechanics or physics of the instrument upon which one is playing. James Holland, in his book Percussion (Yehudi Menuhin Music Guides, pub. Kahn and Averill, London), writes of the problems when playing the xylophone in unison with violinists in the higher registers.

Indeed, the perception of pitch is a complicated process as this extract from Pitch (music) explains:

Before moving on to examine the movement of pitch standards over time it might be instructive to consider an alternative look at pitch. This extract is taken from Music and Your Health - The Relevance of Concert Pitch by Patrick Thilmany which is to be found on the ANAWME (The Association of North American Waldorf Music Educators) web site.

The Schiller Institute, for example, has been in the forefront of attempts to 'return' pitch to a' = 432 Hz, which was chosen as the pitch standard for Italy at a musical congress that took place during the June 1881 Milan Musical Exposition. We quote a relevant passage from their web site.

What one should make of these two extracts, the author leaves to his reader, but it is worth pointing out that while the belief that Verdi espoused the pitch a'=432 Hz. is supported by a letter from Verdi to his librettist, Arrigo Boito, that advocated a lower pitch for colouristic reasons, it is not at all clear that Verdi's suggestion had any philosophical basis at all.

'History of Musical Pitch' - a table prepared by Mr. A. J. Ellis and published in 1880 (with additions from later publications)

Gill Green's History of Piano Tuning tells us that pitch varied from town to town in England as well, providing another tuning headache: in 1880 Henry Fowler Broadwood wrote to George Rose regarding the difficulty of supplying instruments for provincial tours:

A letter to The Pianomaker in 1913 showed the extent to which the problem had escalated: despite an agreement being made in the 1890s to standardize pitch, military bands were a law unto themselves, fuelled by cynical instrument makers in league with bandmasters who changed pitches arbitrarily to force bandsmen to periodically renew their instruments. T.G. Dyson, then the President of the Music Trades Association of Great Britain, wrote:

Considering the 'international pitch' had been in effect for over twenty years at that time, it seemed to have had little or no effect on the music world in general, if Dyson was referring to 'some eight other pitches'.

In 1896 A.J. Hipkins wrote in his History of the Pianoforte that:

Different piano makers had their own pitches: from 1849-1854 Broadwoods used A=445.9 Hz, escalating to A=454.7 Hz in 1874. Collard's 1877 pitch was A=449.9 Hz, Steinway (in England) in 1879 used A=454.7 Hz, Erard used A=455.3 Hz and in 1877 Chappell tuned at to 455.9 Hz.

No wonder that in June 1860 The Society of Arts established a commission to try to establish a single UNIFORM MUSICAL PITCH.

References:

Harmonic or Overtone Series

Sauveur, following on from work, published in 1673, by two Oxford men, William Noble and Thomas Pigot, noted that a vibrating string produces sounds corresponding to several of its harmonics at the same time. The dynamical explanation for this was first published in 1755 by Daniel Bernouilli (1700-1782). He described how a vibrating string can sustain a multitude of simple harmonic oscillations. We call this the 'superposition principle'.

The harmonics are integer multiples of the 'fundamental frequency', also called the 'first harmonic' or 'generator'. So for a string with a fundamental frequency of 440 Hz, that is fixed at both ends, the harmonics are integral multiples of 440 Hz; i.e. 440 Hz (1 times 440 Hz), 880 Hz (2 times 440 Hz), 1320 Hz (3 times 440 Hz), 1760 Hz (4 times 440 Hz) and so on. The term overtone is reserved for those harmonics that lie above the 'fundamental frequency' (also called the 'fundamental' or 'generator').

To see the harmonics of a violin string visit Standing Waves, Medium Fixed At Both Ends which demonstrates visually the 1st, 2nd, 3rd, 4th and 5th harmonics produced by a string on a violin.

The first 15 harmonics (or fundamental plus 14 overtones) are given below, their frequencies set out in the third column. The fourth column, headed 'normalized', is the result of dividing the frequency of the harmonic by powers of 2 (transposing the sound down one octave for each power of 2) so that it lies within a single octave (between 440 Hz and 880 Hz). The nearest note in the chromatic scale on A is given in the fifth column while the last column, headed %, shows how close the normalized frequency is to the frequency of the nearest equal-tempered note diatonic to A.

If you wish to investigate higher harmonics please refer to our interval calculator.

We can extract a complete diatonic scale on A from the first 15 harmonics. The D is somewhat sharp while the F#, in particular, is very flat. It would not be impractical to tune a stringed instrument to play diatonic melodies in the key of A using this scale.

You will see that the perfect fifth appears in this harmonic series as the third harmonic. The ratio of the frequencies of the third and second harmonic is (1320:880) which is (3:2). However the fourth, the note D, which should have a frequency in ratio to A of (4:3) (1.33333), actually comes out as 1.375. A more serious problem is the absence of an interval one could call a tone or a semitone. The Greeks defined their tone as the difference between a perfect fifth and a perfect fourth, but the fourth is not perfect in this scale. There is no way of deriving chromatic scales either by starting from A or by starting from another note, say, the perfect fifth, E.

We notate below the harmonic or overtone series based on C. The overtones shown in brackets are only approximately equivalent to the equal tempered scale notes on the staff. The overtone count (1, 2, 3, etc.) are one more than the harmonic count in the table above (1, 2, 3, etc.). So the first overtone is the second harmonic. Some commentators call the fundamental (or first harmonic) the 'zeroth' overtone.

Inharmonicity

Before leaving the discussion of harmonics, it would be useful to point out that not all systems produce their harmonics as neatly as, say, the strings of a violin. Some instruments produce harmonics that are not integer multiples of the fundamental. The term inharmonicity is used in music for the degree to which the frequencies of the overtones of a fundamental differ from whole number multiples of the fundamental's frequency. These inharmonic overtones are often distinguished from harmonic overtones, which are all whole number multiples, by calling them partials, though partial may also be used to refer to both. Since the harmonics contribute to the sense of sounds as pitched or unpitched, the more inharmonic the content of a sound the less definite it becomes in pitch. Many percussion instruments such as cymbals, tam-tams, and chimes, create complex and inharmonic sounds. However, strings too, become more inharmonic the shorter and thicker they are, which is an important consideration for piano tuners, especially when setting the thick strings of the bass register. Strings on a piano are generally thicker and therefore shorter than those on harpsichords in order, as we learn from the harpsichord and piano maker Johann Andreas Stein writing in 1769, to accommodate "the blow of the hammer."

Inharmonicity is found, also, in the instruments of the gamelan, particularly in the overtones of the free vibrations of the gongs, bells and strings. The inharmonic partials of the instruments require that the octave be stretched by a ratio of 2.02/1, or 17 cents (100ths of a semitone in 12-tone equal temperament) per octave. The stretched octave sounds more harmonious when tuned so frequencies of vibration coincide, rather than when tuned exactly.

Galembo and Cuddy, in their paper to the 134th meeting of the Acoustical Society of America entitled Large grand vs. small upright pianos: factors of timbral difference, write that:

It should also be remembered that, so far as the piano is concerned, it is the sound board that radiates the energy from a vibrating string, and not the string itself, so the natural resonances of the sound board also play a part in determining the tone we hear. This is explained more fully in the reference below.

References:

• Equal temperaments as mathematical series by John Allen
• Pitch, partials and inharmonicity - from The Acoustics of the Piano

Pythagorean Series

Before going any further we should clarify the distinction between tuning and temperament.

We quote below from Pierre Lewis's article Understanding Temperaments.

We should first ask whether the perfect fifth, one of the three intervals (octave, fifth, and fourth) which have been considered to be consonant throughout history by essentially all cultures, form a logical base for building a chromatic scale; for example, one starting from the note C?

We illustrate above a sequence of fifths starting from F, two octaves below middle C. The image is taken from Tuning Systems by Catherine Schmidt-Jones.

A sequence starting from C would progress as follows:

C G D A E B F# C# G# D# A# F C, the first 7 members providing us with the diatonic scale of G major (G A B C D E F# G)

If one applies the ratio (3:2) twelve times, and normalizes the result by dividing by powers of 2, the result is sharp of an octave by a ratio called the Pythagorean or ditonic comma (524288:531441). We find also that if we use these frequencies to construct a scale, the major third (G B) and the octave (G G, the latter generated from the twelfth power of (3/2)) are both too large.

A number of proposals have been considered in order to 'improve' the Pythagorean scale.

For instance, the Greek major tone, represented by the ratio (9:8) could be married to the semitone, represented by the ratio (256:243) and a scale of five whole tones plus two semitones could be formed. Now the octave is exact but the thirds are still sharp and, because the sharps and flats are not enharmonic, there are problems when changing key.

Another solution employed a pure fourth (4:3) and set the octave as a pure fourth above a perfect fifth, before using the ratio (9:8) to fill in the remaining tones. The remaining semitones were chosen on the basis of taste. Unfortunately, the third is still sharp!

A further solution was to slightly narrow the fifth in every or in only some of the notes arising from the circle of fifths, so absorbing the comma of Pythagoras. This kind of solution made it possible to move from one key to any other and formed the basis of the well-tempered system promoted in 1722 and again in 1724 when Bach published his "Well-Tempered Clavier". The series of keyboard preludes and fugues was written as much to show the characteristic colour of different keys as to demonstrate that, using this tuning system, a composer was no longer prevented from exploring every minor and major key.

Meantone Scale

The first mention of temperament is found in 1496 in the treatise Practica musica by the Italian theorist Franchino Gafori, who stated that organists flatten fifths by a small, indefinite amount. This practice was formalised in what is called the mesotonic or meantone (also written mean-tone) scale. It was always particularly favoured by organists and explains why organ music from the period the early sixteenth to the nineteenth century was written in a relatively small number of keys, those that this scale favoured. Arnolt Schlick's Spiegel der Orgelmacher und Organisten (1511) described both the practice of and formulae for mean-tone tuning which makes it clear that it was already in use. Pietro Aron produced a more thorough analysis in Toscanello in Musica (1523), which sufficed for all practical purposes. The earliest complete description was published by Francisco de Salinas in De Musica libri septem (1577).

How was it set?

Based on C, the method relied on using the first five notes from the circle of fifths from C, namely C, G, D, A, E and setting a pure third between C-E by narrowing the fifths by a small amount - from a ratio of (3:2) to a ratio of (2.99:2). D, the note between C and E was set so that the ratio between D and C was identical to that between E and D, so placing D in the mean position between C and E, hence the scale's name. What happened after this to complete the chromatic scale introduced a number of variants which only the more studious of our readers are likely to pursue. Suffice it to point out that the results generally work well in the keys C, G, D, F and B flat but outside these serious problems arise and composers writing for this system avoided keys more distant from C.

Pietro Aron's description of meantone tuning is the best known. All but one of the fifths are flattened from the pure (3:2) ratio by 1/4 of the syntonic comma. The remaining fifth ends up being sharp by 1 3/4 of the syntonic comma (the wolf). The syntonic comma is the ratio (9:8) divided by (10:9), which is the ratio between a pure C-D interval and a pure D-E interval. In a pure harmonic series starting at CCC (CCC is English organ nomenclature: bottom C on a 16' voice), middle C is 8 times the fundamental, middle D is 9 times the fundamental, and middle E is 10 times the fundamental. The result of this procedure is a scale with 8 pure major 3rds and 4 diminished 4ths. But there were other meantone procedures known in the 16th and 17th centuries, especially by 2/7th comma, in which the minor 3rds are pure and the major 3rds beat, and 1/3rd comma. In the the mid-eighteenth century, several instrument-makers and theoreticians used a 1/6th comma meantone temperament, particularly Gottfried Silbermann and Vallotti. A bizarre fact is that equal temperament is really meantone by 1/12th comma, that is every fifth is narrowed by 1/12 of the syntonic comma and the interval between C - D and between D - E are equal. So, all the modern pianos you have ever heard are in one of the many types of meantone temperament!

Equal Temperament

It must have been a brave man who first pointed out to a world wedded to centuries of mean, natural and Pythagorean tuning, that a scale could be formed using a universal ratio for a semitone such that successive application of this ratio generated the notes of a chromatic scale before completing the octave with its harmonic ratio of (2:1), and that using such a system one might play in tune in any key. This universal ratio is the twelfth root of 2.

This tuning system, called 12EDO (Equal Divisions of the Octave), 12-tET by modern tuning theorists or Standard European 1/12 Diatonic Comma Equal Temperament by others, found favour amongst the lutenists of the sixteenth century who, having tuned the instrument's strings to different notes, could fret each at an identical point from the nut to produce parallel equal-tempered scales something that would be impossible using any other temperament. Unfortunately, as Nicola Vicentino, the inventor of the archicembalo with six rows of keys that enabled six different versions of any scale to be performed complete with temperamental adjustment, observed, this produced horrible clashes between the lute tuned to an equal-tempered scale performed with a keyboard tuned using mean-tone temperament.

Evidence of the use of equal temperament in consort singing comes in the madrigal O voi che sospirate a miglior note by Luca Marenzio (c.1553-1599). The composer modulates completely around the circle of fifths within a single phrase, using enharmonic spellings within single chords (for instance, simultaneous C# and Db), which would be impossible to sing unless some approximation of equal temperament is being observed.

At the time, keyboard players found the equal-tempered scale more 'sour' than the other systems in the five keys commonly used, and because most composers worked only in a limited number of keys the benefits to be had from the equal-tempered system in more distant keys were not at all obvious. This probably helped delay its acceptance until such time as enough 'new' ears had become used to it, or enough composers had explored more distant keys with it in mind. In England, it was not until 1842 that the first organ, that of St. Nicholas in Newcastle-upon-Tyne, was tuned to equal temperament.

It is still surprising that the system may have been known in Europe as early as the fifteenth century (some have suggested that equal temperament was first explained by Chu Tsai-yü in a paper entitled A New Account of the Science of the Pitch Pipes published in 1584). However, Henricus Grammateus had already drawn up a fairly close approximation in 1518, and Zarlino corrected Vincenzo Galilei's plan for a twelve-stringed equal-tempered lute (Galilei had invoked Aristoxenus as his inspiration in this project). Even though the mathematician and music theorist Mersenne produced a correct and systematic description in 1635, equal temperament was not adopted until 150 years later in Germany and Austria, while Britain and France delayed for over two centuries. As late as 1879, William Pole was writing in his book The Philosophy of Music, "The modern practice of tuning all organs to equal temperament has been a fearful detriment to their quality of tone. Under the old tuning, an organ made harmonious and attractive music. Now, the harsh 3rds give it a cacophonous and repulsive effect." In 1940, another sceptic, L. S. Lloyd, wrote an article entitled The Myth of equal Temperament in which he described the improbability of singers, or players of any instrument with variable intonation of being able to sing or play in true equal temperament; or, a keyboard instrument actually being tuned to theoretically correct equal temperament. It is worth remembering that Vincenzo Galilei (1520-1591), an Italian lutenist, composer, and music theorist, and the father of the famous astronomer and physicist Galileo, observed that instrumentals and singers failed generally to observed any theoretical tuning or temperament. As Barolsky writes "all intervals, Galilei argued, are natural, not simply those mathematically determined. In performance, a fifth that is a bit off the (3:2) ratio is just as useful as one that is exactly on the mark."

It is interesting to read what the German composer, violinist and conductor Louis Spohr (1784-1859), writing in his Violinschule of 1832, has to say on the subject of equal temperament.

The clavichord maker Peter Bavington, writing to the Yahoo clavichord-list, comments:

The equal-tempered system cannot be derived from rational relationships because the twelfth root of 2, like the square root of 2, is irrational.

Many commentators state that in an equal-tempered scale the fifths are perfect. In fact, the ET-fifth is slightly narrower and the ET-fourth is slightly wider than their just interval equivalents.

The theoretical equal temperament frequencies for the A=440 Hz tuning pitch are:

Just Intonation

Barbour writes, in Tuning and Temperament, "it is significant that the great music theorists ... presented just intonation as the theoretical basis of the scale, but temperament as a necessity". Strict adherence to just intonation could, under certain circumstances, lead to pitch descent by tuning, so-called commatic drift.

Commatic drift is defined by Paul Erlich (and quoted in Joe Monzo's Tonalsoft Encyclopedia of Microtonal Music Theory) as "an immediate change in the pitch of a note, as the note is held or repeated from one harmony into another". He continues, "a drift is an overall pitch change of the entire scale. its effect on the pitch of any note doesn't become evident until an entire "comma pump" chord progression has been traversed. For example, in the classic problem of rendering the I-»vi-»ii-»V-»I progression in strict Just Intonation, one either has a shift (the 2nd scale degree shifts from 10/9 in the ii chord to 9/8 in the V chord) and no drift, or a drift (the final I is lower by 80:81 than the initial I) and no shifts."

However, despite many examples of where, were the intervals defined by Just Intonation strictly to be adhered to, there would be a drift in pitch, musicians actually 'correct the pitch' by tempering various intervals. In fact, it is the pitch of individual notes that vary ever so slightly through chordal progressions, the net effect of which is to hold the overall pitch reasonably constant.

The natural or harmonic scale is being explored again in the twentieth century through the work of Harry Partch, Lou Harrison and others who, with the advantages of modern technology, have sought to explore musical systems that were abandoned more for their practical limitations than for any lack of aesthetic interest. One has only to consider the complexity of a piano built to perform music based on a microtonal system, or remind ourselves of Nicola Vicentino's archicembalo, instruments that have been made and played, to appreciate that the equal-tempered scale brings with it certain advantages.

Following on the ideas of Theodor Adorno, the American composer Ben Johnston believes that music has the power to influence and even control social trends. Johnston believes that an equal tempered tuning system based on irrational intervals contributes to the hectic hyper-activity of modern life. The wildly beating sonorities of equal temperament are thought to resemble (and perhaps foment) the fast-paced, unmeditative current of present-day Western existence. Many just intervals lack the sharp vibrancy of irrational intervals (and higher-order rational intervals) and thus are sometimes felt to convey an affect of stasis and meditative calm. Indeed, cultures whose tuning systems draw heavily on purely tuned intervals (e.g., North Indian classical music) tend to value meditative social attitudes more greatly than in the West.

Reference:

For the reference of tuning enthusiasts, we have included an extended and annotated version of Kyle Gann's Anatomy of An Octave, which contains all pitches that meet any one of the following criteria:

1. All ratios between whole numbers 32 and lower
2. All ratios between 31-limit numbers up to 64 (31-limit meaning that the numbers contain no prime-number factors larger than 31)
3. Harmonics up to 128 (each whole number divided by the closest inferior power of 2) including all ratios between 11-limit numbers up to 128
4. All intervals in the equal-tempered scale
5. All ratios between 5-limit numbers up to 1024
6. Certain historically important ratios such as the schisma and Pythagorean comma

The table is similar to, but much briefer than, that found in Alain Danielou's encyclopedic but long out-of-print Comparative Table of Musical Intervals.

References:

• List of Musical Intervals
• List of Meantone Intervals
• List of Pythagorean Intervals
• Interval calculator that converts frequency ratios to cents

Naming Intervals

The following table lists the names of the most common intervals using a number of modern conventions.

The standard system for comparing intervals of different sizes is with cents, based on a logarithmic scale where the octave is divided into 1200 equal parts. In equal temperament, each semitone is exactly 100 cents. To remind our readers of the formula given earlier, the value in cents for the interval f₁ to f₂ is 1200×log₂(f₂/f₁).

In diatonic set theory, specific and generic intervals are distinguished. Specific intervals are the interval class or number of semitones between scale degrees or collection members, and generic intervals are the number of scale steps between notes of a collection or scale.

Reference:

Historical Temperaments

Prior to the almost universal adoption of the equal temperament system of tuning where the interval between successive semitones is a constant and the ratio for the octave is set at 2:1, musicians and theorists produced numerous solutions for bending the natural Pythagorean scale to practical use. That this was an impossible task, particularly if one wished to modulate to all the possible major or minor scales, was demonstrated time and again by composers such as Willaert who used their works to demonstrate the shortcomings of any of the temperaments then in use. Written in four parts, the vocal work Quid non ebrietas, starts on the key note before taking the singers through a sequence of perfect fifths that, if they use Pythagorean tuning based on perfect fifths, leaves them sharp by the Pythagorean comma when they return to the key note at the end of the piece. If the singers choose instead to use just intonation, they reach the end flat to the desired key note. Of course, most of these problems could be ignored so long as composers chose to remain reasonably close to the key in which the work started. Composers like Willaerts, Nicola Vicentino and Carlo Gesualdo pushed the boundaries of temperament so hard that special instruments had to be invented to handle the complexities of tempered scales as key notes changed. Vicentino invented the archicembalo with its six rows of keys. He also inspired Fabio Colonna's sambuca in which the octave was divided into thirty-one parts.

The temperaments we set out below were commonly used before the widespread introduction of equal temperament. Each was an attempt to rid the 'so-called' natural scale of its problems under modulation. We give some information below about the common 'historical temperaments' used when setting keyboard instruments for historically informed performance all calculations based on a'=440 Hz. We have also provided a calculator below which provides many temperaments at historical pitches where a' is lower or higher than 440 Hz.

Pythagorean

Strictly, not a temperament but a tuning because natural intervals are not adjusted but allowed to fall where they may, it dates back to 500 BC. This simple scale creates eleven pure fifths around the circle, leaving the entire Pythagorean comma between G# and Eb There are four pure major thirds at B-D#, F#-A#, Db-F, and Ab-C, but these are not particularly useful. The remainder are quite harsh.

Those who are paying attention may have noticed that when we tune with pure 5ths the last note in the series, D#, when normalised lies above the Eb at the beginning. However, on modern keyboard instruments, D# and Eb are enharmonically equivalent (that is, they are played with the same key). Some may even be aware that under some tuning systems D# lies below Eb. How can this be?

Clavichord maker Peter Bavington, explains:

van Zwolle

Arnout van Zwolle (1400-1466) modified the Pythagorean scale by placing the comma in the interval B-F#. This moved the thirds to D-F#, A-C#, E-G#, and B-Eb, which were more useful. This gives pure major triads on D, A and E.

Meantone

The best known of the old scales, this scale emphasizes pure thirds by making the fifths narrow. It was certainly in use by the end of the fifteenth century, if not earlier. All whole steps are equally spaced, one half of a major third apart. It also has a very prominent "wolf" between G# and Eb. If the circle is extended down to Ab, the pitch is very different from the G#. Some baroque keyboards had a split pair of black keys that allowed the musician to choose G# or Ab. G# is set two perfect thirds above the lower C and so its ratio is 25:16. Ab is set a perfect third below the higher (octave) C and its ratio is 8:5.

Silbermann I, II

Organ builder Gottfried Silbermann (1678-1734) tried several variants to narrow the "wolf" and make his instruments useable in more keys. None of the intervals of these two scales is pure.

Rameau

Jean Phillipe Rameau (1683-1764) modified the meantone scale to provide three pure fifths. This very pleasant scale almost completely eliminates the harsh "wolf" of the meantone while preserving most of its pure harmony.

Werkmeister I(III), II(IV), III(V), IV(VI)

Organ builder and mathematician Andreas Werkmeister (1645-1706) devoted much of his life to the study of temperament and suggested many, different scales. His goal was to place the best thirds in those keys with the fewest incidentals. It is very likely that Bach (l685-1750) wrote his famous Das Wohltemperierte Klavier, 48 preludes and fugues, for one of these temperaments. The tuning systems are confusingly numbered in two different ways: the first refers to the order in which they were presented as "good temperaments" in Werckmeister's 1691 treatise, the second to their labelling on his monochord. The monochord labels start from III since just intonation is labelled I and quarter-comma meantone is labelled II. Werckmeister IV (VI), the 'so-called' Septenarius tuning is based on a division of the monochord length into 196 parts. Werckmeister I (III) (1/4 ditonic or Pythagorean comma) is one of the three temperaments most commonly used in HIP performance today.

References:

• Werkmeister temperament
• The Septenarius: Werckmeister's mythical tuning, made reality?

Kirnberger II, III

Composer and music theorist Johann Philipp Kirnberger (1721-1783) suggested several temperaments. The two scales here offer a large number of pure fifths. The first has pure thirds at C-E, G-B, and D-F# but the fifths at D-A and A-E are somewhat harsh. Kirnberger later proposed an alternate scale with smoother fifths, but only one pure third at C-E. Kirnberger III (1/5 ditonic or Pythagorean comma) is one of the three temperaments most commonly used in HIP performance today.

Vallotti-Young

Francesco Antonio Vallotti (1697-1780) published his Trattato della scienza teorica e pratica della moderna musica in 1779 although claiming that his theories were set out over 50 years earlier. Discussion of the temperament that bears his name appears in Book 2. Vallotti's temperament was mentioned in England by William Jones in 1781 (Physiological Disquisitions) with a reference to an endorsement by Tartini, who was leader of the professional orchestra in Padua under Vallotti's direction. Thomas Young (1773-1829) was a polymath whose temperament was given as part of an address to the Royal Society in January, 1800 (Philosophical Transactions, pp. 143-47). Young #2 (1800) (1/5 ditonic or Pythagorean comma) is one of the three temperaments most commonly used in HIP performance today.

Kyle Gann in his An Introduction to Historical Tunings, writes: "Thomas Young's Well Temperament [is considered by some] to be the most elegant well temperament, with a fluid variety of tonal colors and a symmetry that matches the piano keyboard: all intervals are symmetrical around D and G# - that is, D-F# and D-Bb are the same size, G#-F# and Ab-Bb the same size, and so on. This is a subtle tuning, quite usable in all keys, and the differences from equal temperament are more evident to the pianist playing in it than to the listener. The best major thirds are grouped in the circle of fifths around C-E, whereas the perfect fifths become more perfect in the black keys, which all have fifths of 702 cents. This gives the keys related to C a sweet, gentle quality, the black-note keys an austere, noble quality, and middle keys like Eb and A a neutral, ambiguous quality."

Italian eighteenth century

One of the many variations commonly in use in the eighteenth century that emphasized a pure third at C-E and distributed the "wolf" around the circle of fifths. There is only one pure interval in this scale.

Temperament Ordinaire

In the near keys, the thirds are much purer and the fifths less so than in the remote keys. In the near keys, irregular temperaments resemble meantone, and in the remote keys, they resemble (the near keys of) Pythagoras' tuning (with the tense thirds). This gives added variety to modulation, which was appreciated in the past. Bradley Lehman, writing to Meantone, on the www.bach-cantatas.com website, argues that temperament ordinaire is more a process (or a strategy) than any specific outcome.

Reference:

• Sources of the Temperament Ordinaire: France, late 17th to late 18th century

Equal Temperament (ET)

This scale is so common in the twentieth century that many musicians and instrument makers tend not to know that there are alternatives.

Dividing the Pythagorean comma equally around the circle of fifths is not a recent idea.

One of the earliest mentions of equal temperament was by the Dutch mathematician Simon Stevin. He published his proposal for equal tempering as early as 1584.

A. D. Fokker, in Simon Stevin's views on music, writes that:

It is from this source that we have it confirmed that while meantone tunings with pure or near-pure thirds were standard on keyboards, ET was standard on fretted instruments such as the lute. However, for a long time it was not considered satisfactory due to the impurity of all the intervals.

Many commentators have suggested that equal temperament (ET) on keyboard instruments is really a twentieth century idea and was not widely used before that.

Peter Bavington, writing on the clavichord mailing list makes the following comment:

Fred Sturm comments on Daniel Gottlob Türk's 1789 Klavier Schule:

In the late 18th- and 19th- centuries, composers began increasingly to explored modulation to more distant keys. They found that most temperaments were unsatisfactory because of the significant tonal changes involved in changing keys. The equal-tempered scale was begrudgingly recognized as an acceptable compromise that worked equally well in every key. It is only through over a century of dominance that this scale has become the one that we are accustomed to - the scale that sounds "in tune" to us today. It is actually one type of 'meantone' tuning because the third lies exactly midway between the root and the fifth.

Temperament/Tuning and Pitch Calculator