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 at 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 song. The monochord, used to check intervals, was too rudimentary a device to be of use as a pitch reference. Margaret Bent (Diatonic ficta revisited: Josquin's Ave Maria in Context) discusses what she calls 'her free-standing, vocally-conceived, Pythagorean, pre-keyboard "medieval" view of pitch, tuning and vocal counterpoint.'
| "Our
musical culture has raised the definition of frequency and pitch-class
to a high status, for analysis, editing and performance. My reading of
a range of early theorists leads me to posit a slightly fuzzier status
both for what we would call pitch-class and for frequency, a status
that places pitch closer to the more flexible view of durations and
tempo that we still have. This reading rests partly on conspicuous
circumlocutions and the late arrival of precise language, notation and
measurement, partly on a pervasive Pythagorean mentality expressed in
the tuning system, partly on my understanding of counterpoint and the
internal evidence of some paradigmatic pieces. We routinely make
rhythmic and durational analyses on the basis of notated values even
though we know that performance fluctuations, some necessary, some
elective, expected but elusive to precise definition, are ignored by
the analyst. We are not necessarily shocked if an analysis disregards
the fact that a piece, any piece, may end slower than it began. A
terminal ritardando needn't affect certain kinds of analysis; nor need the ritardando of pitch caused by a logical downward sequential spiral (Obrecht Libenter gloriabor Kyrie) shock us."
"A rigid frequency stability which, however well established it became in the keyboard-reference era, did not, I believe, govern earlier music. Without cumbersome advance planning, I maintain that it is virtually impossible to sing the Obrecht Libenter gloriabor Kyrie (and about 30 other pieces) from original notation in any other artful way than to let the sequence, indeed, wind smoothly down in its contrapuntal operation (irrespective of the tuning used, even if that were equal temperament). This happened in one of our singing sessions when someone innocent of its notoriety brought a facsimile along. We read it, it descended, as everyone was (and always would have been) well aware as it was happening. The sequence of descending fifths and rising fourths F B E A D G C F is notated only with a few encouraging B and E flats, but its smooth counterpoint locks it into -- in our terms, F Bb Eb Ab Db Gb Cb Fb. Another of us, who had previously been sceptical of "my" solution on paper, exclaimed with surprise that it sounded fine. That is precisely the point. Try it!" |
Even into the 16th 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."
Once we are in an era and a situation where the pitch was, in effect, fixed by the presence of a keyboard instrument, at which pitch musicians played would have varied according to where they were employed. If they performed in a band, an orchestra, at court, in the opera house or in a church they would have experienced several different working pitches. For stringed and keyboard instruments the solution was to retune the instrument. Wind and brass players, however, were faced with very real difficulties which could only be overcome either 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 also had extra crooks, small lengths of tubing called corps de réchange, which could be applied 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. Wind and brass players may have suffered a lower status than stringed and keyboard instrumentalist because of their constant struggle to remain in tune with their fellow musicians and the difficulty they might face when moving from one employer to another, that of differing pitches. For similar reasons, the instruments of even the finest wind-instrument makers tended to travel less widely than those of the finest stringed or keyboard instrument builders.
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') 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, 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 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'=440Hz. proves to be about 753 mm (about 30 inches). 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.
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 16th-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 instrument 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.
'History of Musical Pitch' - a table prepared by Mr. A. J. Ellis and published in 1880 (with additions from later publications) ::
| a' (or A4) (in hertz) | c'' (or C5) (in hertz) | Place | Date | Description |
| 376.3 | 447.5 | Lille, France | 1700 (anté) | Pitch taken by Delezenne from an old dilapidated organ of l'Hospice Comtesse |
| 378.8 | 450.5 | Paris, France | 1766 | Pitch calculated from data given by Dom Bédos in L'Art du Facteur d'Orgues |
| 380.0 | 451.9 | Heidelberg, Germany | 1511 | Pitch calculated from data given by Arnold Schlick |
| 392.2 | 469.1 | St. Petersburg, Russia | 1739 | Euler's clavichord |
| 395.8 | 470.7 | Versailles, France | 1789 | Organ of the palace chapel |
| 398.0 | 473.3 | Berlin, Germany | 1775 | Pitch estimated from a flute described by Jean Henri Lambert in Observations sur les Flûtes, pub. Académie Royal des Sciences, Berlin |
| 400.0 | 475.7 | Paris, France | c. 1756 | Pitch estimated from a flute made by T Lot, one of the five 'maîtres constructeurs' of wind-instruments in Paris, France |
| 401.3 | 477.8 | Paris, France | 1648 | Mersenne's Spinet |
| 404.0 | 480.4 | Paris, France | 1699 | Paris Opera A |
| 405.8 | 482.6 | Paris, France | 1713 | Sauveur's calculation |
| 407.9 | 485.0 | Hamburg, Germany | 1762 | Organ of St. Michael's Church, Hamburg |
| 409.0 | 486.4 | Paris, France | 1783 | Tuning fork of Pascal Taskin, court tuner |
| 415.5 | 494.1 | Dresden, Germany | 1722 | Organ of St. Sophia |
| 419.6 | 499.0 | Seville, Spain | 1785 & 1790 | Organ of Seville cathedral |
| 421.6 | 501.3 | Vienna, Austria | 1780 | supposed to be Mozart's pitch |
| 422.5 | 502.4 | London, England | 1751 | Handel's tuning fork |
| 423.5 | 503.6 | London, England | 1711 | an existing tuning fork of John Shore |
| 425.5 | 506.0 | Paris, France | 1829 | Pianoforte at the Paris Opera |
| 427.6 | 508.5 | Paris, France | 1823 | Opèra Comique |
| 430.8 | 536.4 | Paris, France | 1830 | Opera pitch as related by Drouet, the celebrated French flautist |
| 432.0 | 513.7 | Brussels, Belgium | 1876 | Proposed pitch standard |
| 435.0 | 517.3 | Paris, France | 1859 | The French 'Diapason Normal', set in law by the French government acting with the advice of Halvy, Meyerbeer, Auber, Ambroise Thomas and Rossini, although the mean of several forks set to this pitch lies slightly higher at a'=435.4 which is equivalent to c''=517.8 |
| 437.0 | 519.7 | Paris and Toulouse, France | 1836 & 1859 | The earlier was the pitch of the Italian Opera in Paris, the later that of the Conservatoire in Toulouse |
| 440.0 | 523.25 | Paris, France | 1829 | Orchestral pitch of the Paris Opera |
| 440.0 | 523.25 | Stuttgart, Germany | 1838 | Proposed pitch standard, Stuttgart congress (actually a'=440.2 when corrected to table temperature); also Scheibler's standard. |
| 441.0 | 524.4 | Rome, Italy | 1725 (anté) | Pitch calculated from a flute made by Biglioni and possibly brought from Rome by J. J. Quantz when he left Rome in 1725 |
| 444.0 | 528.0 | London, England | 1860 | Standard intended for the Society of Arts - (however a fork set to this standard by J.H. Griesbach has a measured pitch of a'=445.7, equivalent to a c''=530.1) |
| 444.5 | 528.6 | Madrid, Spain | 1858 | Theatre Royale, Madrid |
| 444.5 | 528.6 | London, England | c. 1810 | Pitch of a flute made by Henry Potter |
| 444.6 | 528.7 | London, England | 1877 | Organ in St. Paul's Cathedral |
| 444.8 | 528.9 | Turin, Weimer, Würtemberg | 1859 | Measurements made for the French Commission |
| 445.7 | 530.1 | London, England | 1860 | see 440.0 above |
| 446.0 | 530.4 | Paris, France; Dresden and Pesth, Germany, | 1859 | Pleyel's Piano taken by Delezenne and the pitches at the Opera houses of Dresden and Pesth |
| 447.11 | 531.7 | London, England | 1845 | Pitch calculated from a fork said to be at the pitch of the Royal Philharmonic Society |
| 448.0 | 532.8 | Hamburg, Germany | 1839 & 1840 | Opera |
| 448.0 | 532.8 | Paris, France | 1854 | Opéra Comique |
| 448.0 | 532.8 | Paris, France | 1858 | Grand Opèra |
| 448.0 | 532.8 | Liège, Belgium | 1859 | Conservatoire |
| 450.0 | 535.1 | London, England | 1850 to 1885 | An average of the pitches of London orchestras during this period |
| 450.5 | 536.7 | Lille, France | 1848 & 1854 | Lille Opera, measured during performance |
| 451.0 | 536.3 | Brussels, Belgium | 1879 | Pitch standard proposed for the Begian Army |
| 451.5 | 536.9 | St. Petersburg, Russia | 1858 | Opera |
| 451.7 | 537.2 | Milan, Italy | 1867 | La Scala Opera |
| 451.8 | 537.3 | Berlin, Germany | 1859 | Opera |
| 451.9 | 537.4 | London, England | 1878 | British Army Regulations |
| 452.0 | 537.5 | Lille, France | 1859 | Conservatoire |
| 452.0 | 537.5 | London, England | 1889 | Official Pitch at the 'Inventions' Exhibition in 1885 - the highest pitch used intentionally by English orchestras up to 1890 |
| 452.5 | 538.2 | London, England | 1846 to 1854 | Mean pitch of the Philharmonic Band under Sir Michael Costa. His Majesty's Rules and Regulations required Army Bands to play at the Philharmonic pitch, and a fork tuned to a'=452.5 in 1890 is preserved as the standard for the Military Training School at Kneller Hall |
| 453.3 | 539.0 | London, England | 1837 (anté) | Pitch calculated from a flute made by Rudall and Rose possibly as early as 1827 |
| 454.08 | 540.0 | London, England | 1874 | Old Philharmonic Pitch, instigated by Sir Charles Hall |
| 454.7 | 540.8 | London, England | 1874 | Fork representing the highest pitch adopted for Philharmonic concerts |
| 454.7 | 540.8 | London, England | 1879 | Steinway's English pitch; also Messrs. Bryceson's pitch |
| 455.3 | 541.5 | London, England | 1879 | Messrs. Erard's pitch |
| 455.5 | 541.7 | Brussels, Belgium | 1859 | Band of the Guides |
| 456.1 | 542.4 | London, England | 1857 | Fork set to the French Society of Pianoforte Makers |
| 457.2 | 543.7 | New York, USA | 1879 | Pitch used by Messrs. Steinway in America |
| 456.0 | 542.30 | Vienna, Austria | 1859 | Viennese 'high pitch' |
| 457.6 | 544.2 | Vienna, Austria | c. 1640 | Great Franciscan organ |
| 460.0 | 547.05 | Vienna, Austria | 1880 | Old Austrian Military Pitch |
| 461.0 | 548.3 | London, England | 1838 (anté) | Actual pitch of a flute said to be tuned to a'=453.3 |
| 474.1 | 563.8 | Durham, England | 1683 | Cathedral Organ by Bernhardt Smith |
| 474.1 | 563.8 | London, England | 1708 | Organ of the Chapel Royal by Bernhardt Smith |
| 480.8 | 571.8 | Hamburg, Germany | 1543 & 1879 | Organ at the church of St. Catherine |
| 484.1 | 575.7 | Lübeck, Germany | 1878 | Cathedral, small organ |
| 489.2 | 581.8 | Hamburg, Germany | 1688 & 1693 | Organ at the church of St. Jacob |
| 505.6 | 601.4 | Paris, France | 1636 | Mersenne's church pitch |
| 506.9 | 602.9 | Halberstadt, Germany | 1361 | Cathedral Organ |
| 567.6 | 675.2 | Paris, France | 1636 | Mersenne's chamber pitch |
| 570.7 | 678.7 | Germany | 1619 | Pitch called Kammerton (chamber pitch) by Praetorius; also called North German church pitch |
| pitches in use in England in the 1920s : taken from Notes on Concertina Pitch | |||||
| note | Normal (-20 cents to ISO) | New Philharmonic (-4 cents to ISO) | Stuttgart/ISO | Society of Arts (+22 cents to ISO) | Old Philharmonic (+54 cents to ISO) |
| A | 434.91 | 438.95 | 440 | 445.68 | 454.08 |
| A# | 460.77 | 465.05 | 466.16 | 472.18 | 481.09 |
| B | 488.17 | 492.70 | 493.88 | 500.25 | 509.69 |
| C | 517.20 | 522.00 | 523.25 | 530.00 | 540.00 |
| C# | 547.95 | 553.04 | 554.37 | 561.52 | 572.11 |
| D | 580.54 | 585.93 | 587.33 | 594.90 | 606.13 |
| D# | 615.06 | 620.77 | 622.25 | 630.28 | 642.17 |
| E | 651.63 | 657.68 | 659.26 | 667.76 | 680.36 |
| F | 690.38 | 696.79 | 698.46 | 707.47 | 720.81 |
| F# | 731.43 | 738.22 | 739.99 | 749.53 | 763.68 |
| G | 774.92 | 782.12 | 783.99 | 794.10 | 809.09 |
| G# | 821.00 | 828.62 | 830.61 | 841.32 | 857.20 |
| A | 869.82 | 877.90 | 880.00 | 891.35 | 908.17 |
| A# | 921.55 | 930.10 | 932.33 | 944.35 | 962.17 |
| B | 976.34 | 985.40 | 987.77 | 1000.51 | 1019.38 |
| C | 1034.40 | 1044.00 | 1046.50 | 1060.00 | 1080.00 |
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 multiples of the 'fundamental frequency', also called the 'first harmonic' or 'generator'. So for a string with a fundamental frequency of 440Hz., that is fixed at both ends, the harmonics are integral multiples of 440Hz.; i.e. 440Hz. (1 times 440Hz.), 880Hz. (2 times 440Hz.), 1320Hz. (3 times 440Hz.), 1760Hz. (4 times 440Hz.) and so on. The first 15 harmonics are given below, their frequencies set out in the second column. The third 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 440Hz. and 800Hz.). The nearest note in the chromatic scale on A is given in column 4 while the column headed % shows how close the normalized frequency is to the frequency of the nearest equal-tempered note diatonic to A.
| Harmonic | Frequency | Normalized | Note name | Closeness in % |
| 1 | 440Hz. | 440Hz | A | 100% |
| 2 | 880Hz. | 440Hz | A | 100% |
| 3 | 1320Hz. | 660Hz | E | 100% |
| 4 | 1760Hz. | 440Hz | A | 100% |
| 5 | 2200Hz. | 550Hz | C# | 99% |
| 6 | 2640Hz. | 660Hz | E | 100% |
| 7 | 3080Hz. | 770Hz | G | 98% |
| 8 | 3520Hz. | 440Hz | A | 100% |
| 9 | 3960Hz. | 495Hz | B | 100% |
| 10 | 4400Hz. | 550Hz | C# | 99% |
| 11 | 4840Hz. | 605Hz | D | 103% |
| 12 | 5280Hz. | 660Hz | E | 100% |
| 13 | 5720Hz. | 715Hz | F# | 97% |
| 14 | 6160Hz. | 770Hz | G | 98% |
| 15 | 6600Hz. | 825Hz | G# | 99% |
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.
Can 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? Such a sequence would progress as follows:
C G D A E B F# C# G# D# A# F C
If one applies the ratio (3:2) twelve times to 440 and normalizes the result by dividing by powers of 2 the result is sharp by a ratio called the Pythagorean or ditonic comma (524288:531441). This scheme unfortunately substitutes one problem for another. Here, the third and the octave are both too large.
To clarify the distinction between tuning and temperament we quote from Pierre Lewis's article Understanding Temperaments.
A tuning is laid out with nothing but pure intervals, leaving the Pythagorean or ditonic comma to fall as it must. A temperament involves deliberately mistuning some intervals to obtain a distribution of the comma that will lead to a more useful result in a given context. Solutions can be grouped into three main classes:
Temperaments are further classified as circulating or closed if they allow unlimited modulation, i.e. enharmonics are usable (equal temperament, most irregular temperaments), non-circulating or open otherwise (tunings, most regular temperaments). The choice of a particular solution depends on many factors such as
|
| Pythagorean intervals and their derivations (also called by modern theorists, the 3-limit system because all ratios are powers only of 2 and/or 3) | |||
| Interval | Ratio | Derivation | Cents* |
| Unison | (1:1) | Unison 1:1 | 0.00 |
| Minor Second | (256:243) | Octave - Major Seventh | 90.22 |
| Major Second | (9:8) | (3:2)^2 | 203.91 |
| Minor Third | (32:27) | Octave - Major Third | 294.13 |
| Major Third | (81:64) | (3:2)^4 | 407.82 |
| Fourth | (4:3) | Octave - Fifth | 498.04 |
| Augmented Fourth | (729:512) | (3:2)^6 | 611.73 |
| Fifth | (3:2) | (3:2)^1 | 701.96 |
| Minor Sixth | (128:81) | Octave - Major Third | 792.18 |
| Major Sixth | (27:16) | (3:2)^3 | 905.87 |
| Minor Seventh | (16:9) | Octave - Major Second | 996.09 |
| Major Seventh | (243:128) | (3:2)^5 | 1109.78 |
| Octave | (2:1) | Octave (2:1) | 1200.00 |
| *
The cent is a logarithmic measure of a musical interval invented by
Alexander Ellis. It first appears in the appendix he added to his
translation of Helmhotz's 'On the Sensations of Tone' [1875]. A cent is
the logarithmic division of the equitempered semitone into 100 equal
parts. It is therefore the 1200th root of 2, a ratio approximately
equal to (1:1.0005777895) The formula for calculating the 'cents-value' of any interval ratio is: cents = log10(ratio) * [1200 / log10(2)] or cents = 1200 × log2 (ratio) Intervals expressed in cents are added while those expressed in ratio form must be multiplied: for example, a perfect fourth plus a perfect fifth equals an octave. In ratio form, (4:3) times (3:2) = (12:6) = (2:1), in cents, 498.04 + 701.96 = 1200 | |||
A number of proposals were adopted 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.
Historical Temperaments are considered in more detail at the end of this lesson.
Sometimes called the mesotonic scale, the meantone (also written mean-tone) scale was particularly favoured by organists and explains why organ music from the period 1500 to the 19th 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 (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-18th 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 meantone 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) by modern tuning theorists, found favour amongst lutenists 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.
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.
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. Today we take it and its convenience for granted.
The equal-tempered system cannot be derived from rational relationships because the twelfth root of 2, like the square root of 2, is irrational.
The theoretical equal temperament frequencies for the A=440Hz. tuning pitch are:
| Tuning Pitch: A=440Hz. | ||||||||
| A | 27.50Hz. | 55.00Hz. | 110.00Hz. | 220.00Hz. | 440.00Hz. | 880.00Hz. | 1760.00Hz. | 3520.00Hz. |
| A# | 29.13Hz. | 58.27Hz. | 116.54Hz. | 233.08Hz. | 466.16Hz. | 932.32Hz. | 1864.65Hz. | 3729.31Hz. |
| B | 30.86Hz. | 61.73Hz. | 123.47Hz. | 246.94Hz. | 493.88Hz. | 987.76Hz. | 1975.53Hz. | 3951.06Hz. |
| C | 32.70Hz. | 65.40Hz. | 130.81Hz. | 261.62Hz. | 523.25Hz. | 1046.50Hz. | 2093.00Hz. | 4186.00Hz. |
| C# | 34.64Hz. | 69.29Hz. | 138.59Hz. | 277.18Hz. | 554.36Hz. | 1108.73Hz. | 2217.46Hz. | |
| D | 36.70Hz. | 73.41Hz. | 146.83Hz. | 293.66Hz. | 587.33Hz. | 1174.65Hz. | 2349.31Hz. | |
| D# | 38.89Hz. | 77.78Hz. | 155.56Hz. | 311.12Hz. | 622.25Hz. | 1244.50Hz. | 2489.01Hz. | |
| E | 41.20Hz. | 82.40Hz. | 164.81Hz. | 329.62Hz. | 659.25Hz. | 1318.51Hz. | 2637.02Hz. | |
| F | 43.65Hz. | 87.30Hz. | 174.61Hz. | 349.22Hz. | 698.45Hz. | 1396.91Hz. | 2793.82Hz. | |
| F# | 46.24Hz. | 92.49Hz. | 184.99Hz. | 369.99Hz. | 739.98Hz. | 1479.97Hz. | 2959.95Hz. | |
| G | 48.99Hz. | 97.99Hz. | 195.99Hz. | 391.99Hz. | 783.99Hz. | 1567.98Hz. | 3135.96Hz. | |
| G# | 51.91Hz. | 103.82Hz. | 207.65Hz. | 415.30Hz. | 830.60Hz. | 1661.21Hz. | 3322.43Hz. | |
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".
However, 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 only has 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.
An interesting 31-note equal temperament, 31EDO (Equal Divisions of the Octave), produces a scale that is much closer to just intonation than the 12-note equal temperament (12EDO) discussed in the previous section. The thirds, (2(8/31) = 1.1958733 and 2(10/31) = 1.2505655), are much nearer just intonation than those of 12-note equal temperament, although the perfect fourth and fifth are less good than 12EDO but still acceptable (2(18/31) = 1.4955179). The 31 notes can be mapped onto the 35 note names of the Western notational system.
Steps = 31 * log2 (f/f0) where f is the frequency in 31EDO
| Note | Interval above C | Steps | Cents |
| C | Perfect unison | 0 | 0 |
| C# | Augmented unison | 2 | 77 |
| C## | Doubly augmented unison | 4 | 155 |
| Dbb & B## | Diminished second | 1 | 39 |
| Db | Minor second | 3 | 116 |
| D | Major second | 5 | 194 |
| D# | Augmented second | 7 | 271 |
| D## & Fbb | Doubly augmented second | 9 | 348 |
| Ebb | Diminished third | 6 | 232 |
| Eb | Minor third | 8 | 310 |
| E | Major third | 10 | 387 |
| E# | Augmented third | 12 | 465 |
| Fb | Diminished fourth | 11 | 426 |
| F | Perfect fourth | 13 | 503 |
| F# | Augmented fourth | 15 | 581 |
| F## | Doubly augmented fourth | 17 | 658 |
| Gbb & E## | Doubly diminished fifth | 14 | 542 |
| Gb | Diminished fifth | 16 | 619 |
| G | Perfect fifth | 18 | 697 |
| G# | Augmented fifth | 20 | 774 |
| G## | Doubly augmented fifth | 22 | 852 |
| Abb | Diminished sixth | 19 | 735 |
| Ab | Minor sixth | 21 | 813 |
| A | Major sixth | 23 | 890 |
| A# | Augmented sixth | 25 | 968 |
| A## & Cbb | Doubly augmented sixth | 27 | 1045 |
| Bbb | Diminished seventh | 24 | 929 |
| Bb | Minor seventh | 26 | 1006 |
| B | Major seventh | 28 | 1084 |
| B# | Augmented seventh | 30 | 1161 |
| Cb | Diminished octave | 29 | 1123 |
| C | Perfect octave | 31 | 1200 |
It is undeniable, though, that just intonation should be explored in greater detail and I recommend readers wishing to do this go to The Just Intonation Network (check out the references listed below)
Below, for the reference of tuning enthusiasts, is Kyle Gann's Anatomy of An Octave, which contains all pitches that meet any one of the following six criteria:
All ratios between whole numbers 32 and lower
All ratios between 31-limit numbers up to 64 (31-limit meaning that the numbers contain no prime-number factors larger than 31)
Harmonics up to 128 (each whole number divided by the closest inferior power of 2)
All ratios between 11-limit numbers up to 128
All ratios between 5-limit numbers up to 1024
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.
| Interval Ratio | Cents equivalent | Interval Name (if any) |
| (1:1) | 0.000 | tonic |
| (32805:32768) | 1.954 | schisma ((3 to the 8th/2 to the 12th) x 5/8) |
| (126:125) | 13.795 | |
| (121:120) | 14.367 | |
| (100:99) | 17.399 | |
| (99:98) | 17.576 | |
| (81:80) | 21.506 | syntonic comma |
| (531441:524288) | 23.460 | Pythagorean comma (3 to the 12th/2 to the 19th) |
| (65:64) | 26.841 | 65th harmonic |
| (64:63) | 27.264 | |
| (63:62) | 27.700 | |
| (58:57) | 30.109 | |
| (57:56) | 30.642 | |
| (56:55) | 31.194 | Ptolemy's enharmonic |
| (55:54) | 31.768 | |
| (52:51) | 33.618 | |
| (51:50) | 34.284 | |
| (50:49) | 34.977 | |
| (49:48) | 35.698 | |
| (46:45) | 38.052 | inferior quarter-tone (Ptolemy) |
| (45:44) | 38.907 | |
| (128:125) | 41.059 | diminished second (16/15 x 24/25) |
| (525:512) | 43.408 | enharmonic diesis (Avicenna) |
| (40:39) | 43.831 | |
| (39:38) | 44.970 | superior quarter-tone (Eratosthenes) |
| (77:75) | 45.561 | |
| (36:35) | 48.770 | superior quarter-tone (Archytas) |
| (250:243) | 49.166 | |
| (35:34) | 50.184 | equal temperament (ET) 1/4-tone approximation |
| (34:33) | 51.682 | |
| (33:32) | 53.273 | 33rd harmonic |
| (32:31) | 54.964 | inferior quarter-tone (Didymus) |
| (125:121) | 56.305 | |
| (31:30) | 56.767 | superior quarter-tone (Didymus) |
| (30:29) | 58.692 | |
| (29:28) | 60.751 | |
| (57:55) | 61.836 | |
| (28:27) | 62.91 | inferior quarter-tone (Archytas) |
| (80:77) | 66.170 | |
| (27:26) | 65.337 | |
| (26:25) | 67.900 | 1/3-tone (Avicenna) |
| (51:49) | 69.261 | |
| (126:121) | 70.100 | |
| (25:24) | 70.672 | minor 5-limit semitone (half-step) |
| (24:23) | 73.681 | |
| (117:112) | 75.612 | |
| (23:22) | 76.956 | |
| (67:64) | 79.307 | 67th harmonic |
| (22:21) | 80.537 | hard semitone (1/2-step) (Ptolemy, Avicenna, Safiud) |
| (21:20) | 84.467 | |
| (81:77) | 87.676 | |
| (20:19) | 88.801 | |
| (256:243) | 90.225 | Pythagorean semitone (half-step) |
| (58:55) | 91.946 | |
| (135:128) | 92.179 | limma ascendant |
| (96:91) | 92.601 | |
| (19:18) | 93.603 | |
| (55:52) | 97.107 | |
| (128:121) | 97.364 | |
| (18:17) | 98.955 | equal temperament (ET) semitone (half-step), approximation |
| 2 to the 1/12th | 100.000 | equal temperament (ET) semitone (half-step), exact |
| (35:33) | 101.867 | |
| (52:49) | 102.880 | |
| (86:81) | 103.698 | |
| (17:16) | 104.955 | overtone semitone (half-step) |
| (33:31) | 108.237 | |
| (49:46) | 109.381 | |
| (16:15) | 111.731 | major 5-limit semitone (half-step) |
| (31:29) | 115.458 | |
| (77:72) | 116.234 | |
| (15:14) | 119.443 | Cowell just semitone (half-step) |
| (29:27) | 123.712 | |
| (14:13) | 128.298 | |
| (69:64) | 130.229 | 69th harmonic |
| (55:51) | 130.726 | |
| (27:25) | 133.238 | alternate Renaissance semitone (half-step) |
| (121:112) | 133.810 | |
| (13:12) | 138.573 | 3/4-tone (Avicenna) |
| (64:59) | 140.828 | |
| (38:35) | 142.373 | |
| (63:58) | 143.159 | |
| (88:81) | 143.498 | |
| (25:23) | 144.353 | |
| (62:57) | 145.568 | |
| (135:124) | 147.145 | |
| (49:45) | 147.433 | |
| (12:11) | 150.637 | undecimal "median" semitone (1/2-step) |
| (59:54) | 153.307 | |
| (35:32) | 155.140 | 35th harmonic |
| (23:21) | 157.493 | |
| (57:52) | 158.940 | |
| (34:31) | 159.920 | |
| (800:729) | 160.897 | |
| (56:51) | 161.916 | |
| (11:10) | 165.004 | |
| (54:49) | 168.219 | |
| (32:29) | 170.423 | |
| (21:19) | 173.268 | |
| (31:28) | 176.210 | |
| (567:512) | 176.646 | |
| (51:46) | 178.642 | |
| (71:64) | 179.697 | 71st harmonic |
| (10:9) | 182.404 | minor whole-tone |
| (49:40) | 186.340 | |
| (39:35) | 187.343 | |
| (29:26) | 189.050 | |
| (125:112) | 190.115 | |
| (48:43) | 190.437 | |
| (19:17) | 192.558 | |
| (160:143) | 194.468 | |
| (28:25) | 196.198 | |
| (121:108) | 196.771 | |
| (55:49) | 199.987 | |
| 2 to the 1/6th | 200.000 | equal-tempered whole-tone, exact |
| (64:57) | 200.532 | |
| (9:8) | 203.910 | major whole-tone |
| (62:55) | 207.404 | |
| (44:39) | 208.843 | |
| (35:31) | 210.104 | |
| (26:23) | 212.253 | |
| (112:99) | 213.598 | |
| (17:15) | 216.687 | |
| (25:22) | 221.309 | |
| (58:51) | 222.667 | |
| (256:225) | 222.463 | |
| (33:29) | 223.696 | |
| (729:640) | 225.416 | |
| (57:50) | 226.840 | |
| (73:64) | 227.789 | 73rd harmonic |
| (8:7) | 231.174 | septimal whole-tone |
| (63:55) | 235.104 | |
| (55:48) | 235.685 | |
| (39:34) | 237.527 | |
| (225:196) | 238.886 | |
| (31:27) | 239.171 | |
| (147:128) | 239.607 | |
| (169:147) | 241.449 | |
| (23:20) | 241.961 | |
| (2187:1900) | 243.547 | |
| (38:33) | 244.240 | |
| (144:125) | 244.969 | diminished third (6/5 x 24/25) |
| (121:105) | 245.541 | |
| (15:13) | 247.741 | |
| (52:45) | 250.313 | |
| (37:32) | 251.344 | 37th harmonic |
| (81:70) | 252.680 | |
| (125:108) | 253.076 | |
| (22:19) | 253.805 | |
| (51:44) | 255.602 | |
| (196:169) | 256.596 | consonant interval (Avicenna) |
| (29:25) | 256.950 | |
| (36:31) | 258.874 | |
| (93:80) | 260.679 | |
| (57:49) | 261.816 | |
| (64:55) | 262.368 | |
| (7:6) | 266.871 | septimal minor third |
| (90:77) | 270.080 | |
| (75:64) | 274.582 | augmented second (9/8 x 25/24) |
| (34:29) | 275.378 | |
| (88:75) | 276.736 | |
| (27:23) | 277.591 | |
| (20:17) | 281.358 | |
| (33:28) | 284.447 | |
| (46:39) | 285.802 | |
| (13:11) | 289.210 | |
| (58:49) | 291.925 | |
| (45:38) | 292.721 | |
| (32:27) | 294.135 | Pythagorean minor third |
| (19:16) | 297.513 | overtone minor third |
| 2 to the 1/4th | 300.000 | equal-tempered minor third, exact |
| (25:21) | 301.847 | |
| (31:26) | 304.508 | |
| (105:88) | 305.777 | |
| (55:46) | 309.368 | |
| (6:5) | 315.641 | 5-limit minor third |
| (77:64) | 320.144 | 77th harmonic |
| (35:29) | 325.562 | |
| (29:24) | 327.622 | |
| (75:62) | 329.550 | |
| (98:81) | 329.832 | |
| (121:100) | 330.008 | |
| (23:19) | 330.761 | |
| (63:52) | 332.208 | |
| (40:33) | 333.041 | |
| (17:14) | 336.130 | |
| (243:200) | 337.148 | |
| (62:51) | 338.125 | |
| (28:23) | 340.552 | |
| (39:32) | 342.483 | 39th harmonic |
| (128:105) | 342.905 | |
| (8000:6561) | 343.304 | |
| (11:9) | 347.408 | undecimal "median" third |
| (60:49) | 350.617 | |
| (49:40) | 351.351 | |
| (38:31) | 352.477 | |
| (27:22) | 354.547 | |
| (16:13) | 359.472 | |
| (79:64) | 364.537 | 79th harmonic |
| (100:81) | 364.807 | |
| (121:98) | 364.984 | |
| (21:17) | 365.826 | |
| (99:80) | 368.914 | |
| (26:21) | 369.747 | |
| (57:46) | 371.194 | |
| (31:25) | 372.408 | |
| (36:29) | 374.333 | |
| (56:45) | 378.602 | |
| (96:77) | 381.811 | |
| (8192:6561) | 384.360 | Pythagorean "schismatic" third |
| (5:4) | 386.314 | 5-limit major third |
| (64:51) | 393.090 | |
| (49:39) | 395.183 | |
| (44:35) | 396.192 | |
| (39:31) | 397.447 | |
| (34:27) | 399.090 | |
| 2 to the 1/3rd | 400.000 | equal-tempered major third, exact |
| (63:50) | 400.108 | |
| (121:96) | 400.681 | |
| (29:23) | 401.303 | |
| (125:99) | 403.713 | |
| (24:19) | 404.442 | |
| (512:405) | 405.866 | |
| (62:49) | 407.384 | |
| (81:64) | 407.820 | Pythagorean major third |
| (19:15) | 409.244 | |
| (33:26) | 412.745 | |
| (80:63) | 413.578 | |
| (14:11) | 417.508 | |
| (51:40) | 420.612 | |
| (125:98) | 421.289 | |
| (23:18) | 424.364 | |
| (32:25) | 427.373 | diminished fourth |
| (41:32) | 429.062 | 41st harmonic |
| (50:39) | 430.160 | |
| (77:60) | 431.875 | |
| (9:7) | 435.084 | septimal major third |
| (58:45) | 439.353 | |
| (49:38) | 440.154 | |
| (40:31) | 441.278 | |
| (31:24) | 443.081 | |
| (1323:1024) | 443.517 | |
| (128:99) | 444.772 | |
| (22:17) | 446.363 | |
| (57:44) | 448.150 | |
| (162:125) | 448.879 | |
| (35:27) | 449.275 | |
| (83:64) | 450.047 | 83rd harmonic |
| (100:77) | 452.484 | |
| (13:10) | 454.214 | |
| (125:96) | 456.986 | augmented third (5/4 x 25/24) |
| (30:23) | 459.994 | |
| (64:49) | 462.348 | |
| (98:75) | 463.069 | |
| (17:13) | 464.428 | |
| (72:55) | 466.278 | |
| (55:42) | 466.867 | |
| (38:29) | 467.936 | |
| (21:16) | 470.781 | septimal fourth |
| (46:35) | 473.152 | |
| (25:19) | 475.114 | |
| (320:243) | 476.539 | |
| (29:22) | 478.259 | |
| (675:512) | 478.492 | |
| (33:25) | 480.646 | |
| (45:34) | 485.286 | |
| (85:64) | 491.269 | 85th harmonic |
| (4:3) | 498.045 | perfect fourth |
| 2 to the 5/12ths | 500.000 | equal-tempered perfect fourth, exact |
| (75:56) | 505.757 | |
| (51:38) | 509.415 | |
| (43:32) | 511.518 | 43rd harmonic |
| (121:90) | 512.412 | |
| (39:29) | 512.905 | |
| (35:26) | 514.612 | |
| (66:49) | 515.621 | |
| (31:23) | 516.761 | |
| (27:20) | 519.551 | |
| (23:17) | 523.319 | |
| (42:31) | 525.745 | |
| (19:14) | 528.687 | |
| (110:81) | 529.812 | |
| (87:64) | 531.532 | 87th harmonic |
| (34:25) | 532.328 | |
| (49:36) | 533.761 | |
| (15:11) | 536.951 | |
| (512:375) | 539.104 | |
| (26:19) | 543.015 | |
| (63:46) | 544.462 | |
| (48:35) | 546.835 | |
| (1000:729) | 547.211 | |
| (11:8) | 551.318 | undecimal tritone (11th harmonic) |
| (62:45) | 554.812 | |
| (40:29) | 556.737 | |
| (29:21) | 558.796 | |
| (112:81) | 561.006 | |
| (18:13) | 563.382 | |
| (25:18) | 568.717 | augmented fourth (4/3 x 25/24) |
| (89:64) | 570.880 | 89th harmonic |
| (32:23) | 571.726 | |
| (39:28) | 573.657 | |
| (46:33) | 575.022 | |
| (88:63) | 578.582 | |
| (7:5) | 582.512 | septimal tritone |
| (108:77) | 585.721 | |
| (1024:729) | 588.270 | low Pythagorean tritone |
| (45:32) | 590.224 | high 5-limit tritone |
| (38:27) | 591.648 | |
| (31:22) | 593.718 | |
| (55:39) | 595.170 | |
| (24:17) | 597.000 | |
| Square root of 2 | 600.000 | equal-tempered tritone, exact |
| (99:70) | 600.088 | |
| (17:12) | 603.000 | |
| (44:31) | 606.304 | |
| (125:88) | 607.623 | |
| (27:19) | 608.352 | |
| (91:64) | 609.354 | 91st harmonic |
| (64:45) | 609.776 | low 5-limit tritone |
| (729:512) | 611.730 | high Pythagorean tritone |
| (57:40) | 613.154 | |
| (77:54) | 614.279 | |
| (10:7) | 617.488 | septimal tritone |
| (63:44) | 621.418 | |
| (33:23) | 624.999 | |
| (56:39) | 626.343 | |
| (23:16) | 628.274 | 23rd harmonic |
| (36:25) | 631.283 | diminished fifth (3/2 x 24/25) |
| (121:84) | 631.855 | |
| (49:34) | 632.719 | |
| (13:9) | 636.618 | |
| (81:56) | 638.994 | |
| (55:38) | 640.141 | |
| (42:29) | 641.204 | |
| (29:20) | 643.263 | |
| (45:31) | 645.211 | |
| (93:64) | 646.991 | 93rd harmonic |
| (16:11) | 648.682 | |
| (51:35) | 651.794 | |
| (729:500) | 652.789 | |
| (35:24) | 653.185 | |
| (19:13) | 656.985 | |
| (375:256) | 660.896 | |
| (22:15) | 663.049 | |
| (47:32) | 665.507 | 47th harmonic |
| (72:49) | 666.258 | |
| (25:17) | 667.672 | |
| (81:55) | 670.188 | |
| (28:19) | 671.313 | |
| (31:21) | 674.255 | |
| (189:128) | 674.691 | |
| (34:23) | 676.681 | |
| (40:27) | 680.449 | dissonant "wolf" 5-limit fifth |
| (46:31) | 683.263 | |
| (95:64) | 683.827 | 95th harmonic |
| (49:33) | 684.403 | |
| (52:35) | 685.412 | |
| (58:39) | 687.095 | |
| (125:84) | 688.160 | |
| (112:75) | 694.243 | |
| (121:81) | 694.816 | |
| 2 to the 7/12ths | 700.000 | equal-tempered perfect fifth, exact |
| (3:2) | 701.955 | perfect fifth |
| (121:80) | 716.322 | |
| (50:33) | 719.380 | |
| (97:64) | 719.895 | 97th harmonic |
| (1024:675) | 721.508 | |
| (44:29) | 721.766 | |
| (243:160) | 723.461 | |
| (38:25) | 724.886 | |
| (35:23) | 726.865 | |
| (32:21) | 729.219 | |
| (29:19) | 732.064 | |
| (84:55) | 733.149 | |
| (55:36) | 733.748 | |
| (26:17) | 735.572 | |
| (75:49) | 736.931 | |
| (49:32) | 737.652 | 49th harmonic |
| (23:15) | 740.006 | |
| (192:125) | 743.014 | diminished sixth (8/5 x 24/25) |
| (20:13) | 745.786 | |
| (77:50) | 747.516 | |
| (54:35) | 750.752 | |
| (125:81) | 751.121 | |
| (17:11) | 753.637 | |
| (99:64) | 755.228 | 99th harmonic |
| (48:31) | 756.946 | |
| (31:20) | 758.722 | |
| (45:29) | 760.674 | |
| (14:9) | 764.916 | septimal minor sixth |
| (120:77) | 768.125 | |
| (39:25) | 769.855 | |
| (25:16) | 772.627 | augmented fifth |
| (36:23) | 775.636 | |
| (11:7) | 782.492 | undecimal minor sixth |
| (63:40) | 786.422 | |
| (52:33) | 787.283 | |
| (101:64) | 789.854 | 101st harmonic |
| (30:19) | 790.756 | |
| (128:81) | 792.180 | Pythagorean minor sixth |
| (49:31) | 792.644 | |
| (405:256) | 794.134 | |
| (19:12) | 795.558 | |
| (46:29) | 798.726 | |
| (100:63) | 799.892 | |
| 2 to the 2/3rds | 800.000 | equal-tempered minor sixth, exact |
| (27:17) | 800.910 | |
| (62:39) | 802.553 | |
| (35:22) | 803.822 | |
| (51:32) | 806.910 | 51st harmonic |
| (8:5) | 813.686 | 5-limit minor sixth |
| (6561:4096) | 815.640 | Pythagorean "schismatic" sixth |
| (77:48) | 818.189 | |
| (45:28) | 821.427 | |
| (103:64) | 823.801 | 103rd harmonic |
| (29:18) | 825.667 | |
| (50:31) | 827.600 | |
| (121:75) | 828.053 | |
| (21:13) | 830.253 | |
| (55:34) | 832.706 | |
| (34:21) | 834.175 | |
| (81:50) | 835.193 | |
| (125:77) | 838.797 | |
| (13:8) | 840.528 | overtone sixth |
| (57:35) | 844.328 | |
| (44:27) | 845.483 | |
| (31:19) | 847.523 | |
| (80:49) | 848.662 | |
| (49:30) | 849.413 | |
| (18:11) | 852.592 | undecimal "median" sixth |
| (105:64) | 857.095 | 105th harmonic |
| (64:39) | 857.517 | |
| (23:14) | 859.448 | |
| (51:31) | 861.905 | |
| (400:243) | 862.852 | |
| (28:17) | 863.870 | |
| (33:20) | 866.959 | |
| (38:23) | 869.239 | |
| (81:49) | 870.168 | |
| (48:29) | 872.409 | |
| (53:32) | 873.505 | 53rd harmonic |
| (58:35) | 874.438 | |
| (63:38) | 875.223 | |
| (128:77) | 879.856 | |
| (107:64) | 889.760 | 107th harmonic |
| (5:3) | 884.359 | 5-limit major sixth |
| (57:34) | 894.513 | |
| (52:31) | 895.524 | |
| (42:25) | 898.153 | |
| (121:72) | 898.726 | |
| 2 to the 3/4ths | 900.000 | equal-tempered major sixth, exact |
| (32:19) | 902.487 | |
| (27:16) | 905.865 | Pythagorean major sixth |
| (49:29) | 908.107 | |
| (22:13) | 910.790 | |
| (39:23) | 914.208 | |
| (56:33) | 915.553 | |
| (17:10) | 918.641 | |
| (109:64) | 921.821 | 109th harmonic |
| (46:27) | 922.442 | |
| (75:44) | 923.264 | |
| (29:17) | 924.622 | |
| (128:75) | 925.418 | diminished seventh (16/9 x 24/25) |
| (77:45) | 929.920 | |
| (12:7) | 933.129 | septimal major sixth |
| (55:32) | 937.632 | 55th harmonic |
| (31:18) | 941.126 | |
| (441:256) | 941.562 | |
| (50:29) | 943.084 | |
| (19:11) | 946.195 | |
| (216:125) | 946.924 | |
| (121:70) | 947.496 | |
| (45:26) | 949.730 | |
| (26:15) | 952.259 | |
| (111:64) | 953.299 | 111th harmonic |
| (125:72) | 955.031 | augmented sixth (5/3 x 25/24) |
| (33:19) | 955.760 | |
| (40:23) | 958.039 | |
| (54:31) | 960.864 | |
| (96:55) | 964.323 | |
| (110:63) | 964.896 | |
| (7:4) | 968.826 | septimal minor seventh |
| (58:33) | 976.304 | |
| (225:128) | 976.537 | |
| (51:29) | 977.368 | |
| (44:25) | 978.725 | |
| (30:17) | 983.313 | |
| (113:64) | 984.215 | 113th harmonic |
| (99:56) | 986.402 | |
| (23:13) | 987.747 | |
| (62:35) | 989.896 | |
| (39:22) | 991.165 | |
| (55:31) | 992.631 | |
| (16:9) | 996.090 | Pythagorean small minor seventh |
| (57:32) | 999.468 | 57th harmonic |
| 2 to the 5/6ths | 1000.000 | equal-tempered minor seventh |
| (98:55) | 1000.020 | |
| (25:14) | 1003.802 | |
| (34:19) | 1007.442 | |
| (52:29) | 1010.986 | |
| (88:49) | 1013.666 | |
| (115:64) | 1014.588 | 115th harmonic |
| (9:5) | 1017.596 | 5-limit large minor seventh |
| (56:31) | 1023.790 | |
| (38:21) | 1026.732 | |
| (29:16) | 1029.577 | 29th harmonic |
| (49:27) | 1031.823 | |
| (20:11) | 1034.996 | |
| (51:28) | 1038.121 | |
| (729:400) | 1039.103 | |
| (31:17) | 1040.080 | |
| (42:23) | 1042.507 | |
| (117:64) | 1044.438 | 117th harmonic |
| (64:35) | 1044.860 | |
| (4000:2187) | 1045.266 | |
| (11:6) | 1049.363 | undecimal "median" seventh |
| (90:49) | 1052.572 | |
| (57:31) | 1054.432 | |
| (46:25) | 1055.684 | |
| (81:44) | 1056.502 | |
| (35:19) | 1057.627 | |
| (59:32) | 1059.172 | 59th harmonic |
| (24:13) | 1061.427 | |
| (50:27) | 1066.772 | |
| (63:34) | 1067.780 | |
| (13:7) | 1071.702 | |
| (119:64) | 1073.781 | 119th harmonic |
| (54:29) | 1076.326 | |
| (28:15) | 1080.557 | |
| (58:31) | 1084.542 | |
| (15:8) | 1088.269 | 5-limit major seventh |
| (62:33) | 1091.763 | |
| (32:17) | 1095.045 | |
| (49:26) | 1097.163 | |
| (66:35) | 1098.133 | |
| 2 to the 11/12ths | 1100.000 | equal-tempered major seventh, exact |
| (17:9) | 1101.045 | |
| (121:64) | 1102.636 | 121st harmonic |
| (125:66) | 1105.668 | |
| (36:19) | 1106.397 | |
| (256:135) | 1107.821 | |
| (55:29) | 1108.094 | |
| (243:128) | 1109.775 | Pythagorean major seventh |
| (19:10) | 1111.199 | |
| (40:21) | 1115.533 | |
| (61:32) | 1116.885 | 61st harmonic |
| (21:11) | 1119.463 | |
| (44:23) | 1123.084 | |
| (23:12) | 1126.319 | |
| (48:25) | 1129.338 | |
| (121:63) | 1129.900 | |
| (123:64) | 1131.017 | 123rd harmonic |
| (25:13) | 1132.100 | |
| (77:40) | 1133.830 | |
| (52:27) | 1134.703 | |
| (27:14) | 1137.039 | septimal major seventh |
| (56:29) | 1139.249 | |
| (29:15) | 1141.308 | |
| (60:31) | 1143.233 | |
| (31:16) | 1145.036 | 31st harmonic |
| (64:33) | 1146.727 | |
| (33:17) | 1148.318 | |
| (243:125) | 1150.834 | |
| (35:19) | 1151.230 | |
| (39:20) | 1156.169 | |
| (125:64) | 1158.941 | augmented seventh (15/8 x 25/24) |
| (88:45) | 1161.094 | |
| (45:23) | 1161.991 | |
| (96:49) | 1164.303 | |
| (49:25) | 1165.066 | |
| (51:26) | 1166.424 | |
| (108:55) | 1168.233 | |
| (55:28) | 1168.847 | |
| (57:29) | 1169.891 | |
| (63:32) | 1172.736 | 63rd harmonic |
| (160:81) | 1178.494 | |
| (99:50) | 1182.601 | |
| (125:63) | 1186.205 | |
| (127:64) | 1186.422 | 127th harmonic |
| (2:1) | 1200.000 | octave |
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.
| # semitones |
Interval class |
Generic interval |
Common diatonic name |
Comparable just interval |
Comparison of interval width in cents (to nearest integer) | |||
| equal temperament |
just intonation |
quarter-comma meantone |
Pythagorean tuning |
|||||
| 0 | 0 | 0 | perfect unison | (1:1) | 0 | 0 | 0 | 0 |
| 1 | 1 | 1 | minor second | (16:15) | 100 | 112 | 117 | 90 |
| 2 | 2 | 1 | major second | (9:8) | 200 | 204 | 193 | 204 |
| 3 | 3 | 2 | minor third | (6:5) | 300 | 316 | 310 | 294 |
| 4 | 4 | 2 | major third | (5:4) | 400 | 386 | 386 | 408 |
| 5 | 5 | 3 | perfect fourth | (4:3) | 500 | 498 | 503 | 498 |
| 6 | 6 | 3 4 |
augmented fourth diminished fifth |
(45:32) (64:45) |
600 600 |
590 610 |
579 621 |
612 |
| 7 | 5 | 4 | perfect fifth | (3:2) | 700 | 702 | 697 wolf fifth 737 |
702 |
| 8 | 4 | 5 | minor sixth | (8:5) | 800 | 814 | 814 | 792 |
| 9 | 3 | 5 | major sixth | (5:3) | 900 | 884 | 889 | 906 |
| 10 | 2 | 6 | minor seventh | (16:9) | 1000 | 996 | 1007 | 996 |
| 11 | 1 | 6 | major seventh | (15:8) | 1100 | 1088 | 1083 | 1110 |
| 12 | 0 | 0 | perfect octave | (2:1) | 1200 | 1200 | 1200 | 1200 |
Reference: 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. PYTHAGOREAN - see table below 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. van ZWOLLE - see table below Arnout van Zwolle (1400-1466) modified the Pythagorean scale by placing the comma
between B and 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 - see table below The
best known of the old scales (see more below), this scale emphasizes
pure thirds by making the fifths narrow. It was certainly in use by the
end of the 15th century, if not earlier. It has the greatest number of
pure triads of any of the scales on this disk. 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. SILBERMANN I, II - see table below 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 are pure. RAMEAU - see table below 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 III, IV, V, VI - see table below Organ
builder and mathematician Andreas Werkmeister (1645-1706) devoted much
of his life to the study of temperament and suggested many. different
scales. The best known of these are included on this disk. 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 "Well-tempered
Klavier" pieces for one of these temperaments. KIRNBERGER II, lII - see table below 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. ITALIAN 18th Century One
of the many variations commonly in use in the 18th 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. EQUAL TEMPERED - see table above This
scale is so common in the 20th 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. Equal temperament was probably known in
the 1700s or earlier, but was not considered a satisfactory scale due
to the impurity of all intervals. In the late 18th and 19th centuries,
composers increasingly explored modulation to many different 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. Table from Alternate Temperaments: Theory and Philosophy by Terry Blackburn and calculated from a = 440 Hz.
Information on Temperaments ::
Books:
Online
All
musical instrument have acoustical properties determined by their form
and material of construction. Musical instruments require intervention
from an actuator (or performer) to provide the energy that will
initiate the production of sound. Sound is a form of mechanical energy
that requires a medium through which to propagate or travel. A sound
travels from a source, through a medium to a detector. For us the
detector is the human ear. If the sound is to be considered musical
with a specific pitch or tone quality, rather than just 'noise', the
mechanical energy has to radiate from the instrument as regular
disturbances, what we call 'periodic' vibrations. The vibrator
producing fluctuations, oscillations, pulsations or undulations (these
terms are all equivalent) will be different on different instruments
and the initiation and resonance may arise from two separate processes.
We say that the sound producing system has two parts - the initiator and the resonator. Examples of initiators: Examples of resonators: The
character of the sound each instrument produces is, therefore, partly
due to vibrations associated with the process of initiation and partly
due to the characteristic vibrations that are generated by the
resonator, initially sustained but usually decaying once energy is no
longer supplied to the system. If, on a stringed instrument, the bow is
continuously drawn across a string, the instrument is described as
being in continuous-control mode; i.e. onset - sustain. If, however, on
the same instrument, a string is plucked with a finger, the instrument
is then said to be in envelope-based mode; i.e. onset - sustain -
decay. In general and when the process of initiation is mechanical and
occurs over a relatively short time, short relative to the persistance
of the resonance response that follows, a note has a clear starting or
'onset' sound (arising from the initiator) which is distinguishable
from the sound that follows (that arising from the resonator). For
example, the 'tonguing' sound that begins notes produced on
wind-instruments is distinguishable from the sustained resonance
associated with the remainder of the note. The percussive initiation of
a note produced on a piano, the sound of the hammer striking the
string, is distinguishable from the sound that rings on should you keep
the piano key depressed for any length of time. The mechanical
processes involved in sound production on musical instruments include
plucking or bowing (on violin, viola, cello string bass, harpsichord),
blowing (on clarinet, oboe, trumpet, trombone, recorder, voice) or
striking (on drums, piano, clavichord, xylophone). It has been found
that if the onset is removed from recordings of sounding musical
instruments it become much more difficult to distinguish one from
another. External factors, too, can influence 'timbre' - for example,
if an instrument moves in a room relative to the listener. To
summarise, timbre is the spectrotemporal pattern of a generated sound
indicating the way the energy in the system is distributed between
different harmonics or frequency components and the way that
distribution is changing over time. The instruments of the orchestra, viewed as mechanical systems, can be classified in the following manner:
Reference:
Classification of Common Musical Instruments ::
References:
Kirnberger II c 262.37 c# 276.40 d 295.16 d# 310.95 e 327.96 f 349.82 f# 368.95 g 393.55 g# 414.60 a 440.00 a# 466.43 b 491.93 c 524.73
Kirnberger III 263.18 277.26 294.25 311.92 328.98 350.91 370.10 393.55 415.89 440.00 467.88 493.47 526.36
Werckmeister III 263.40 277.50 294.33 312.18 330.00 351.21 369.99 393.77 416.24 440.00 468.27 495.00 526.81
Werckmeister IV 263.11 275.93 294.66 311.83 330.00 350.81 369.58 392.88 413.90 440.00 469.86 492.77 526.21
Werckmeister V 261.63 276.56 294.33 311.13 328.88 350.02 369.99 392.44 413.43 440.00 466.69 493.33 523.25
Werckmeister VI 262.77 276.83 292.77 312.03 330.00 350.36 370.53 393.39 415.24 440.00 468.05 495.00 525.54
Van Biezen 262.51 277.18 294.00 311.83 329.26 350.81 369.58 392.88 415.77 440.00 467.75 492.76 525.03
Bach (Klais) 262.76 276.87 294.30 311.46 328.70 350.37 369.18 393.70 415.30 440.00 467.18 492.26 525.53
Just (Barbour) 264.00 275.00 297.00 316.80 330.00 352.00 371.25 396.00 412.50 440.00 475.20 495.00 528.00
Pythagorean c 260.74 c# 278.44 d 293.33 d# 309.03 e 330.00 f 347.65 f# 371.25 g 391.11 g# 417.66 a 440.00 a# 463.54 b 495.00 c 521.48
van Zwolle 260.74 274.69 293.33 309.03 330.00 347.65 366.25 391.11 417.66 440.00 463.54 495.00 521.48
Meantone (-1/4) 263.18 275.00 294.25 314.84 328.98 352.00 367.81 393.55 411.22 440.00 470.79 491.93 526.36
Silbermann(-1/6) 262.37 276.14 293.94 312.89 329.32 350.55 368.95 392.73 413.35 440.00 486.36 492.25 524.73
Salinas (-1/3) 264.00 273.86 294.55 316.80 328.64 353.46 366.67 394.36 409.10 440.00 473.24 490.92 528.00
Zarlino (-1/7) 263.53 274.51 294.38 315.68 328.83 358.63 367.32 393.90 410.31 440.00 471.84 491.50 527.06
Rossi (-1/5) 262.69 275.68 294.06 313.67 329.18 351.13 368.49 393.06 412.50 440.00 469.33 492.55 525.38
Rossi (-1/9) 262.91 275.38 294.14 314.19 329.09 351.51 368.19 393.28 411.93 440.00 469.98 492.27 525.82
Rameau(syntonic) 263.18 276.71 294.25 310.31 328.98 352.00 368.95 393.55 415.07 440.00 467.39 491.93 526.36
(Wind Instruments - Vibrating Air)
(Stringed Instruments - Vibrating Strings)
(Vibrating Instruments)
(Vibrating Membrane Instruments)
(Electronically Created Sounds) Free Aerophones Free Aerophones
(Moving object vibrates air to create sound) Bull-roarer
Buzzer Free Reed Aerophones
(Vibrating reeds without resonators) Accordion
Harmonica
Harmonium
Mouth Organ
Sheng Flutes (Flue Voiced)
(Air column split by lip of of the instrument) Open Tube End Blown Single Flutes Kaval End Blown Multiple Flutes Panpipes
Antara Whistle Blown Boatswains Whistle
Flageolet
Recorder Side Blown Flute (transverse) Closed Tube Ocarina Keyboard Organ Reedpipes
(Vibration of reeds) Double Pipes
Triple Pipes
Hornpipes
Bladder pipe Crumhorn Bagpipes Musette Shawm Oboe
Cor Anglais or English Horn Rackett Sordone
Sordun Bassanelli
Bassoon Curtal
Bassoon
Contrabassoon
High-pitched Bassoon
Single-reed Bassoon Sarrusophone
Single-reed Reedpipes Clarinet
Saxophone Free-reed Aerophones Lip Vibrated Aerophones
(Vibration of Lips) Horn
Fingerhole Horns Cornett
Serpent Trumpet Conical Bore Alphorn
Bugle
Cornet
Euphonium
Flugelhorn
Tuba Cylindrical Bore English Baritone Horn
Sousaphone
Trombone Zithers Ground Zithers
Musical Bows Aeolian Bows
Pluriare Stick Zithers Vinã
Bladder and String Raft Zither
Trough Zither
Frame Zither
Tube Zither
Board Zither
Long Zither Individually Bridged Long Zither
Fretted Long Zither Box Zither Monochord
String Drum
Trumpet Marine
Psaltery
Aeolian Harp
Fretted Zither
Bell Harp Autoharp
Dulcimer
Zither Keyboard Chordophones Keyboard Experimental Keyboard
Transposing Keyboard Chekker
Dulce Melos
Clavichord Cimbal d'amour Harpsichord Clavicytherium
Gut-strung Harpsichords &
Enharmonic Harpsichords
Spinet
Virginal
Claviorganum
Piano-organ Bowed Keyboard Instruments Pianoforte Harpsichord Piano
Tangent Piano
Sustaining Piano
Pedalboard Piano
Enharmonic Piano
Player Piano
Fortepiano
Piano Lyres Lyre
Crwth Harps Ground Bows
Harps Harp Harp Zither
Harp Lute Lutes Long-necked Lute
Short-necked Lute
Archlute Theorbo
Theorbo Lute
Chitarrone
Angelica Mandora
Mandolin
Sitar
Cittern Archcittern Bandora Orpharion
Penorcon
Polyphant
Balalaika
Charango
Colascione Guitar Gittern
Vihuela
Spanish Guitar
Bass Guitar
Bandurria
Ukulele Organistrum
(hurdy-gurdy) Bowed Chordophones Bow
Fiddle
Rebec Kit
Folk Rebec
Polnische Geige Lira da Braccio Lira da Gamba Viola da Gamba
Violone
Baryton
Viola Violino d'amore Violin Viola
Tenor Violin
Violoncello or 'Cello
Double Bass Concussion Idiophones Clappers Claves
Slapstick Castanets
Cymbals Percussion Idiophones Stamped Idiophones
Percussion Beams Marimba
Bell Lyre
Celesta
Gender
Glockenspiel
Orchestra Bells
Saron
Vibraphone Percussion Disks Gong Percussion Sticks Triangles Xylophone Crystallophones
Lithophones
Metallophones Percussion Tubes Stamping Tubes
Slit Drums
Tubular Bells & Chimes
Angklung Percussion Vessels Percussion Gourds & Pots
The Echeion
Steel Drums
Bells Temple Blocks
Wood Block Shaken Idiophones
(Rattles) Vessel Rattles Pellet Bells Gourd Rattles Maracas Basketry Rattles
Hollow Ring Rattles Suspension Rattles Stick Rattles
Sistrum
Strung Rattles Frame, Pendant, and Sliding Rattles Sistro Scraped Idiophones Scrapers Güiro Cog Rattles Cog Rattle
Ratchet
Washboard Split Idiophone Plucked Idiophones Jew's Harp Thumb Piano Music Box
Sansa
Mbira Friction Idiophones Friction Sticks
Friction-bar Pianos
Friction Vessels Musical Glasses
Glass Armonica
Musical Saw Predrum Membranophones Ground drums
Pot drums Tubular Drums Frame Drums
Shallow Drums
Cylinder Drums
Hourglass Drums
Conical Drums
Goblet Drums
Barrel Drums
Rattle Drums
Water Drums
Drum Kit / Drum Set
Talking Drums
Drum Chimes Bass Drum
Bongos
Conga
Snare Drum
Tenor Drum
Timbales
Tom-tom Kettledrums Timpani Friction Drums Mirliton Kazoo
Synthesizers Moog
Theremin
Ondes Martenot
Trautonium