Pitch :: Key word: pitch 1 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 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." 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. It is, I believe, a fact that most people tend to hear sharp in the upper octaves, and of course violinists are known to play sharp in the upper registers, where the fingering becomes so close that it becomes almost impossible to play rapid passages without this tendency. These two factors together have led to many professionals' having the xylophone tuned 'brighter', or very slightly sharp, in the upper octaves. Even so, the xylophonist, with his instrument of fixed pitch, will fequently find himself flat compared to the violins in any very high unison passages. In these instances there is nothing that can be done, apart from trying to make the violins section aware of their tendency to be too high - no easy matter! 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 earliest conventions of Western music held that "Music on earth was a reflection of the greater 'music of the spheres', a harmony created by relative distances and rates of motions of the planets - a harmony that was constantly present, if only people were sufficiently sensitive to hear it" (Yudkin, Jeremy, Music in Medieval Europe, 1989). If we as individuals can identify with the concepts presented in this statement, and we accept that we as human beings are multifaceted creatures who must live in harmony with our environment in order to maintain our health and reach our full potential, we can begin to see into the secrets of music and its impact on our health. Such a philosophy would indicate that music should be based in nature and the cosmic rhythms of the universe, if it is to be beneficial to humanity. From this standpoint one can extrapolate that the standard used to determine concert pitch should have an organic foundation as well. One theory of setting the standard for a concert A at 432 Hz attempts to utilize the argument that 432 Hz is based in nature. This theory would indicate by deduction that 440 Hz would then lend itself to generating an unhealthy effect in the environment. To be sure, this debate becomes a very heady and esoteric conversation. Some of the more radical proponents of 432 Hz as the true basis for concert pitch would indicate that everything in nature has a basis in 432 vibrations per second, most of which has not been verified and/or is not verifiable. There is one realm of nature that does support the idea that 432 Hz has an organic basis - that is the movement of the sun. Without going into a lengthy technical monologue we can ascertain that the note C of a scale based on 432 Hz can be reduced to a vibration rate of one vibration per second. We can further establish that the true origination for the measure of one second is based on the movement of the sun. There are further, more in depth, studies based on planetary motion and the harmonic overtones and undertones which do lend further support to the "organic" basis of 432 Hz as a solid foundation for musical structure. The tuning of a scale based on 440 Hz does not lend itself to a reduction on any basis which corresponds to a cosmic movement or rhythm. The difference between 440 Hz and 432 Hz is only 8 vibrations per second, but it is a perceptible difference in the human experience. 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. The Schiller Institute has become known internationally for its initiative to lower the international standard musical pitch to C = 256 Hz (A = 432 Hz.), in order to preserve the human voice and return the performance of Classical music to the pitch for which it was written. The Institute's 1992 publication of A Manual on the Rudiments of Tuning and Registration, Vol. I, Introduction and Human Singing Voice, is creating an educated leadership in the music world to return the pitch to that for which all the great Classical music was written - known as the “Verdi pitch" - and to save the human voice. No less than a revolution in musical history was unleashed on April 9, 1988 in Milan, Italy, when the Schiller Institute brought together some of the world's most highly regarded Classical singers and instrumentalists, to demand a return to rationality in musical tuning and performance. At a conference held at the Casa Giuseppe Verdi, conference speakers, including Lyndon H. LaRouche, Jr., who had conceived the initiative, called for an end to the high-pitched tuning, which has been literally destroying all but the most gifted voices during the past century, and for a return to the principles of Classical aesthetics, according to which the process of musical composition is just as lawful as are the orbits of the planets in the solar system. To underline this call, the conference resolved to introduce legislation into the Italian parliament which would require a return to the natural tuning at which middle-C equals precisely 256 cycles per second - significantly lower than the current tuning which sets A at 440 cps, or frequently even higher 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 :: Key word: history of pitch 1 'History of Musical Pitch' - a table prepared by Mr. A. J. Ellis and published in 1880 (with additions from later publications) units, hertz or Hz, are equivalent to vibrations per second c'' (one octave above middle C, C5 in scientific notation) is calculated from a' (A4 in scientific notation) using equal temperament c' (middle C, C4 in scientific notation) is calculated from a' (A4 in scientific notation) using equal temperament if another note was originally measured this has been converted to a' using equal temperament all pitches assume an ambient temperature of 59° Fahrenheit (15° centigrade) The speed of sound in air increases as the square-root of temperature The speed of sound in air at 0° centigrade is 331.5 m/s, and it increases by 0.6 m/s for each increase of 1° centigrade a' (A4, la4) (in hertz) c'' (C5, do5) (in hertz) c' (C4, do4) (in hertz) Place Date Description 376.3 447.5 223.75 Lille, France 1700 (anté) Pitch taken by Delezenne from an old dilapidated organ of l'Hospice Comtesse 378.8 450.5 225.25 Paris, France 1766 Pitch calculated from data given by Dom Bédos in L'Art du Facteur d'Orgues 380.0 451.9 225.95 Heidelberg, Germany 1511 Pitch calculated from data given by Arnold Schlick 392.2 469.1 234.55 St. Petersburg, Russia 1739 Euler's clavichord 395.8 470.7 235.35 Versailles, France 1789 Organ of the palace chapel 398.0 473.3 236.65 Berlin, Germany 1775 Pitch estimated from a flute described by Jean Henri Lambert in Observations sur les Flûtes, pub. Académie Royal des Sciences, Berlin 400.0 475.7 237.85 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 401.3 477.8 238.9 Paris, France 1648 Mersenne's Spinet 404.0 480.4 240.2 Paris, France 1699 Paris Opera A 405.8 482.6 241.3 Paris, France 1713 Sauveur's calculation 407.9 485.0 242.5 Hamburg, Germany 1762 Organ of St. Michael's Church, Hamburg 409.0 486.4 243.2 Paris, France 1783 Tuning fork of Pascal Taskin, court tuner 415.5 494.1 247.05 Dresden, Germany 1722 Organ of St. Sophia 419.6 499.0 249.5 Seville, Spain 1785 & 1790 Organ of Seville cathedral 421.6 501.3 250.65 Vienna, Austria 1780 supposed to be Mozart's pitch 422.5 502.4 251.2 London, England 1751 Handel's tuning fork 423.5 503.6 251.8 London, England 1711 an existing tuning fork of John Shore 425.5 506.0 253.0 Paris, France 1829 Pianoforte at the Paris Opera 427.6 508.5 254.25 Paris, France 1823 Opèra Comique 430.8 512.3 256.15 Paris, France 1830 Opera pitch as related by Drouet, the celebrated French flautist 432.0 513.7 256.85 Brussels, Belgium 1876 Proposed pitch standard Milan, Italy 1881 at a musical congress in Milan as part of the Musical Exposition held in June 1881, it was decided to stabilise pitch in Italy at a'=432. [quoted in a message on http://launch.groups.yahoo.com/group/earlyflute/message/8074] 432.54 512.00 256.00 Paris, France c.1700 'Joseph Sauveur's Philosophical Pitch, C-512', also called 'scientific pitch', fixed middle C at exactly 256 Hz (arrived at by computing the ninth power of 2) and resulted in the A above it (a') being tuned to approximately 430.54 Hz. It gained some popularity due to its mathematical convenience (the frequencies of all the Cs being a power of two) but it never received the same official recognition as diapason normal (a'=435 Hz) and was not as widely used. Joseph Sauveur (1653-1716) was a French physicist and mathematician 435.0 517.3 258.65 Paris, France 1859 The French 'Diapason Normal', set in a law passed on 16 February 1859 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 or c' = 258.9. This pitch was adopted outside France. For example, several Italian institutions, including the Istituto Musicale di Firenze (Florence) and the Teatro San Carlo in Naples adopted the French 'Diapason Normal'. 437.0 519.7 258.85 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 440.0 523.25 261.625 Paris, France 1829 Orchestral pitch of the Paris Opera 440.0 523.25 261.625 Stuttgart, Germany 1838 Proposed pitch standard, Stuttgart congress (actually a'=440.2 when corrected to table temperature); also Scheibler's standard. 441.0 524.4 262.2 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 444.0 528.0 264.0 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 c''=530.1 or c'=265.05) 444.5 528.6 264.3 Madrid, Spain 1858 Theatre Royale, Madrid 444.5 528.6 264.3 London, England c. 1810 Pitch of a flute made by Henry Potter 444.6 528.7 264.35 London, England 1877 Organ in St. Paul's Cathedral 444.8 528.9 262.45 Turin, Weimer, Würtemberg 1859 Measurements made for the French Commission 445.7 530.1 265.05 London, England 1860 see comment for a'=444.0 (above) 446.0 530.4 265.2 Paris, France; Dresden and Pesth, Germany, 1859 Pleyel's Piano taken by Delezenne and the pitches at the Opera houses of Dresden and Pesth 447.11 531.7 265.85 London, England 1845 Pitch calculated from a fork said to be at the pitch of the Royal Philharmonic Society 448.0 532.8 266.4 Hamburg, Germany 1839 & 1840 Opera 448.0 532.8 266.4 Paris, France 1854 Opéra Comique 448.0 532.8 266.4 Paris, France 1858 Grand Opèra 448.0 532.8 266.4 Liège, Belgium 1859 Conservatoire 450.0 535.1 267.55 London, England 1850 to 1885 An average of the pitches of London orchestras during this period mid-nineteenth century Rome, Italy Accademia di Santa Cecilia in Rome adopted as its own pitch a'=450. Other musical institutions adopted the French 'Diapason Normal' while others used a'=432 450.5 535.7 267.85 Lille, France 1848 & 1854 Lille Opera, measured during performance 451.0 536.3 268.15 Brussels, Belgium 1879 Pitch standard proposed for the Belgian Army 451.5 536.9 268.45 St. Petersburg, Russia 1858 Opera 451.7 537.2 268.6 Milan, Italy 1867 La Scala Opera 451.8 537.3 268.65 Berlin, Germany 1859 Opera 451.9 537.4 268.7 London, England 1878 British Army Regulations 452.0 537.5 268.75 Lille, France 1859 Conservatoire 452.0 537.5 268.75 London, England 1889 Official Pitch at the 'Inventions' Exhibition in 1885 - the highest pitch used intentionally by English orchestras up to 1890 452.5 538.2 269.1 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 453.3 539.0 269.5 London, England 1837 (anté) Pitch calculated from a flute made by Rudall and Rose possibly as early as 1827 454.08 540.0 270.0 London, England 1874 Old Philharmonic Pitch, instigated by Sir Charles Hall 454.7 540.8 270.4 London, England 1874 Fork representing the highest pitch adopted for Philharmonic concerts 454.7 540.8 270.4 London, England 1879 Steinway's English pitch; also Messrs. Bryceson's pitch 455.3 541.5 270.75 London, England 1879 Messrs. Erard's pitch 455.5 541.7 270.85 Brussels, Belgium 1859 Band of the Guides 456.1 542.4 271.2 London, England 1857 Fork set to the French Society of Pianoforte Makers 457.2 543.7 271.85 New York, USA 1879 Pitch used by Messrs. Steinway in America 456.0 542.30 271.15 Vienna, Austria 1859 Viennese 'high pitch' 457.6 544.2 272.1 Vienna, Austria c. 1640 Great Franciscan organ 460.0 547.05 273.525 Vienna, Austria 1880 Old Austrian Military Pitch 461.0 548.3 274.15 London, England 1838 (anté) Actual pitch of a flute said to be tuned to a'=453.3 474.1 563.8 281.9 Durham, England 1683 Cathedral Organ by Bernhardt Smith 474.1 563.8 281.9 London, England 1708 Organ of the Chapel Royal by Bernhardt Smith 480.8 571.8 285.9 Hamburg, Germany 1543 & 1879 Organ at the church of St. Catherine 484.1 575.7 287.85 Lübeck, Germany 1878 Cathedral, small organ 489.2 581.8 290.9 Hamburg, Germany 1688 & 1693 Organ at the church of St. Jacob 505.6 601.4 300.7 Paris, France 1636 Mersenne's church pitch 506.9 602.9 301.45 Halberstadt, Germany 1361 Cathedral Organ 567.6 675.2 337.6 Paris, France 1636 Mersenne's chamber pitch 570.7 678.7 338.35 Germany 1619 Pitch called Kammerton (chamber pitch) by Praetorius; also called North German church pitch 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: I will not send out new non-concert instruments, therefore the regular concert instruments form our only resource - then again I will not send these packed - but only in a van - and accompanied by a tuner. 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: So long as the military bands retain their present pitch (C=537.5 Hz), it must be recognized; but there is no reason why some eight other pitches should not be swept out of the way for musical purposes, leaving the international pitch (C=517.3 Hz) which is now the only recognized pitch in America as well as on the Continent, and the military band pitch as the low and high pitch of this country. 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: The French pitch, or Diapason Normal, is now generally adopted on the Continent and has made its way to the United States of America. In this country, with the exception of the Italian opera, which has been at the low pitch for the last 15 years, we may say the high or Philharmonic pitch has, from 1846 to 1895, prevailed. ... The Philharmonic Society, has, however, for 1896, relinquished its high pitch and adopted the Diapason Normal. 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. pitches in use in England in the 1920s : frequencies in Hz : 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 References: The Rise and Fall of English Pitch History of Piano Tuning The Rise and Fall of English Pitch Society of Arts Pitch: Part I - A Meeting is convened Society of Arts Pitch: Part II - Report of the Committee Society of Arts Pitch Part III - Discussion and Action Arising Harmonic or Overtone Series :: Key words: harmonics fundamental overtones 1 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. harmonic overtone frequency normalized note name (in ET) closeness in % to note (in ET) 100%, very close or exact less than 100%, harmonic is flat greater than 100%, harmonic is sharp 1 Fundamental 440 Hz 440 Hz A 100% 2 1 880 Hz 440 Hz A 100% 3 2 1320 Hz 660 Hz E 100% 4 3 1760 Hz 440 Hz A 100% 5 4 2200 Hz 550 Hz C# 99% 6 5 2640 Hz 660 Hz E 100% 7 6 3080 Hz 770 Hz G 98% 8 7 3520 Hz 440 Hz A 100% 9 8 3960 Hz 495 Hz B 100% 10 9 4400 Hz 550 Hz C# 99% 11 10 4840 Hz 605 Hz D or D# D# slightly closer D 103% D# 97.2% 12 11 5280 Hz 660 Hz E 100% 13 12 5720 Hz 715 Hz F or F# F slightly closer F 102.4% F# 97.6% 14 13 6160 Hz 770 Hz G 98% 15 14 6600 Hz 825 Hz G# 99% 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 :: Key words: inharmonicity harmonic overtones inharmonic overtones 1 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: Schuck and Young in 1943 were the first to measure the spectral inharmonicity in piano tones. They found that the spectral partials in piano tones are progressively stretched and hypothesized that the lower inharmonicity of longer strings in the bass range explains why musicians prefer grand piano tone quality over that of uprights. Nineteen years later, Harvey Fletcher with collaborators found that the spectral inharmonicity is important for tones to sound piano-like. They proposed that inharmonicity is responsible for the "warmth" property common to real piano tones. Fletcher et al.'s statement about the importance of inharmonicity for timbre provided a perceptual basis to the hypothesis of Shuck and Young. Since then, experts have commonly attributed the primary difference in the quality of bass tones in small vs. large pianos to the difference in the inharmonicity between short and long strings. The influences of other acoustical or design factors have never been given serious experimental consideration. 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 :: Key words: tuning temperament Pythagoras 1 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. 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: tunings (Pythagorean, just intonation) regular temperaments where all fifths but the wolf fifth are tempered the same way; note: regular meantone implies that all major thirds are identical irregular temperaments where the quality of the fifths around the circle changes, generally so as to make the more common keys more consonant 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 the needs of the music (harmonic vs melodic, modulations) the tastes of the musicians and listeners the instrument to be tuned (organ vs harpsichord - tuning the former is much more work so one needs a more convenient solution), aesthetic (Gothic's tense thirds and pure fifths vs the stable, pure thirds of the Renaissance and Baroque) and theoretical considerations, and ease of tuning (equal temperament is one of the more difficult) 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. 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 0.000 Minor Second (256:243) Octave - Major Seventh 90.225 Major Second (9:8) (3:2)2 203.910 Minor Third (32:27) Octave - Major Third 294.135 Major Third (81:64) (3:2)4 407.820 Fourth (4:3) Octave - Fifth 498.045 Augmented Fourth (729:512) (3:2)6 611.730 Fifth (3:2) (3:2)1 701.955 Minor Sixth (128:81) Octave - Major Third 792.180 Major Sixth (27:16) (3:2)3 905.865 Minor Seventh (16:9) Octave - Major Second 996.090 Major Seventh (243:128) (3:2)5 1109.775 Octave (2:1) Octave (2:1) 1200.000 * 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 Herman von Helmholtz's 'On the Sensation of Tone As a Physiological Basis for the Theory of Music' [in German, 1863; Ellis's English translation with Appendix, 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, 498.045 + 701.955 = 1200 We have provided an online calculator that converts frequency ratios to cents 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 :: Key word: meantone scale 1 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 :: Key word: equal temperament 1 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. "By pure intonation is naturally meant that of equal temperament [gleichschwebenden], since in modern music no other exists ... The budding violinist needs to know only this one intonation. ..." The clavichord maker Peter Bavington, writing to the Yahoo clavichord-list, comments: Spohr was the central figure in a school of violinists who advocated playing in strict equal temperament (ET): he was certainly not an ivory-tower theorist. However, not every violin teacher agreed with him. Violin intonation was the subject of an interesting article by Patrizio Barbieri in Early Music a few years ago ('Violin intonation: a historical survey', Early Music, vol. XIX No. 1 (February 1991)). Barbieri, the author of Acustica, accordatura e temperamento nell'illuminismo veneto: con scritti inediti di Alessandro Barca, Giordano Riccati e altri autori, examined many sources and concluded that there was a distinct change from 'harmonic' to 'Pythagorean-expressive' intonation around 1780-1800. If playing in the 'harmonic' intonation, one's aim is to produce good harmony, particularly with the bass, which might lead one, for example, to play the major third of a common chord lower than one might in ET so as to get a nearer-to-pure third. On the other hand, in the 'Pythagorean-expressive' intonation one would seek to emphasise the melodic tendency of a note: thus one might play the leading note in a common chord on the dominant higher than pure, perhaps as high as a Pythagorean third, because it is tending upwards so as to resolve on the tonic. This in turn would lead to a wide third in the chord, perhaps a dissonant third, but that would be resolved. Barbieri quotes a lot of evidence for this change, which we might perhaps associate with the transition from late-baroque to classical, and in keyboard instruments from mean-tone and other unequal tunings to the universal acceptance of equal temperament. Two issues later, incidentally, there was another article, this time by Bruce Haynes, covering some of the same ground. Reference: Bruce Haynes: 'Beyond temperament: non-keyboard intonation in the 17th and 18th centuries' in Early Music, vol XIX No. 3 (August 1991) 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: tuning pitch: a'=440 Hz A 27.500 Hz 55.000 Hz 110.000 Hz 220.000 Hz 440.000 Hz 880.000 Hz 1760.000 Hz 3520.000 Hz A# 29.135 Hz 58.271 Hz 116.541 Hz 233.082 Hz 466.164 Hz 932.328 Hz 1864.656 Hz 3729.312 Hz B 30.868 Hz 61.735 Hz 123.471 Hz 246.942 Hz 493.883 Hz 987.766 Hz 1975.532 Hz 3951.064 Hz C 32.703 Hz 65.407 Hz 130.813 Hz 261.626 Hz 523.252 Hz 1046.504 Hz 2093.008 Hz 4186.016 Hz C# 34.648 Hz 69.296 Hz 138.592 Hz 277.183 Hz 554.366 Hz 1108.732 Hz 2217.464 Hz 4434.928 Hz D 36.708 Hz 73.416 Hz 146.833 Hz 293.665 Hz 587.330 Hz 1174.660 Hz 2349.320 Hz 4698.640 Hz D# 38.891 Hz 77.782 Hz 155.564 Hz 311.127 Hz 622.254 Hz 1244.508 Hz 2489.016 Hz 4978.032 Hz E 41.204 Hz 82.407 Hz 164.814 Hz 329.628 Hz 659.256 Hz 1318.512 Hz 2637.024 Hz 5274.048 Hz F 43.654 Hz 87.307 Hz 174.614 Hz 349.228 Hz 698.456 Hz 1396.912 Hz 2793.824 Hz 5587.648 Hz F# 46.249 Hz 92.499 Hz 184.997 Hz 369.994 Hz 739.988 Hz 1479.976 Hz 2959.952 Hz 5919.904 Hz G 48.999 Hz 97.999 Hz 195.998 Hz 391.995 Hz 783.990 Hz 1567.980 Hz 3135.960 Hz 6271.920 Hz G# 51.913 Hz 103.826 Hz 207.653 Hz 415.305 Hz 830.610 Hz 1661.220 Hz 3322.440 Hz 6644.880 Hz Just Intonation :: Key word: just intonation 1 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: Ben Johnston 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 name 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) Kyle Gann's Anatomy of An Octave :: Key word: anatomy of an octave 1 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: 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) including all ratios between 11-limit numbers up to 128 All intervals in the equal-tempered scale 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(s) (if any) (1:1) 0.000 tonic, unison, 1st harmonic (fundamental of the harmonic series), normalised 2nd harmonic (4375:4374) 0.40 ragisma [ref. List of Musical Intervals] (2401:2400) 0.72 breedsma [ref. List of Musical Intervals] (21/1200:1) 1.00 cent (21/1000:1) 1.20 millioctave [ref. List of Musical Intervals] (32805:32768) 1.954 skhisma, schisma ((3 to the 8th/2 to the 12th) x 5/8) (the difference between five octaves and eight justly tuned tuned fifths plus one justly tuned major third; the difference between the Phythagorean and syntonic commas) ((1+(1/100)log10/(2)):1) 3.99 Savart [ref. List of Musical Intervals] (1224440064:1220703125) 5.292 parakleisma (225:224) 7.712 septimal kleisma or marvel comma [ref. List of Musical Intervals] (15625:15552) 8.107 kleisma (393216:390625) 11.445 Würschmidt comma (126:125) 13.795 septimal semicomma or starling comma [ref. List of Musical Intervals] (121:120) 14.367   (100:99) 17.399   (99:98) 17.576   (2048:2025) 19.553 minor comma, diaschisma (81:80) 21.506 major comma, syntonic comma, comma of Didymus (difference between four justly tuned perfect fifths and two octaves plus a major third) There are 55.79763 syntonic commas in the octave (21/53:1) 22.64 Arabian comma or Holder's comma [ref. List of Musical Intervals] (531441:524288) 23.460 Pythagorean or ditonic comma (312/219) (difference between twelve justly tuned perfect fifths and seven octaves) There are 51.15087 Pythagorean commas in the octave (65:64) 26.841 65th harmonic (64:63) 27.264 septimal comma, comma of Archytas (the difference between the 3-limit or Pythagorean seventh and the harmonic seventh) (20000:19683) 27.660 minimal diesis (63:62) 27.700   (3125:3072) 29.614 small diesis (58:57) 30.109   (57:56) 30.642   (56:55) 31.194 Ptolemy's enharmonic (55:54) 31.767   (52:51) 33.617   (51:50) 34.283   (50:49) 34.976 septimal sixth-tone or jubilisma [ref. List of Musical Intervals] (49:48) 35.697 septimal diesis or slendro diesis [ref. List of Musical Intervals] (46:45) 38.051 inferior quarter-tone (Ptolemy) (45:44) 38.906   (128:125) 41.059 diminished second (16/15 x 24/25), enharmonic diesis (the difference between three justly tuned major thirds and one octave), great diesis, enharmonic diesis (Vincentino), 5-limit diesis, limma (525:512) 43.408 enharmonic diesis (Avicenna) (40:39) 43.831 difference between major and minor semitones (39:38) 44.970 superior quarter-tone (Eratosthenes) (77:75) 45.561   (36:35) 48.770 superior or septimal quarter-tone [ref. List of Musical Intervals] (250:243) 49.166 maximal diesis (21/24:1) 50.00 equal-tempered quarter tone (35:34) 50.184 equal temperament (ET) 1/4-tone approximation (34:33) 51.682   (50331648:48828125) 52.504   (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   (648:625) 62.565 major diesis (28:27) 62.961 inferior quarter-tone (Archytas) (80:77) 66.170   (27:26) 65.337   (26:25) 67.900 1/3-tone (Avicenna) (20480:19683) 68.719 comma that is associated with super-Pythagorean temperament (51:49) 69.259   (126:121) 70.100   (25:24) 70.672 minor 5-limit semitone (half-step), chromatic diesis, semitone minimus, lesser or just chromatic semitone, minor chroma [ref. List of Musical Intervals] (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 major diesis, septimal chromatic semitone [ref. List of Musical Intervals] (81:77) 87.676   (20:19) 88.801   (256:243) 90.225 Pythagorean diatonic semitone (half-step), minor or Pythagorean limma, minor semitone [ref. List of Musical Intervals] (58:55) 91.946   (135:128) 92.179 limma ascendant, greater chromatic semitone, semitone medius, chromatic semitone, major chroma (96:91) 92.601   (6442450944:6103515625) 93.563   (19:18) 93.603   (55:52) 97.104   (128:121) 97.364   (18:17) 98.955 equal temperament (ET) semitone (half-step) (approximation familiar to makers of lutes, who used it when fixing frets on lute fingerboards) (21/12:1) 100.000 equal temperament (ET) semitone (half-step), equal-tempered minor second, exact [ref. List of Musical Intervals] (35:33) 101.867   (52:49) 102.876   (86:81) 103.698   (17:16) 104.955 overtone semitone (half-step) (33:31) 108.237   (49:46) 109.377   (16:15) 111.731 major 5-limit semitone (half-step), just diatonic semitone [ref. List of Musical Intervals] (2187:2048) 113.685 apotome or apotome Pythagorica, Pythagorean major semitone, Pythagorean chromatic semitone the difference between a Pythagorean diatonic semitone and a Pythagorean chromatic semitone is the Pythagorean or ditonic comma (31:29) 115.458   (77:72) 116.234   (15:14) 119.443 Cowell just semitone (half-step), septimal diatonic semitone (29:27) 123.712   (14:13) 128.298 major semitone (69:64) 130.229 69th harmonic (55:51) 130.721   (27:25) 133.238 alternate Renaissance semitone (half-step), semitone maximus, minor second, large limma (121:112) 133.810   (13:12) 138.573 3/4-tone (Avicenna), minor tone (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) 147.143   (49:45) 147.428   (12:11) 150.637 undecimal "median" semitone (1/2-step), lesser undecimal neutral second [ref. List of Musical Intervals] (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) 161.915   (11:10) 165.004 greater undemical neutral second [ref. List of Musical Intervals] (54:49) 168.213   (32:29) 170.423   (21:19) 173.268   (31:28) 176.210   (567:512) 176.646   (51:46) 178.636   (71:64) 179.697 71st harmonic (10:9) 182.404 minor whole-tone, just minor tone, smaller step [ref. List of Musical Intervals] (49:40) 186.338   (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) 199.980   (21/6:1) 200.000 equal-tempered whole-tone, exact, equal-tempered major second [ref. List of Musical Intervals] (64:57) 200.532   (9:8) 203.910 major whole-tone, major tone, greater step, just major second, sesquioctave, tonus, 5th harmonic (normalised) (62:55) 207.404   (44:39) 208.835   (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) 226.841   (73:64) 227.789 73rd harmonic (8:7) 231.174 septimal whole-tone, septimal major second [ref. List of Musical Intervals] (63:55) 235.104   (55:48) 235.677   (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) 243.545   (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) 250.304   (37:32) 251.344 37th harmonic (81:70) 252.680   (125:108) 253.076   (22:19) 253.805   (51:44) 255.592   (196:169) 256.596 consonant interval (Avicenna) (29:25) 256.950   (36:31) 258.874   (93:80) 260.677   (57:49) 261.816   (64:55) 262.368   (7:6) 266.871 septimal minor third, subminor third the named interval is only approximately equal to 7:6 frequency ratio. In terms of cents, it is 267 cents, a quartertone of size 36/35 flatter than a 5-limit just minor third of 6:5. In the meantone era the interval made its appearance as the alternative minor third in remote keys, under the name augmented second [ref. List of Musical Intervals] (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) 285.792   (13:11) 289.210   (58:49) 291.925   (45:38) 292.711   (32:27) 294.135 Pythagorean minor third, Pythagorean semiditone (19:16) 297.513 overtone minor third (21/4:1) 300.000 equal-tempered minor third, exact (25:21) 301.847   (31:26) 304.508   (105:88) 305.777   (55:46) 309.357   (6:5) 315.641 5-limit just minor third, sesquiquintan, semiditonus [ref. List of Musical Intervals] (19683:16384) 317.595   (77:64) 320.144 77th harmonic (35:29) 325.562   (29:24) 327.622   (75:62) 329.547   (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 (major third minus a minor diesis) (128:105) 342.905   (8000:6561) 343.301   (11:9) 347.408 undecimal "median" third, undecimal neutral third [ref. List of Musical Intervals] (60:49) 350.617   (49:40) 351.338   (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) 365.825   (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 just major third, sesquiquartan, ditonus (64:51) 393.090   (49:39) 395.169   (44:35) 396.178   (39:31) 397.447   (34:27) 399.090   (21/3:1) 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, Pythagorean ditone (that is two 9:8 tones) (19:15) 409.244   (33:26) 412.745   (80:63) 413.578   (14:11) 417.508 undecimal major third [ref. List of Musical Intervals] (51:40) 420.597   (125:98) 421.289   (23:18) 424.364   (32:25) 427.373 diminished fourth (41:32) 429.062 41st harmonic (50:39) 430.145 major third plus a minor diesis (77:60) 431.875   (9:7) 435.084 septimal major third, supermajor third the named interval interval is exactly or approximately equal to a 9:7 frequency ratio. In terms of cents, it is 435 cents, a quartertone of size 36/35 sharper than a just major third of 5/4. In the early meantone era the interval made its appearance as the alternative major third in remote keys, under the name diminished fourth. [ref. List of Musical Intervals] (58:45) 439.353   (49:38) 440.139   (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 perfect fourth minus a minor diesis (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) 466.851   (38:29) 467.936   (21:16) 470.781 septimal fourth (46:35) 473.135   (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)</