December 7, 2008
Thanks to
John Thompson
http://www.silkqin.com/index.html
who first suggested this
page.
There are some links to sites with historical pictures at the very end of this page.
I have put some of the math into the following animations which are linked together: "Origins 12TET/EDO ", "Origins 12TET/EDO_2 " and "Origins 12TET/EDO 3". These include some ideas on the mesolabium which appears to be the first geometrically correct 12Tet division of the octave in Western literature though it did not replace the luthier's "Rule of 18" in practical application.
The concern here is 12 tone equal temperament on the guitar. (12 Equal Divisions of the Octave - The common Western guitar)
The idea is to refer you to what may be relevant as quickly as possible. Dates are insufficiently accurate for serious research. I'm going to have to print this out and take a trip to the library myself.
Tentative Conclusion: (everything is tentative here)
From a review of Mark Lindley, Lutes, Viols and Temperaments - 1984 " ...equal temperament on lutes and viols is never anachronistic." - meaning generally it is never chronologically out of place, meaning I suppose, it could have been in use at any time.
The work of V.Galilei, G. Zarlino (Italy), Zhu Zaiyu (China), J. Napier (Scotland), Simon Stevin (Belgian/Dutch - unpublished manuscript), and the subsequent success of logarithms in the "Scientific Revolution" all point to an academic interest in equal temperament on the lute from about the last half of the 16th century, apparently following its use by practicing musicians in Europe a bit earlier in the century. It is from this time period I would date the earliest practical acceptance of the 12tet scale though ideas about it apparently existed earlier, especially in China.
There were some mathematical ideas antecedent to logarithims as far back as Archimedes and perhaps academic discussions of equal tempered music from Aristoxenes (4th century B.C.) In 1373 an English organ builder suggested ratios of 18:17:16, (an arithmetic progression derived by expanding the whole tone ratio 9:8) roughly suggestive of the luthier's "Rule of 18" (a geometric progression) for equal temperament. The "Rule of 18" was advocated by Vincenzo Galilei (1581) but it was in some use earlier (according to him) and, also, in 1555 Nicolo Vicento had written of equal temperament in lute music. The "Rule of 18" (refined version [17.817 more or less]) is still in use by luthiers today who have many practical in addition to theoretical considerations when they build these instruments..
The Greek Aristoxenes in the 4th century B.C. is sometimes credited with suggesting equal temperament which is at least doubful though he certainly criticized rigorous Pythagorean ratios. There was a very early approximation in China with Ho Tcheng-tien (370-447 CE) creating a series of string lengths for a scale of twelve approximately equal semitones - the maximum deviation from today's Even Temperament was less than 0.1 semitone!
Some approaches to equal temperament were being tried in practice from the early 16th century in Europe. The theoretical mathematical solution for fret placement was approached from a mechanical apparatus, the mesolabium, invented apparently by Eratosthenes (ca 284-192 B.C.) to solve an entirely different problem (doubling the cube) and first applied to mean-tone fret placement by Zarlino (1558), later suggested for equal temperament by Salinas (1577) with some subsequent confirmation by Zarlino (1588).
There was strong advocacy of equal temperament on the lute by Vincenzo Galilei, a lutenist and the father of Galileo (also a lutenist), and strong philosophical disagreement with Zarlino on some more general issues of polyphony versus monody, tuning, scales, etc. Apparently they eventually agreed on equal temperament on lutes as a necessary compromise for that instrument and the theoretical problem was a rigorous mathematical derivation.
Shortly thereafter, more rigorous and exact methods of derivation (geometric or complex roots - I don't know what to call them, exactly) were put forth by Stevin in Europe (unpublished; 1585?) and Zhu Zaiyu in China (published; 1584-1596?). This could have been a simultaneous coincidence but I might look for some influence through Matteo Ricci who may have renewed interest in the problem in China where it was solved by Zhu Zaiyu, rejected there as unsuited to the ritual music, and possibly sent back to Europe. Likely no one will ever know for certain, that would be my guess as I write this. I know of no solid facts and nothing that would have prevented either culture from developing it independently of the other.
Logarithims were conceived by Napier almost immediately thereafter and no one seems to know exactly where he was for a period of time prior to this either. Apparently he began work on logarithms about 1590 or a bit earlier. The logarithmic tables required such immense work that they were not published till 1614. They were a mainstay of scientific calculation untill the electronic calculators of say about 1960 and remain a principle tool for describing equal-temperament mathematically.
In 1619 Descartes wrote his first work, a private research paper on music mentioning certain aspects from Zarlino's book. Descartes subsequently worked out his method for the discovery of truth, appending his work on Analytic Geometry which featured his version of the mesolabium and a number of other mechanical proceedures by which dynamic geometry could be rendered into algebra. All of this a virtual guidebook to the modern scientific method (influential but not the only such method). Descartes also had considerable music theory correspondence with Mersenne who in 1639 wrote an important treatise on music, deriving (among many other things) a rigorous equal temperament solution therein.
Actually, many of the people who had to do with the early "Scientific Revolution" in Europe seem to have been in correspondence with each other - many through letters to Mersenne, so much as the Reformation and Counter-Reformation would allow, and apparently most had some connection with music. The first to use logarithms for the calculation of interval sizes were Faulhaber (1630) Bonaventura Cavalieri (1639), Juan Caramel de Lobkowitz (1647), Brouncker (1653), and Lemme Rossi (1666). Christian Huygens, the son of a musician, did the best known work published in 1691 though he apparently had it in manuscript much earlier and he preferred 31 equal divisions of the octave.
Practical musicianship and historical instruments seem to indicate, in addition, a variety of temperaments (primarily mean-tone) achieved by a variety of fret adjustments right up to perhaps the early 19th century (at least one luthier indicates the earliest embedded frets on guitars were mean-tone.) By the early 20th century when piano technology was sufficient, and many instruments had been specifically designed for 12 tone equal temperament (and the mass media all powerful) there was almost no acknowledgement of anything other than 12 tone equal temperament in the West.
| Date | Name | Statement | Source | Comments |
| 41,000 to 80,000 B.C. |
An ancient bone flute segment, estimated at about 43,ooo up to 82,ooo years old, was found recently at a Neanderthal campsite by Dr. Ivan Turk, a paleontologist at the Slovenian Academy of Sciences in Ljubljana. It's the first flute ever to be associated with Neanderthals and its confirmed age makes it the oldest known musical instrument.
|
http://www.greenwych .ca/fl-compl.htm |
There is controversy here - and probably should be. Reasonable arguments are made both for and against. In any event, and regardless of how you may feel about dates, a solid argument can be made that some flute would have been around before the confirmed, rather developed Chinese bone flutes noted next. | |
| 7-9000 B.C. |
Brookhaven Lab Expert Helps Date Flute
Thought to be Oldest Playable Musical Instrument... Recent excavations at the early Neolithic site of Jiahu, located in Henan province, China, have yielded six complete boneflutes between 7,000 and 9,000 years old... Tonal analysis of the |
http://www.bnl.gov/bnlweb/ (scroll to September |
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| late fourth millennium B.C. | Richard Dumbrill | a (Sumerian) cylinder seal ... BM WA
1996-10-2,1, ... with lutanist...
a series of ratios - 6:5:4:3 - which coincides with the god numbers of
60 for Anu, 50 for Enlil, 40 for Ea, and 30 for Sin. This is the basic
infrastructure for the Babylonian scale... |
November 26, 2008 - the original site no longer exists, a copy has been preserved on Wayback: http://web.archive.org/web original site http://members.aol.com
|
He wrote the book; The Musicology and Organology of the Ancient Near East, by Richard Dumbrill
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| Ernest G. McClain | More on Sumerian music theory - and more, |
http://www.new-universe. his home page: |
The Harmony of the Spheres idea is traced back to Mesopotamian roots. I find some of this controversial as well. | |
| twenty-seventh century B.C. | Ling Lun - China | per Jorgensen, the first known person to have
formulated equal temperament |
This is all by itself with no further collaboration. It could use some. A possible start, in a popular style: By Cheng-Yih Chen
| |
| twenty-sixth century B.C. | This starts with early Sumerian tablets and continues with later music manuscripts. | http://www. schoyencollection.com/ music.htm#2340 |
A great online collection of images. "Commentary: The earliest known record of music and musical instruments in history." Whether they remain the earliest by the time you read this, they are certainly early and available thanks to the owner's permission. | |
| 6th century B.C. | Pythagoras | Section 5. It is crucial to recognize from the beginning that the Pythagoras of the Middle Ages and Renaissance is the Pythagoras of the Neopythagorean tradition (detailed in section 4), in which he is regarded as either the most important or one of the most important philosophers in the Greek philosophical tradition. ...This is a far cry from the Pythagoras that can be reconstructed by responsible scholarship. |
http://plato.stanford.edu Pythagoreanism |
Many myths and falacies are debunked in this article and sources are given so it should be possible for anyone to check it out. (December 3, 2007) I have a problem with the Musemath website in that I am proceeding on the basis of the monochord and it is not traced back to the original Pythagoreans in this article (section 3.3). If you go to the page and use the "Edit>Find" feature, the word monochord is mentioned only once and very equivocally. I cannot really do without it as a number line is needed for the math so I may need to emphasize it as possibly a "Neopythagorean" demonstration. Perhaps a little more research into early use of a monochord for acoustic reearch is needed since I cannot imagine the early work on mean proportions being done without one and the apparatus was clearly in place everywhere in the form of strnged musical instruments which had been around for millenia.. Section 3.3 Also," ...neither does any ancient source connect Hippasus to the discovery of the irrational..." - the story is too good to leave out entirely however - maybe a phrase like "legendary, apocryphical, and untrue; but illustrative... " will do. The concluding sentence below remains the decisive idea for the Musemath website: "On the other hand, many modern scientists accept the basic tenet that knowledge of the natural world is to be expressed in mathematical formulae, which is rightly regarded as a central Pythagorean thesis, since it was first rigorously formulated by the Pythagoreans Philolaus ( Fr. 4 — see Huffman 1993) and Archytas (Huffman 2005, 65 ff.) and may, in a rudimentary form, go back to Pythagoras himself." |
| 6th century B.C. | Pythagoras |
Scale based on 2:1 and 3:2 ratios The most powerful and influential precept of Pythagoreanism was that all is ordered according to the order inherent in whole numbers. Recent scholarship finds no ground to doubt that this doctrine came through Pythagoras himself, though he may well, as his biographers claimed, have gotten it from Babylonia and/or Egypt |
http://www.ex-tempore.org Means, Meaning, and Music: Pythagoras, Archytas, and Plato Scott Makeig |
In the Western Traditon this is where music and science first meet. This is not entirely true as the records noted above indicate. Nevertheless, the methods of questioning, logic and proof developed by the Greeks (and this obsession with numbers) would eventually, despite the lingering magic and perhaps a bit too much love of argument for its own sake, (philosargue) become the method of science used today to understand the physical world. |
| c. 470-385 B.C. | Philolaus |
The next step in harmonic theory was to describe an entire octave length scale in terms of mathematical ratios. The earliest such description of a scale is found in Philolaus. Philolaus recognizes that, if we go up the interval of a fourth from any given note, and then up the interval of a fifth, the final note will be an octave above the first note. Thus, the octave is made up of a fourth and a fifth. In mathematical terms, the ratios that govern the fifth (3 : 2) and fourth (4 : 3) are added by multiplying the terms and thus produce an octave (3 : 2 x 4 : 3 = 12 : 6 = 2 : 1). The interval between the note that is a fourth up from the starting note and the note that is a fifth up was regarded as the basic unit of the scale, the whole tone, which corresponded to the ratio of 9 : 8 (subtraction of ratios is carried out by dividing the terms, or cross multiplying: 3 : 2 / 4 : 3 = 9 : 8). The fifth was thus regarded as a fourth plus a whole tone, and the octave can be regarded as two fourths plus a whole tone. |
Stanford Encyclopedia of Philosophy 2003, 2007 Carl Huffman Section 2.2 |
It is sometimes stated that Philolaus was the first to write down the Pythagorean philosophy which had previously been an oral tradition, and sometimes it is added that Plato had a copy of this book. |
| ...In the Ancient and Medieval world, Music was more than just intentional organized sound, as we now tend to think; it was above all, the theoretical knowledge of the principles illustrated by organized sound. These are proportional, mathematical principles coextensive with those which were thought to rule the created world; Music was therefore regarded as fit to lift the soul from sensorial experience to the contemplation of eternal, cosmic truth. |
Mathematics and Music: A Diderot Mathematical Forum
Published 2002 |
November 29, 2007 - This is a selection from Google.books brought up on a keyword search. I am experimenting to see if these can function in internet research. The link originally brought up the following: p. 3 - Pythagoras' original tetractys relatiomships pp.7-9 - A good bit of information on Archytas's expansion and application of superparticular ratios. p. 10 - some reference to Aristoxenus' rejection of mathematical ratios. If it stays in place for any length of time, it is decent enough as a resource for information not otherwise obtainable in digital form. (a lot of typing if you want to keep or edit though).
Another source http://www.ex-tempore.org The Musical System of Archytas
| ||
| 5th-4th century B.C. | Plato (428-348 B.C.) | The teachers of harmony compare the sounds and
consonances which are heard only, and their labor, like that of the
astronomers, is in vain.” “Yes, by heaven!” he said; “and ‘tis as good as a play to hear them talking about their condensed notes, as they call them; they put their ears close alongside of the strings like persons catching a sound from their neighbor’s wall— one set of them declaring that they distinguish an intermediate note and have found the least interval which should be the unit of measurement; the others insisting that the two sounds have passed into the same—either party setting their ears before their understanding.” “You mean,” I said, “those gentlemen who tease and torture the strings and rack them on the pegs of the instrument; I might carry on the metaphor and speak after their manner of the blows which the plectrum gives, and make accusations against the strings, both of backwardness and forwardness to sound; but this would be tedious and therefore I will only say that these are not the men, and that I am referring to the Pythagoreans, of whom I was just now proposing to inquire about harmony. For they too are in error, like the astronomers; they investigate the numbers of harmonies which are heard, but they never attain to problems—that is they never reach the natural harmonies of number, or reflect why some numbers are harmonious and others not.” PP417,418 |
Plato (selection from The Republic, Book VII) Translated by B. Jowett Edited, with introduction by Louis Ropes Loomis Published for the Classics Club by Walter J. Black, Inc. Roslyn, NY 1942 |
http://plato The following may apply here: There is an important
Also, The Cambridge History of Western Music Theory By Thomas Street Plato citing Socrates - p.1 |
| 4th century B.C. | Aristoxenus |
...thought that the judgment of the ear more important than the math and protested against the rigidity of the mathematical theories. Aristoxenus had many scales, one of which was equal temperament. He suggested that since "pitch was a continuum it could be divided into equal intervals even if the mathematics of the Pythagoreans could not express them as string lengths." |
>Papers> The |
The reference to the ear being more important than the math is of value in itself, whether you agree or not, and this appears as the earliest statement of the idea I have found so far. I have seen it questioned whether Aristoxenus had suggested equal temperament.
Another source: google.books Aristoxenus made an enormous contribution to the development of music theory in antiquity. Despite his Pythagorean upbringing, he rejected Pythagorean methods of harmonics which focused on the mathematical significance and instead applied a scientific methodology appropriated from Aristotle. |
| 4th century B.C. | ...we had two basic musical approaches in opposition. (Pythagoras) said the ideal tuning came from a mathematical ratio,(Aristoxenus) said that the ideal tuning was one that came from the human sensual reaction. |
From a |
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| 78–37 BC | Jing Fang | ... by using the Pythagorean comma, Jing observed that 53 just fifths approximates to 31 octaves. This would later lead to the discovery of 53 equal temperament, | http://en.wikipedia.org /wiki/History_of_ mathematics |
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| Unlike other tunings such as 19-TET and 31-TET, 53-TET is not suitable as an approximation for meantone temperament for various reasons.. . does not produce a near-just major third | http://en.wikipedia.org /wiki/Jing_Fang |
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| 83 – 161 A.D. | Ptolemy | Ptolemy ...said that "tuning is best when the ear and ratios are in accord," ... |
>Papers> The |
http://en.wikipedia.org Ptolemy ...wrote an influential work, Harmonics, on music theory and the mathematics of music. After criticizing the approaches of his predecessors, Ptolemy argued for basing musical intervals on mathematical ratios (in contrast to the followers of Aristoxenus and in agreement with the followers of Pythagoras) backed up by empirical observation (in contrast to the overly theoretical approach of the Pythagoreans).
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| 83 – 161 A.D. | Ptolemy | The data of perception are rough and ready, but reason has no right to dismiss them as wholly false. its task is to 'bring them into accuracy', on the assumption that counterparts of the kinds of relation broadly gestured at by perceptual impressions are indeed there in mathematical structures to which the perceived relations approximately or exactly correspond. |
Scientific Method p.69 |
November 29, 2007 - google.books selection |
| 4th -5th Century A.D. | Several amazingly close attempts (at equal temperament) were made by the Chinese, with Ho Tcheng-tien (370-447 CE) creating a series of string lengths for a scale of twelve approximately equal semitones - the maximum deviation from today's Even Temperament was less than 0.1 semitone! | http://mathforum.org/ library/drmath/view/ 52470.htm |
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| ca. 520 A.D. |
Boethius (ca. A.D. 480-524), De institutione musica, |
...transmitter of ancient Greek philosophy, aesthetics, and theory of
music to ...Boethius seems to have been the first to use the term quadrivium... Music, as taught at medieval universities, was |
http://etext.virginia.edu/
Dictionary of the History of Ideas Musical Genius ... author of the article. |
It was this tradition which Vincenzo Galilei, a lutenist had to break through to gain academic acceptance for his musical ideas 1000 years later. It was similar with other disciplines, there was a long period of reliance on antiquity as absolute authority. More than just a rebirth or rediscovery of ancient ideas, the Renaissance fostered hands-on, practical experiment and proof by both reason and results. The idea was to move beyond the ancients, to surpass them, and to gain respect and fame through intellectual achievment. A goal opened to all who would apply themselves to it by the printing press. The quest for fame might have been toned down a little I suppose but there were few enough financial rewards in the endeavour. In any event, the concept of requiring results which match the reasoning could be seen as an advance in science. |
| 600 A.D. | Arab Lute Frettings by Joseph Monzo - The European instrument was derived from the Arab 'ud ("al-'ud" --> "lute"), whose first flowering occurred during the 600s and 700s in Syria and Iraq...The Arab lute's open strings and ancient fretting were entirely Pythagorean (that is, 3-limit), derived most likely from Sumerian/Babylonian, Indian, and Greek precedents. |
http://sonic-arts.org /monzo/arablute /arablute.htm |
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| 711-1492 A.D. | The Arab 'Ud was introduced into Europe by the Moors during their conquest and occupation of Spain | http://www.ncconsort.org/ instruments-5.htm#Lute |
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| 10th-11th centuries A.D. | ...The Golden Age of Arab classical music came in the 10th and 11th centuries, with the work of al-Farabi, Safi al-Din, and Ibn Sina (Avicenna)......The first and greatest name in Arabic music is al-Farabi, actually Persian, born in Khorasan (north-eastern Iran) in 882 A.D. He originated the majority of the music theory that still forms the backbone of both the Arabic and Persian maqams (dastgah in Iran). He defined the intervals of the musical scale in his Kitab al-Musiqa al-Kabir (Grand Treatise on Music), intervals that are still the standard in Arab, Persian, and Turkish music, and that also influenced scholars in Medieval Europe until equal temperament came into general use......The oud (or al-'ud) has been the most important instrument in the Arab world since its introduction in the 7th century. The oud originated in the barbat, a large, carved instrument with a short neck and four silk strings that originated in what is now Tajikistan. Depictions of the barbat have been found dating from the 1st century A.D., and it may well go back further. It spread along the Silk Road trade routes, both east where it evolved into the Chinese pipa and Japanese biwa, both also with additional variants in Vietnam and Korea, and west to Persia and Arabia... |
offline July 7, 2006 http://voxclamantis.org |
To what extent may the Persian 24 tone scale be explored as equal temperament? (quarter tones) | |
| 11th century | Ibn Jounis | Arab mathematician Ibn Jounis proposed a method based on trigonometric functions which uses addition to multiply numbers, "Prostapheresis". This is something of a precursor for logarithms apparently |
The connection with logarithms is simply that it inspired Napier to invent them. I do not believe it has anything to do with tuning per se. See "Prosthaphaeresis" if interested. (The subject has some utility in sound engineeering (beats) which I have not explored) Equal Temperament is a logarithmic scale, but approximations existed in both theory and practice before logarithims were invented, perhaps discovered would be a better description. I was interested in the earliest possible mathematical solution. Prostapheresis or something similar came into use in Europe in the 16th century, was used extensively by Tycho Brahe and through him came to the attention of Napier who then invented logarithms, which later became a mathematical foundation for 12 Tone Equal Temperament. (Logarithmic tables became the essential calculating tool for handling the large numbers of scientific measurement right up to the invention of electronic calculators. A tuning system with such mathematical authority and success may have a certain appeal.) | |
| 12th century A.D. |
The earliest reports of the technique of "drawing" wire—that is, forcing a piece of metal through succeedingly smaller holes, increasing its length and tensile strength in the process—tell of its origin in Paris in the twelfth century. Wire strings were used extensively on certain plucked instruments, such as the cittern, the Irish harp, and the harpsichord. | http://www.cumpiano.com /Home /Articles/Articles /stringmaking.htm |
Wire strung citterns (often?) had embedded metal frets - their tuning systems were fixed. I was interested to know their earliest possible date. (November 9, 2007 - A tuning system this early would likely have been Pythagorean, later Renaissance tuning would as likely have been "meantone". I leave the details for the experts, my concern is with 12TET and I have found here no reason to believe there was any equal tempered fretting system this early.)
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| 1373 |
Around 1373, an English source on organ design suggests taking the average length of two pipes a whole-tone (9:8) apart to add a pipe at the semitone between them. This simple rule produces a string ratio (or pipe ratio) of 18:17:16
Curiously, the 18:17:16 division of the whole-tone is mentioned by early 14th-century writers including Jacobus of Liege, who offers it as a proof that this interval cannot be divided into two equal semitones. Jacobus is, of course, quite correct as long as we restrict ourselves to divisions based on integer ratios. |
http://www.medieval.org Section 5.6.5 |
This is the earliest date I've seen for something like an approximate mathematical solution for Equal Temperament in the West. It is an arithmetical progression. Today, the traditional luthier may still use the Rule of 18" to build a guitar. (a geometric progression formula which closely approximates the 12th root of 2 - exact numbers, details and practical adjustments omitted here, see 1569 and following. There is as at least as much skill, as math in practical, hand made construction. The math is the guide, the ear makes the final decisions. That at least is my understanding. This appears as possibly a first attempt to derive the mathematical rule. | |
| 1450 A.D. |
The earliest vihuelas and violas... Much viol music predates the adoption of equal temperament tuning by musicians. The moveable nature of the tied-on frets permit the viol player to make adjustments to the tempering of the instrument and some players and consorts adopt meantone temperaments which are arguably more suited to Renaissance music. There are several recognised fretting schemes in which the frets are spaced unevenly, in order to give "better-sounding" chords in a limited number of keys. In some of these schemes, the two strands of gut which comprise the fret are separated so that the player can finger a slightly sharper or flatter version of a note, to suit different circumstances. |
http://www.pepysdiary .com/p/461.php |
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| 1450-1500 A.D. | Tinctoris |
During the second half of the l5th century, there was a change to playing (the Euorpean lute) with the fingertips, though, as Page (1981) pointed out, the two methods continued for some time side by side. Tinctoris (c.1481-3) wrote of holding the lute 'while the strings are struck by the right hand either with the fingers or with a plectrum, but did not imply that the use of the fingers was a novelty... | http://www.ncconsort.org/ instruments-5.htm#Lute |
Allows different notes to be played simultaneously (with precision). |
| 1452 A.D. | Leon Battista Alberti (1404 – 1472) |
De re aedificatoria (1452, Ten Books of Architecture), were patterned after the De architectura by the Roman architect and engineer Vitruvius (fl. 46-30 B.C.) Alberti can symbolize the Renaissance ideal of the "Universal Man" with interest and skill in may areas. His application of classical ideas of proportion were derived from music theory and of considerable influence. The Vitruvius work contains a possible source of knowledge of the mesolabium later used to establish mathematical equal temperament and some interesting other uses of music theory needed by architects and military engineers for theaters and tuning catapults. |
http://en.wikipedia.org/
http://www-history.mcs
http://www.aboutscotland Note that the architect Palladio (1508-1580) also had some influence with these ideas (1570 The Four Books of Architecture...chose measurements which reflect musical consonances) and can be referred to in the last named article. |
A source for the Vitruvius work: http://penelope.uchicago.edu/ The mesolabium is also mentioned but not described in Plutarch who was also popular: http://classics.mit.edu/
The fact that these books were in print (i.e., no longer restricted to manuscript), and often in the common language (not just Latin) should help explain why the ideas took hold outside academic circles. The printing press was quite the internet of its day. The Vitruvius work mentioned but did not describe the mesolabium. Some more probable sources are cited in google.books: Descartes's Mathematical Thought Page 119
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| Late 15th century A.D. | Bartolomeo Ramos de Pareja (ca. 1440-1491?). | The earliest theorist to publish a complete just tuning | http://www.tonalsoft .com /enc/just.htm |
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| 1496 A.D. |
Gafurius’ Practica music |
per Riemann first mention of temperament | http://www.terraworld .net/c- jasmussen/thesis _asmussen.pdf |
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Because it relies on precise measurement, music was considered until fairly modern times, indeed until around 1650, a branch of science. In late antiquity it began to be included in the four mathematical disciplines of the quadrivium along with arithmetic, geometry, and astronomy. But actually only theoretical music was accorded this place. No singing or playing was included in this curriculum. Practical music making went its own way, maintaining only limited contact with theoretical music, drifting farthest from it in the Middle Ages and approaching nearer during the Renaissance. The musical component of the mathematical curriculum in the universities never went beyond the heritage of Greek music theory. Only the Renaissance humanists succeeded in making this relevant to Western musical art. |
http://etext.lib.
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| 16th century A.D. | As no lutes from before the 16th century have survived, information must be gathered from pictures, sculpture and written descriptions. | http://www.ncconsort.org/ instruments-5.htm#Lute |
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| 16th century A.D. | ...In the sixteenth century, the rediscovery of
Greek writings on music, especially the writings of Ptolemy, gave
considerable added ammunition to the advocates of consonant thirds and
sixths based on ratios involving five...The adoption of twelve-tone equal
temperament was strictly a matter of expediency. Equal temperament allowed
composers to explore increasingly complex chromatic harmonies and remote
modulations without increasing the complexity of instrument design or the
difficulty of playing techniques. These benefits, as we shall see, were
not without costs... |
http://www.justintonation .net /primer2.html |
What is the Ptolemy source? | |
| 1518 A.D. | Henricus Grammateus |
drew up a fairly close approximation to equal temperament in 1518
In 1518, Henricus Grammateus (Heinrich Schreiber) published an "amusing reckoning" using Euclidean geometry to calculate the length of organ pipes for a Pythagorean temperament with diatonic whole-tones divided into two equal semitones; Jacques Lefevre d'Etaples had described this Euclidean method in a music treatise of 1496. |
http://www.dolmetsch
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source? analysis? |
| 1530 A.D. | Adrian Willaert | choral piece Quid non ebrietas, which is impossible to perform in any tuning system other than equal temperament |
http://www.tnr.com Site may be offline. |
This has become a questionable source for me. I have not seen it verified by independent reliable sources. |
| 1532-1568 A.D. | Gerle (1532), Bermudo (1555), the anonymous
author of Discours non plus mélancholique (1557), Vincenzo
Galilei (Il Fronimo, 1568) and John Dowland put forward various
systems, many of which were based on Pythagorean intervals |
http://www.vanedwards .co.uk/history4.htm |
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| 1533 |
Lanfranco Scintille de musica |
The first tuning rules that might be interpreted as equal temperament were given by Giovani Maria Lanfranco. Scintille de musica, (Brescia, 1533), p. 132 |
Tuning and Temperament:
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| Dowland, Cutting, Pilkington, Holborne,
Bacheler, Allison and the host of other names associated with the lute
repertory seem to have bypassed the keyboard altogether. Apart from the
vast differences in playing techniques, the only reason to have been
advanced for this apparent polarity that seems plausible is that whereas
the lute by its nature had to be equally tempered in tuning, keyboards
still used unequal temperaments, and it is possible that the resulting
differences in tonalities and key colours may have made them
incompatible...
The constant pairing of the lute and voice in lists of royal musicians reflects the primarily accompanimental role that it was expected to occupy, a role emphasised by the lute-song industry. In fact, the quantity of lute songs with not only high quality affective poetry but also superb musical settings may suggest that the voice came first, and a good singer took up the lute to accompany himself, rather than simply for the purpose of playing a harmony instrument. |
http://www.ramesescats .co.uk/thesis/ |
The phrase: "whereas the lute by its nature had to be equally tempered in tuning, " might benefit here with some discussion. Certainly straight, across-the-neck frets most naturally divide all the strings into the same ratios and some form of equal temperament is the simplest scale construction on the instrument. But tied gut frets and string tension were adjustable on the instrument, extra frets could be added by the player and "tastini" (temporary wood frets) were also used for a variety of tunings, inconvenient though it might be. Most of the people reviving this early musc use some form of meantone. Musicians have to play something the audience is familiar with or they don't get paid. They also have to play things differently from time to time or the audience will get bored and stop paying them. They usually move about because people get bored anyway, so its good to have something easier to carry than a keyboard . They also have to play with other musicians from time to time in order to get paid so its good to be playing the same scale. A self accompanied singer-songwriter on a lute has a lot of expressive control and independence in the instrument. Power (and even more expression and control) would be in the voice. It would simplify things to tend towards equal temperament block chords on an instrument with single courses and a narrower neck which is more or less what eventually happened.
| ||
| The lute or lutenist was often used to portray the sense of hearing in allegorical representations of the five senses.[2] It was a physical example of geometric perfection and scientific precision in its shape, dimensions and the positions of its frets, and was frequently used as the representative of all music and artistic invention. Its geometrical accuracy and symmetry were also seen as symbolic of the perfection of nature.[3] More than any other instrument, the lute has come to symbolize the renaissance in the modern mind. |
http://www.ramesescats Chapter 1 |
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| 1550 A.D. |
fretted instruments reported tempered in equal semitones - Renaissance lutenists playing a burgeoning polyphonic repertory often encountered vertical progressions moving a diatonic semitone on one fret and a chromatic semitone on another. Equalizing all semitones would simplify life for builders and players alike.
|
http://www.medieval.org /emfaq/harmony/pyth5 .html#6.5 |
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| 1550-1650 A.D. |
Praetorius and Mersenne |
Late I6th-century theorists in Italy, as well as
17th-century writers such as Praetorius and Mersenne, habitually assumed
that the intonation of the lute (and other fretted instruments)
represented equal temperament, whereas keyboard instruments were tuned to
some form of mean-tone temperament... |
http://www.vanedwards .co.uk/history4.htm |
direct quotes and cites needed here |
| ...With fretted string instruments, like the guitar, lute and theorbo, and the various viols, the design of the instrument changes the rules. Because each fret goes all the way across the fingerboard, it creates equal intervals on each string. So equal temperament was advocated for tuning lute strings as early as the 16th century. Yet lutenists have learned tricks to ''temper'' their instruments in ensembles, like tying pieces of gut just above the fret, or attaching bits of toothpick to the fingerboard... | http://gfhandel.org /bleissa/pipe/nyt.htm |
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|
All citterns used a meantone tuning somewhere between equal temperament and quarter comma. Fret positions were marked on a rule and transferred to the fingerboard. |
http://www.cittern. Wood and Wire — |
http://www.cittern. None of the fingerboards that I have measured fit exactly to a set of calculations. Perhaps only those dismissed as unplayable have survived; or more likely fret placement was dictated by the local repertoire, so copying from an original fingerboard will not suit a different time or place. | ||
| 1555 A.D. |
Nicola Vicentino Ancient Music Adapted to Modern Practice |
Nicola Vicentino's treatise was one of the most influential music theory texts of the sixteenth century. ...His best-known contribution is the adaptation of the ancient Greek chromatic and enharmonic genera to modern polyphonic practice. |
http://yalepress.yale.edu/ Ancient Music Adapted to Modern Practice |
...the 31-tone octave division (equal tempered)... is implicit in the
works of authors such as Nicola Vicentino (1555)
This was the Renaissance, a "rebirth" of classical knowledge, round arches instead of gothic pointed arches, etc...everyone was trying to recreate and then improve antiquity - sometimes successfully. |
| 1558 |
Gioseffo Zarlino (1517 - 1590) Le institutione harmoniche |
endorsement of the ratio 5:4 as the basis of the
major third...
...but he also embraced meantone temperament for the keyboard,(ftnote 20) as shown in Table 3, as well as tempered tuning for the lute. |
http://www.terraworld .net/c- jasmussen/thesis _asmussen.pdf |
His meantone ideas are summarized in this animation: |
| 1563 A.D ? | Giambattista Benedetti | argued that since sound consists of air waves or
vibrations, in the more consonant intervals the shorter more frequent
waves concurred with the longer less frequent waves at regular intervals…
(much later apparently)...Marin Mersenne (1588-1648), who sought René Descartes' (1596-1650) opinion of it. Descartes declined to judge the goodness of consonances by such a rational method, protesting that the ear prefers one or another according to the musical context rather than because of any concordance of vibrations... |
http://etext.lib.virginia .edu /cgi-local/DHI /dhi.cgi?id=dv3-32 |
|
| 1567 | Giacomo Gorzanis | composer/lutenist writes a collection of 24 dance suites - apparently for Equal Temperament | http://www.medieval.org /emfaq/harmony/pyth5 .html#6.5 |
|
| 1569 A.D |
Vincenzo Galilei 1525-1591 The Fronimo Dialogo di Vincentio Galilei 1st edition |
...an instructional book on playing, composing and intabulating vocal music for the lute |
http://www.cs.dartmouth. edu/ ~lsa/associated /Galilei /index.html |
A second edition of the complete Dialogo, with significant revisions, was printed in 1584... I do not yet know if this first edition advocated equal temperament. I believe the more heated 16th century blogging and flaming with Zarlino will be found in the 2nd edition and other writings from around the later date. |
| Vincenzio Galilei |
Vincenzio Galilei (1525-1591), (the father of Galileo, was one of the founding members of the Florentine Camerata, a group of men who met in the home of Count Giovanni Bardi to discuss topics mainly related to musical theory, but also touching on science and the arts. The modern art form known as opera was created by this group, and the very first operas were composed by its members as part of their campaign to restore what they believed to have been the classical Greek forms in music, with simple monodal melodies emphasizing the words Throughout his writings, Vincenzio criticized the tendency to mathematize music, and he argued against the kind of numerological and idealist reasoning associated with Plato and the Pythagoreans. |
http://www.mathpages .com/ home/kmath217 /kmath217.htm |
I have seen some of this disputed - as to the actual impact on opera and singing and as to their understanding of the music of classical antiquity - I'll leave this to others.
http://books.google.com/ Encyclopedia Britanica p. 771 Regarding the earlier Pythagorean tuning ratios: "The point at issue was, that neither in the polyphonic school, in which Zarlino was educated, nor in the later monodic school, of which his recalcitrant pupil, Vincenzo Galilei, was the most redoubtable champion, could those proportions be tolerated in practice, however attractive they might be to the theorist in their mathematical aspect." (ror - It does appear that Galilei was an early advocate for equal temperament on the lute and that Zarlino eventually agreed to it, reluctantly perhaps in (Sopplimenti musicali, 1588) but for the lute only. Also, that Galilei published the first approach to the "Rule of 18" a practical method to derive equal tempered fret position by using a geometric progression. Lutenists and luthiers were apparently using some idea of it well before this publication though. In it's most basic form it consists of taking the open string length and subtracting 1/18th of that length, placing the first fret there, next taking 1/18th of the remaining length etc. A fairly recent book and many internet sites refine the number to 17.817 and suggest three decimal places but practical construction has many considerations and realities. I have also seen some reference to the ratio of 17/18 (0.9444...) or 18/17 (1.0588-) as the, or as a variation of "The Rule of 18". I leave this to the luthiers and their artistry, each of the handmade instruments sings its own song. Some understanding of the difference between theory and performance is generally necessary in physics, performance, and guitar construction.) | |
| 1572 A.D. | Girolamo Mei |
letter to Vincenzo Galilei of 1572 - "The true end of science is altogether different from that of art..." |
http://etext.lib
|
|
| 1577 A.D. |
Francisco Salinas De musica libri VII, p.173 |
The first sixteenth century writer to suggest a geometrical or mechanical means of solving equal temperament. The mesolabium had been previously advocated by Zarlino for meantone temperament, and later Zarlino was to follow Salina's lead in recomending it for equal temperament. |
Tuning and Temperament: 1953, 2004 |
Another source indicated Salinas advocated 19 Tone Equal Temperament (unable to verify yet). Zarlino eventually advocated 12 equal divisions, but only for fretted stringed instruments. Summarized in the following animation: |
| 1581 | Vincenzo Galilei |
(V. Galilei,influenced by Girolamo Mei's views on [tuning... and the means by which music moves the affections] questioned the counterpoint practices of the late 16th century) and brought him into open conflict with Zarlino. Galilei expressed his views in the Dialogue on Ancient and Modern Music (Dialogo della musica antica, et della moderna, 1581). Zarlino replied in his Sopplimenti musicali of 1588, and Galilei continued the exchange in his Discorso intomo all'opere di Messer Giosoffo Zarlino of 1589. |
Source Readings in Music History |
|
| 1581 A.D |
Vincenzo Galilei Dialogo della musica antica e moderna, Firenze |
Before Stevin, he wanted intervals all to be equal...Vincenzo Galilei's rule seems to have been commonly accepted at the end of the sixteenth century, as it is to this day, but only for the lute, the viol, and similar instruments ___________________ Galilei is best known for his rejection of modern polyphonic music in favor of Greek monophonic song. The treatise sheds new light on his importance, both as a musician who advocated a new philosophy of music history and theory based on an objective search for the truth, and as an experimental scientist who was one of the founders of modern acoustics |
http://www.xs4all.nl
__________________________ Dialogue on Ancient and Modern Music Translated by Claude V. Palisca |
The second link here is to google.books to the Galilei book itself. |
| Music and Science in the Age of Galileo By Victor Coelho Published 1992, Springer 247 pages ISBN:079232028X at page 57 |
...Kepler, for reasons of mathematical harmony, preferred the just scale, and he felt that Galilei's attempts at tempering the scale were, from a theoretical viewpoint, "ruinous": "See another very clever tempering of this sort by Vincenzo Galilei,
made not in ignorance of the mathematical size of the notes, but with a
particular intention. And I indeed recognize its mechanical function, so
that in instruments we can enjoy almost the same freedom of tuning as can
the human voice. However for theorizing and even more for investigating
the nature of melody, I consider it ruinous; and the effect of it is that
the instrument never truly attains the nobiliity of the human
voice." |
online, the quote may be found here: http://www.wlym.com |
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| 1584 |
Zhu Zaiyu A New Account of the Science of the Pitch Pipes. |
Chinese sources claim that the mathematical calculations for 12 equal intervals were first published there, in 1584, by the Chinese prince Zhu Zaiyu (Chu Tsai-yü) in a paper entitled A New Account of the Science of the Pitch Pipes. |
The publication of the second edition of Fromino by V.Galilei in the same year together with some indications that Napier's notes from this period indicate he was working out logarithms at about this time seems a bit too much to be coincidence. No matter who finally gets credit for it this is probably an important time frame for 12 tone equal temperament. There is an animation which includes what I know or think I know about the math: | |
| The Discovery of Musical Equal Temperament in
China and Europe in the Sixteenth Century Cho, Gene Jinsiong ISBN10: 0-7734-6941-9 ISBN13: 978-0-7734-6941-9 Pages: 332 Year: 2003 USA List Price: $119.95 UK List Price: £ 74.95 |
http://www.mellenpress This study provides little-known mathematical, musicological and scientific facts regarding the discovery of musical equal temperament, and narrates the circumstances of the discovery in the historical and cultural contexts of the period (mid 16th to early 17th centuries) which, in turn, is placed in the intellectual chronology of the Eastern and Western worlds. |
I haven't seen this book and it comes at a high cost. Might be a good reference to this time period but I haven't seen anything online (July 26, 2008.) | ||
| 1584 A.D. |
Vincenzo Galilei The Fronimo Dialogo di Vincentio Galilei 2nd edition |
(Fronimo) reporting the modifications introduced by some lutenists to obtain purer thirds Although often an iconoclastic theorist, Galilei defends equal temperament on the lute as a time-honored arrangement: if it were indeed possible to obtain purer thirds without compromising many other intervals more seriously, it would have been done long ago.
|
http://www.medieva l.org /emfaq/harmony /pyth5.html#6.5 |
(revised 2nd edition - see 1569 for 1st edition) |
| 1588 | Roselli |
...treatise advocating equal temperament for singers and instrumentalists Two problems: |
http://www.medieva l.org /emfaq/harmony /pyth5.html#6.5 |
|
| 1588 A.D. |
Gioseffo Zarlino Sopplimenti musicali |
Zarlino gave three methods by which "to divide the octave directly into 12 equal and proportional parts or semitones." |
Sopplimenti musicali Tuning and Temperament: page 51 Also: |
The mesolabium was one of the methods... several other mechanical methods by other authors are noted by Barbour. Consider as a possible source: Pappus in the first half of the fourth century A.D., provided a minute explanation of that instrument in Book III of his "Collection" clearly mentioning the names of both the instrument and its inventor. In Commandino's Latin translation of 1588, the names of the mathematician and his mathematical device appeared as "in Eratosthenis mesolabo." Chikara Sasaki |
| 1589 A.D | Vincenzo Galilei | All scales are man-made, with no basis in nature
whatsoever |
http://www.tnr.com Site may be offline. |
They are based on natural segmentation of vibrating bodies (strings, eardrums, etc.) but pitch is not the only element in music and there are some benefits in not being too precise. |
|
...The ear, Galilei argued, had no regard for systems. Only the octave itself could be critically determined. All other intervals were flexible, part of a continuum. |
http://www.dolmetsch .com/poshistory.htm |
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| 1600? |
Simon Stevin (1548-1620) unpublished manuscript |
De Spiegheling der Singconst (Theory of the art of singing) ...usually seen as the first correct theory of the division of the octave into twelve equal intervals - ...the core issue of the science of music, namely, the problem of consonance |
http://www-groups.dcs. st-and.ac.uk/%7Ehistory /Mathematicians /Stevin.html |
This is an unpublished manuscript re-discovered in the late 19th century. |
| Simon Stevin | ... But to explain the unexplained cause why this fifth on the aforesaid instruments cannot be hit off as right as the natural singing of human voices testifies it should be, the common imperfection should be understood of practical operation in all matters, which cannot be performed as perfectly as mathematical operations... | http://www.xs4all.nl /~huygensf/doc/singe.html |
The difference between theory and practice is explaineded here as well as anywhere. | |
| Late I6th-17th century A.D |
Praetorius Mersenne |
Late I6th-century theorists in Italy, as well as 17th-century writers such as Praetorius and Mersenne, habitually assumed that the intonation of the lute (and other fretted instruments) represented equal temperament, whereas keyboard instruments were tuned to some form of mean-tone temperament | http://www.ncconsort.org/ instruments-5.htm#Lute |
The source material will have to be consulted and quoted directly... |
|
1614 Latin 1616 English |
John Napier | John Napier publishes Mirifici Logarithmorum Canonis Descriptio (A Description of the Wonderful Canon Of Logarithms) where, using a kinematic approach, he connects a geometrical series and an arithmetic series. |
November 26, 2008 offline http://www.ru.nl/w-en-s/gmfw/bronnen/ There is an animation with some explanation by A. Audsley: There is another book published in his name at or shortly before his death in 1617 using, I believe base 10 logs. http://www-history.mcs. 2. Mirifici Logarithmorum Canonis Constructio. (Edinburgh; Andrew Hart, 1619). This work, though written several years before the Descriptio, only appeared after Napier's death and contained Notes by Briggs The definition of a logarithm is very different from that now so familiar. In the first place Napier speaks always of the logarithm of a sine, not of a number; it was the simplification of Trigonometric calculations he had primarily in view and therefore it was the sine that bulked most largely. It is to be remembered that in his day, and for long after, the sine was a line, not a ratio, and the whole sine meant the radius of the circle whose half-chords were the sines. In the second place he makes use of two moving points to define a logarithm. The fundamental rule on which all the others depend is that if a : b = c : d
|
This is pretty much where logarithms start (but cf 11th century - Ibn Jounis) and they were very important to the developement of the Scientific Method. When I was creating "Logarithims 3" animation I wondered if he had gotten his ideas from fretboard intonation - it's not impossible - he is thought to have been touring Europe for an education sometime between 1566-1571 where Stevins, Zarlino and V. Galilei were soon working on equal temperament. The ideas at least were "in the air". |
| 17th century |
Faulhaber (1630) - first calculation of equal temperament with logarithms Brouncker (1653) described the nature of hearing as "geometrical." Hence, he believed that the division of intervals should also be geometrical.
|
In the calculation of temperaments, logarithms serve especially well in the geometric division of intervals, where they replace root extraction by division. ...Tables of logarithms were published from about the late 1620's, and from that time onwards we see the application of logarithms to the calculation of temperament. ...at least five scholars tried it independently of one another. In the end, the description by Christiaan Huygens became best known, being published in French in a relatively widely disseminated publication. (1691)
The first wide-ranging application of the logarithmic method (to music) in print was provided by the French scientist Joseph Sauveur (1653-1716), who had first explained his ideas in a manuscript treatise dated 1697 |
The Cambridge History By Thomas Street 2002 page 210 |
My opinion would be that the continuing use and success of logarithms in the scientific revolution may have generated increasing interest in equal temperament but for the construction of fretted stringed instruments luthiers used some variety of the "Rule of 18" which is a practical approximation. I would doubt there was ever a time when people did not experiment where they could but sometime in say the 18th century fixed and embedded metal frets in the popular instruments put severe restrictions on such experiments. |
| 1620 A.D + | ... After c1620 the harmonic demands of composers on their instruments required the alteration of one or more of those intervals, and a period in which a large number of different tunings came into use resulted. These tunings are now collectively described as transitional tunings, and are particularly associated with the French seventeenth-century repertory. | http://www.ramesescats .co.uk/thesis/ |
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| 1639 A.D |
Marin Mersenne |
Harmonie Universelle ...is (sometimes) credited with the invention of equal temperament |
offline July,2006 http://www.casaninja.com |
The book was at least influential in spreading the idea - An important treatise. My current understandnig is that this was the first publication in the West of a mathematically rigorous derivation of equal temperament. Mersenne was corresponding with many of the great scientific minds of his day including Galileo and Descartes. A very important figure in the early scientific revolution. |
| I believe, however, that by presenting disparate views, Mersenne is attempting to reconcile humanist musical thought with scientific musical inquiry…Mersenne believed, for example, that tuning systems and the differing intervals produced by meantone temperament, just intonation, and equal temperament each produce unique resonance's in the mind, a humanist construction. |
http://www.arts.uci.edu article by Elisabeth Honn - |
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| 1661-1691 | Christiaan Huygens |
Christiaan Huygens ...in 1661 he had already made notes in which he accomplished the following:
described the 31-tone system in his Lettre touchant le cycle harmonique (Rotterdam 1691) |
http://www.xs4all.nl /~huygensf/ english/huygens.html |
|
| Christiaan Huygens1629 - 1695 | (his) most noted tuning system divides the octave into 31 logarithmically equal parts. | http://www.terraworld .net/c- jasmussen/thesis _asmussen.pdf |
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|
...Huygens, like Galileo, was the son of a musician... He explained how logarithms could be used to calculate the division of the octave, although the first to use logarithms for the calculation of interval sizes were Bonaventura Cavalieri (1639), Juan Caramel de Lobkowitz (1647), and Lemme Rossi (1666). |
http://www.dolmetsch .com/poshistory.htm |
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| 1664 A.D |
... The discovery that gut, wire, or silk could
act as a core material, around which extremely fine wire could be spun,
allowed for the final increase of elasticity that instrument makers needed
to solve their range problems. What is probably the earliest documented
reference to over-spun strings appears in a manuscript written in
1664... |
http://www.cumpiano.com /Home /Articles/Articles /stringmaking.htm |
String technology affects instrument design. | |
| 1650-1750 A.D ? | By its very nature, the guitar encourages the player to think about harmony in the new way, rather than in terms of traditional counterpoint. The ability to play block chords with relative ease made it an ideal tool for the early monodists, and the invention of alfabeto enabled them to notate, and hence transmit widely, a style of guitar accompaniment that may well have been a traditional, unwritten, Neapolitan accompanying practice. | http://www.sscm-jscm .org /jscm/v9/no1 /Tyler.html |
(The transition from lute to vihuela and guitar; and from movable gut frets to fixed, embedded metal frets, and this style of block chords and monody mentioned here, and the long term relation between Spain and Naples (14th-19th century?) may all play some part here. So near as I can tell, the transition to fixed equal temperament on guitars was pretty much completed during this period. None of the early guitar afficionado web sites I have seen have anything to say about anything other than 12 tone equal temperament. In fact, they don't even mention it, they just assume it. | |
| ...it may be argued that the guitar in our sense
of the word was regarded as the descendant on European soil of a late
Roman instrument, to wit the cithara, whose name it shares; whereas the
lute was known to be what it always has been regarded as by common
consent, an Oriental instrument transferred to medieval Europe from the
Persian-Arabian civilization during the Moorish occupation of
Spain... |
http://us.geocities.com The site apparently has |
( This argument is based largely on the shape of the body of the
instrument and the tone quality produced by such shape). It is not in
context | ||
| 1700? | There was one more trick the vihuela guitars were experimenting with that would later prove to be equally or even more important than simply adding frets. They were experimenting with ways of permanently embedding their frets (rather than using tied gut frets). The routine practice of embedding metal frets on European instruments may have happened first on instruments called citterns, and out of necessity. Citterns used courses of metal strings (bronze), and gut frets simply wouldn’t have held up, they would have been cut through quickly, sliced by the metal strings. Citterns were similar in some respects to vihuela guitars but one big difference is that many of them used diatonic fretting rather than chromatic. | http://www.thecipher.com /viola_da_gamba _cipher-3.html |
Note: the site contains some extensive images and arguments emphasising the author's belief in the nature of the viola (4th tuned - fretted) family as essentially a guitar as opposed to the violin (5th tuned - fretless). I rather like the arguments made though you may feel otherwise, in any event, the collection of images is much appreciated and the nature of how chords are played is certainly relevant. | |
| 1743 | Daniel Strahle | Strahle’s method – based not on mathematics, but on his expertise as a craftsman – gives a practical method for fret placement extraordinary close to the theoretical equal tempered placement. | http://www.mth.pdx.edu/ ~caughman/Laura.doc |
See animation "Strahle" I put a few of my own ideas in this as well as some additional references. So far as tuning history, I think the episode shows an increasing interest in Equal Temperament at this time and place. So far as math, check your work, this animation contains a few of my own errors in addition some image distortion problems and some possibly untenable thinking.
|
| 1747 | d'Alembert | The traveling-wave solution of the wave equation was first published by d'Alembert in 1747 | http://www.dsprelated.com /dspbooks/pasp /Traveling_Wave_Solution.html |
...The website gives me a few ideas on what needs to be considered for the goal of a simplified fourier animation on the guitar string... |
| 1750 A.D ? |
By the middle 18th century, the guitar had
become an amateur’s instrument, and the stringing was consequently made
simpler: the 5 pairs of strings made way for the 6 single strings, tuned
E0 A0 d0 g0 b0 e1, as they are today... |
http://us.geocities.com The site apparently has an |
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| 1757-1790 | Friedrich Wilhelm Marpurg (1718-95) |
(his)... many works on tuning and temperament ...represent, in a way, the culmination of the historical theory of tuning and temperament. Despite the great variety of unequal temperaments described, Marpurg basically adhered to equal temperament. |
The Cambridge History By Thomas Street 2002 page 219 |
"His three major works on the subject were the "Anfangsgrunde der theoretischen Musik (1757), the "Versuch uber die musikalischen Temperatur" (1776) and Neue Methode allerley Arten con Temperaturen dem Claviere aufs bequemste mitzutheilen (1790)." |
| John Harrison (1693-1776) | In his book Concerning Such Mechanism,
(after "slagging off" his contemporaries), he very clearly states his
conclusions: The Natural Notes of Melody may be derived mathematically
from pi. He gives no experimental details, except that he used monochords,
and clearly understood and criticises WNR (whole number ratio) logic and
practice... |
http://www.lucytune.com /academic/manuscript _search.html |
Since I have been a navigator, I am not inclined to ignore anything done by Mr Harrison. Nothing but perfection satisfied him and he was relentless in its persuit. | |
| 1800 A.D. | "All the early 19th century makers used different systems of unequal fretting, rather than the equally scaled fretting used today." | http://www.earlyromantic guitar.com/erg/ components.htm |
||
| 1800-1900 A.D | ...More and more in the course of the 19th
century, the significance of enharmonic modulations lay not so much in
their momentary effect as in the way they enabled composers to exploit the
same physical scale in terms of two systems at once: harmonic and
equal-division... |
http://www. societymusictheory.org /mto/issues/mto.93.0.3 /mto.93.0.3.lindley.art |
| |
| 1870 A.D |
Twelve tone equal temperament was introduced in the West to permit the playing of music in all keys with an equal amount of mis-tuning in each, without having to provide more than 12 pitches per octave on instruments, while still roughly approximating just intonation intervals. This allows much more facile harmonic motion, while losing some subtlety of intonation. True equal temperament was not available to musicians before about 1870 because scientific tuning and measurement was not available. | http://en.wikipedia.org /wiki/Equal_temperament |
Caveat - with regard to guitar-like instruments, fixed frets per the logarithmic, equal tempered formula appear much earlier, few if any instruments survive unaltered. 1870 may be about right for the piano but some put it at about 1917 for that instrument. An 1890 Encyclopedia Britanica article stated that keyboards had been using equal temperament universally for "the last 35 years" i.e. since 1855. | |
| 1885 A.D | Ellis's findings | Jorgensen analysis - clearly show a strong bias for the same style of inequality that had been recognized for over 150 years. | ||
| 1917 A.D | William Braid White |
"discovered" beat rates. Up until that time, tuners tuned by fifths, chords and colors. Equal Temperament was only a theory that tuners strove for, had the mathematical knowledge for, but did not have the ability to achieve, because they could not count beats yet | http://www.radfordpiano .com/historical.html |
This brings the range for an increasing acceptance of 12 tone equal temperament on keyboards from 1664 (wound strings) to1917. |
| 1984 A.D | Mark Lindley, Lutes, Viols and Temperaments | (his)...conclusion was that equal temperament
was probably the most usual way to tune a lute though for some periods and
places (e.g., early 16th century Spain) some adjustment of the frets to
obtain better tuning might have been used.
Since Lindley's book was written, it has become (almost) common amongst serious lute players to use unequal temperaments, largely for practical reasons: when tuned in this way the lute sounds sweeter, louder, and more in tune with ensembles |
I'll have to read the book to continue with any serious study - however, the current practice of lutenists is a Very convincing argument for me. Mark Lindley, Lutes, Viols and Temperaments, Cambridge University Press, 1984, ISBN:0521246709 | |
| Mark Lindley, Lutes, Viols and Temperaments |
JSTOR critique on Lindley book Some conclusions: ...equal temperament on lutes and viols is never anachronistic.
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http://links.jstor.org /sici?sici= 0027-4224 %28198607 %2967%3A3% 3C313%3ALVAT%3E 2.0.CO%3B2-T&size =LARGE&origin=JSTOR -enlargePage |
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This seems a good article- with links and sources: Temperaments and Tunings: A Guide for Lute Players ...The mathematical and theoretical placement of frets is really just a guide to follow, after which one may decide to alter this fret or that in order to please one's own aesthetic sense. |
http://www.theaterofmusic .com /temperaments.html |
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| 1991 | Owen H. Jorgensen | TUNING Containing The Perfection of Eighteenth-Century Temperament The Lost Art of Nineteenth-Century Temperament and the Science of Equal Temperament Complete With Instructions for Aural and Electronic Tuning by Owen H. Jorgensen © Owen H. Jorgensen 1991 Michigan State University Press East Lansing, Michigan 48823-5202 ISBN 0-87013-290-3 | http://mmd.foxtail.com /Tech/jorgensen.html |
Another source for serious researchers |
| PICTURES - back to top of page | ||||
| This webpage includes a considerable number of rare musical manuscripts. | MS 2340 LEXICAL LIST OF 9 TYPES OF MUSICAL STRINGS, 23 TYPES OF MUSICAL INSTRUMENTS AND MUSIC, INCLUDING DIFFERENT TYPES OF STRINGED INSTRUMENTS SUCH AS HARP AND LYRE, AS WELL AS HITHERTO UNKNOWN INSTRUMENTS... Commentary: The earliest known record of music and musical instruments in history. ... and terms for 3rds and 6ths that appear to have been used to fine tune (or temper in some way) the 7 notes generated for each scale... The oldest musical instruments known are a ca. 41 000 BC flute made of bear bone, found in 1995 at a Neanderthal site in Slovenia, (controversial - see first entry this page) and 6 intact and 30 fragmentary crane bone flutes from Jiahu, in the
Chinese province of Henan, dated to 9000-7700 BC. One crane bone flute is
still in playing order, the earliest instrument possible to
play. |
http://www. schoyencollection.com/ music.htm#2340 |
Also, MS 2340 4 string fretted and: MS 5105 | |
| ancient egyptian instruments |
http://nefertiti.iwebland |
andré dollinger | ||
| ancient egyptian instruments | http://www.touregypt.net /featurestories/music.htm |
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| Ancient Greek Instruments | http://www.mlahanas.de /Greeks/Music.htm |
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| Roger E. Blumberg - The Cypher | In general, on this page and throughout the entire Cipher for Viola da Gamba and Lute section on this site, we’ll be recapturing a full 3/4’s of the greater guitar family, the viols, lutes, violas vihuelas, integrating and linking the greater guitar family in as many ways as possible, reintroducing viola da gambas, lutes, and violas and vihuela’s to the relevant playing public, the 21st century guitar players | http://www.thecipher .com/viola_da_gamba _cipher.html |
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| Great lute pictures Offline December 4, 2006
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http://www.xs4all.nl /~amarin/Page1.html |
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| The Metropolitan Museum of Art history_musical _instruments |
http://www.metmuseum.org/ toah/hi/hi_mi.htm |
November 26, 2008 changed url link | ||