Enharmonic
In modern musical notation and tuning, an enharmonic equivalent is a note, interval, or key signature, equivalent to some other note, interval, or key signature but "spelled", or named differently. Thus, the enharmonic spelling of a written note, interval, or chord is an alternative way to write that note, interval, or chord. For example, in twelve-tone equal temperament, the notes C♯ and D♭ are enharmonic notes. Namely, they are the same key on a keyboard, thus they are identical in pitch, although they have different names and different roles in harmony and chord progressions. Arbitrary amounts of accidentals can produce further enharmonic equivalents, such as B, although these are much rarer and have less practical use. In other words, if two notes have the same pitch but are represented by different letter names and accidentals, they are enharmonic. "Enharmonic intervals are intervals with the same sound that are spelled differently…, of course, from enharmonic tones."Prior to this modern meaning, "enharmonic" referred to notes that were close in pitch—closer than the smallest step of a diatonic scale—but not identical in pitch, such as F♯ and a flattened note such as G♭, as in an enharmonic scale.
"Enharmonic equivalence is peculiar to post-tonal theory." "Much music since at least the 18th century, exploits enharmonic equivalence for purposes of modulation and this requires that enharmonic equivalents in fact be equivalent." Some key signatures have an enharmonic equivalent that represents a scale identical in sound but spelled differently. The number of sharps and flats of two enharmonically equivalent keys sum to twelve. For example, the key of B major, with 5 sharps, is enharmonically equivalent to the key of C♭ major with 7 flats, 5 + 7 = 12. Keys past 7 sharps or flats exist only theoretically and not in practice; the enharmonic keys are six pairs, three major and three minor: B major/C♭ major, G♯ minor/A♭ minor, F♯ major/G♭ major, D♯ minor/E♭ minor, C♯ major/D♭ major and A♯ minor/B♭ minor. There are no works composed in keys that require double sharps or double flats in the key signature. In practice, musicians learn and practice 15 major and 15 minor keys, three more than 12 due to the enharmonic spellings.
Enharmonic equivalents can be used to improve the readability of a line of music. For example, a sequence of notes is more read as "ascending" or "descending" if the noteheads are on different positions on the staff. Doing so may reduce the number of accidentals that must be used. Thus, in the key of B♭ major, the sequence B♭-B♮-B♭ is more read using the enharmonic spelling C♭ instead of B♮. For example, the intervals of a minor sixth on C, on B♯, an augmented fifth on C are all enharmonic intervals Play; the most common enharmonic intervals are the augmented fourth and diminished fifth, or tritone, for example C–F♯ = C–G♭. Enharmonic equivalence is not to be confused with octave equivalence, nor are enharmonic intervals to be confused with inverted or compound intervals. In principle, the modern musical use of the word enharmonic to mean identical tones is correct only in equal temperament, where the octave is divided into 12 equal semitones. In other tuning systems, enharmonic associations can be perceived by listeners and exploited by composers.
In Pythagorean tuning, all pitches are generated from a series of justly tuned perfect fifths, each with a frequency ratio of 3 to 2. If the first note in the series is an A♭, the thirteenth note in the series, G♯ is higher than the seventh octave of the A♭ by a small interval called a Pythagorean comma; this interval is expressed mathematically as: twelve fifths seven octaves = 12 2 7 = 3 12 2 19 = 531 441 524 288 = 1.013 643 264... ≈ 23.46 cents In quarter-comma meantone, on the other hand, consider G♯ and A♭. Call middle C's frequency x. High C has a frequency of 2x; the quarter-comma meantone has just major thirds, which means major thirds with a frequency ratio of 4 to 5. To form a just major third with the C above it, A♭ and high C must be in the ratio 4 to 5, so A♭ needs to have the frequency 4 5 = 8 5 x = 1.6 x. To form a just major third above E, however, G♯ needs to form the ratio 5 to 4 with E, which, in turn, needs to form the ratio 5 to 4 with C, thus the frequency of G♯ is 2 x = 25 16 x = 1.5625 x Thus, G♯ and A♭ are not the same note.
The difference is the interval called the enharmonic diesis, or a frequency ratio of 128/125. On a piano tuned in equal temperament, both G♯ and A♭ are played by striking the same key, so both hav
The Well-Tempered Clavier
The Well-Tempered Clavier, BWV 846–893, is a collection of two sets of preludes and fugues in all 24 major and minor keys, composed for solo keyboard by Johann Sebastian Bach. In Bach's time Clavier was a generic name indicating a variety of keyboard instruments, most a harpsichord or clavichord – but not excluding an organ either; the modern German spelling for the collection is Das wohltemperierte Klavier. Bach gave the title Das Wohltemperirte Clavier to a book of preludes and fugues in all 24 major and minor keys, dated 1722, composed "for the profit and use of musical youth desirous of learning, for the pastime of those skilled in this study"; some 20 years Bach compiled a second book of the same kind, which became known as The Well-Tempered Clavier, Part Two. Modern editions refer to both parts as The Well-Tempered Clavier, Book I and The Well-Tempered Clavier, Book II, respectively; the collection is regarded as being among the most important works in the history of classical music. Each set contains twenty-four pairs of fugue.
The first pair is in C major, the second in C minor, the third in C♯ major, the fourth in C♯ minor, so on. The rising chromatic pattern continues until every key has been represented, finishing with a B minor fugue; the first set was compiled in 1722 during Bach's appointment in Köthen. Bach recycled some of the preludes and fugues from earlier sources: the 1720 Klavierbüchlein für Wilhelm Friedemann Bach, for instance, contains versions of eleven of the preludes of the first book of the Well-Tempered Clavier; the C♯ major prelude and fugue in book one was in C major – Bach added a key signature of seven sharps and adjusted some accidentals to convert it to the required key. In Bach's own time just one similar collection was published, by Johann Christian Schickhardt, whose Op. 30 L'alphabet de la musique, contained 24 sonatas in all keys for alto recorder or flute or violin and basso continuo. Although the Well-Tempered Clavier was the first collection of worked keyboard pieces in all 24 keys, similar ideas had occurred earlier.
Before the advent of modern tonality in the late 17th century, numerous composers produced collections of pieces in all seven modes: Johann Pachelbel's magnificat fugues, Georg Muffat's Apparatus Musico-organisticus of 1690 and Johann Speth's Ars magna of 1693 for example. Furthermore, some two hundred years before Bach's time, equal temperament was realized on plucked string instruments, such as the lute and the theorbo, resulting in several collections of pieces in all keys: a cycle of 24 passamezzo–saltarello pairs by Giacomo Gorzanis 24 groups of dances, "clearly related to 12 major and 12 minor keys" by Vincenzo Galilei 30 preludes for 12-course lute or theorbo by John Wilson One of the earliest keyboard composers to realize a collection of organ pieces in successive keys was Daniel Croner, who compiled one such cycle of preludes in 1682, his contemporary Johann Heinrich Kittel composed a cycle of 12 organ preludes in successive keys. J. C. F. Fischer's Ariadne musica neo-organoedum is a set of 20 prelude-fugue pairs in ten major and nine minor keys and the Phrygian mode, plus five chorale-based ricercars.
Bach borrowed some of the themes from Fischer for the Well-Tempered Clavier. Other contemporary works include the treatise Exemplarische Organisten-Probe by Johann Mattheson, which included 48 figured bass exercises in all keys, Partien auf das Clavier by Christoph Graupner with eight suites in successive keys, Friedrich Suppig's Fantasia from Labyrinthus Musicus, a long and formulaic sectional composition ranging through all 24 keys, intended for an enharmonic keyboard with 31 notes per octave and pure major thirds. A lost collection by Johann Pachelbel, Fugen und Praeambuln über die gewöhnlichsten Tonos figuratos, may have included prelude-fugue pairs in all keys or modes, it was long believed that Bach had taken the title The Well-Tempered Clavier from a similarly-named set of 24 Preludes and Fugues in all the keys, for which a manuscript dated 1689 was found in the library of the Brussels Conservatoire. It was shown that this was the work of a composer, not born in 1689: Bernhard Christian Weber.
It was in fact written in 1745–50, in imitation of Bach's example. Bach's title suggests that he had written for a well-tempered tuning system in which all keys sounded in tune; the opposing system in Bach's day was meantone temperament in which keys with many accidentals sound out of tune. Bach would have been familiar with different tuning systems, in particular as an organist would have played instruments tuned to a meantone system, it is sometimes assumed that by "well-tempered" Bach intended equal temperament, the standard modern keyboard tuning which became popular after Bach's death, but modern scholars suggest instead a form of well temperament. There is debate whether Bach meant a range of similar temperaments even altered in practice from piece to piece, or a single specific "well-tempered" solution for all purposes. During much of the 20th century it was assumed that Bach wanted equal temperament, described by t
Francesco Antonio Vallotti
Francesco Antonio Vallotti was an Italian composer, music theorist, organist. He was born in Vercelli, he studied with G. A. Bissone at the church of St. Eusebius, joined the Franciscan order in 1716, he was ordained as a priest in 1720. In 1722 he became an organist at St. Antonio in Padua, would become maestro there in 1730, succeeding maestro Calegari, would hold that position for the next fifty years. Here he would work with another theorist and composer named Giuseppe Tartini. Vallotti died in Padua on 10 January 1780. Vallotti spent a great deal of thought on the theory of counterpoint, his theoretical endeavours would culminate in 1779 with the publishing of his 167-page, four volume work, Della scienza teorica e pratica della moderna musica, just before the end of his life. One of his most cited contributions to theory was his development of a system of Well temperament, known today as Vallotti temperament, one of many systems of instrumental tuning for the accommodation of composition in every key.
The six diatonic fifths F-C-G-D-A-E-B are all tuned 1/6 of a comma flat, while the remaining six fifths B-F#-C#-G#-D#-A#-F are all tuned pure. This leads to thirds quite close to pure in the'home' keys of F, C and G major and in D, A, E and B minor. 1/6 comma extended. On the other hand, there were impure Pythagorean thirds in the remote keys of B, F# and C# major and in G#, Eb, Bb and F minor; the keys in between progressed from meantone to Pythagorean as the number of accidentals increased, with Eb and A major having thirds the same width as in modern 12-tone equal temperament. In effect, the'simpler' the key, the closer it is to meantone intonation, the more remote the key, the closer it is to Pythagorean intonation, the'middle' keys are similar to equal temperament. Young temperament No.2 has the same structure as Vallotti's, except that there the split happens at C rather than F: that is, the block of fifths C-G-D-A-E-B-F# are all tuned 1/6 of a comma flat, F#-C#-G#-D#-A#-F-C are all tuned pure.
Vallotti's extant compositions are sacred in nature. They include: Responsorial for four voices accompanied by harpsichord Responsorial for sabbato sancto Responsorial for coena dominiMany of his works remain only in manuscript; these include: 12 Introits for 5 and 8 voices 24 Kyries, 24 Glorias, 21 Credos for 4 and 5 voices 68 Psalms for 2 and 8 voices and instruments 46 Hymns 10 Responsorials 3 Dies Irae for 4 voices and instruments 2 Pange lingua 15 Tantum ergo 2 Te Deum 2 De profundis 1 Sepulto domino and other compositionsHe orchestrated 43 sacred pieces by his former master Calegari, an Introit in 5 voices by Porta. This page incorporates material from the Italian Wikipedia article as of 23 November 2006. Biography at "Here of a Sunday Morning"Specific
Meantone temperament
Meantone temperament is a musical temperament, a tuning system, obtained by compromising the fifths in order to improve the thirds. Meantone temperaments are constructed the same way as Pythagorean tuning, as a stack of equal fifths, but in meantone each fifth is narrow compared to the perfect fifth of ratio 3:2. Equal temperament, obtained by making all semitones the same size, each equal to one-twelfth of an octave (with ratio the 12th root of 2 to one, narrows the fifths by about 2 cents or 1/12 of a Pythagorean comma, produces thirds that are only better than in Pythagorean tuning. Equal temperament is the same as 1/11 comma meantone tuning. Quarter-comma meantone, which tempers the fifths by 1/4 comma, is the best known type of meantone temperament, the term meantone temperament is used to refer to it specifically. Four ascending fifths tempered by 1/4 comma produce a perfect major third, one syntonic comma narrower than the Pythagorean third that would result from four perfect fifths.
Quarter-comma meantone has been practiced from the early 16th century to the end of the 19th. In third-comma meantone, the fifths are tempered by 1/3 comma, three descending fifths produce a perfect minor third one syntonic comma wider than the Pythagorean one that would result from three perfect fifths. Third-comma meantone can be approximated by a division of the octave in 19 equal steps; the name "meantone temperament" derives from the fact that all such temperaments have only one size of the tone, while just intonation produces a major tone and a minor one, differing by a syntonic comma. In any regular system the tone is reached after two fifths, while the major third is reached after four fifths: the tone therefore is half the major third; this is one sense. In the case of quarter-comma meantone, in addition, where the major third is made narrower by a syntonic comma, the tone is half a comma narrower than the major tone of just intonation, or half a comma wider than the minor tone: this is another sense in which the tone in quarter-tone temperament may be considered a mean tone, it explains why quarter-comma meantone is considered the meantone temperament properly speaking.
"Meantone" can receive the following equivalent definitions: The meantone is the geometric mean between the major whole tone and the minor whole tone. The meantone is the mean of its major third; the family of meantone temperaments share the common characteristic that they form a stack of identical fifths, the tone being the result of two fifths minus one octave, the major third of four fifths minus two octaves. Meantone temperaments are described by the fraction of the syntonic comma by which the fifths are tempered: quarter-comma meantone, the most common type, tempers the fifths by 1⁄4 of a syntonic comma, with the result that four fifths produce a just major third, a syntonic comma lower than a Pythagorean major third. A meantone temperament is a linear temperament, distinguished by the width of its generator, as shown in the central column of Figure 1. Notable meantone temperaments, discussed below, occupy a narrow portion of this tuning continuum, with fifths ranging from 695 to 699 cents.
While the term meantone temperament refers to the tempering of 5-limit musical intervals, temperaments that approximate 5-limit intervals well, such as Quarter-comma meantone, can approximate 7-limit intervals well, defining septimal meantone temperament. In Figure 1, the valid tuning ranges of 5-limit, 7-limit, 11-limit tunings are shown, can be seen to include many notable meantone tunings. Meantone temperaments can be specified in various ways: by what fraction of a syntonic comma the fifth is being flattened, what equal temperament has the meantone fifth in question, the width of the tempered perfect fifth in cents, or the ratio of the whole tone to the diatonic semitone; this last ratio was termed "R" by American composer and theoretician Easley Blackwood, but in effect has been in use for much longer than that. It is useful because it gives us an idea of the melodic qualities of the tuning, because if R is a rational number N/D, so is 3R + 1/5R + 2 or 3N + D/5N + 2D, the size of fifth in terms of logarithms base 2, which tells us what division of the octave we will have.
If we multiply by 1200, we have the size of fifth in cents. In these terms, some notable meantone tunings are listed below; the second and fourth column are corresponding approximations to the first column. The third column shows how close the second column's approximation is to the actual size of the fifth interval in the given meantone tuning from the first column. Neither the just fifth nor the quarter-comma meantone fifth is a rational fraction of the octave, but several tunings exist which approximate the fifth by such an interval. Equal temperaments useful as meantone tunings include 19-ET, 50-ET, 31-ET, 43-ET, 55-ET; the farther the tuning gets away from quarter-comma meantone, the less related the tuning is to harmonic timbres, which can be overcome by tempering the timbre to match the tuning. A whole number of just perfect fifths
Diatonic and chromatic
Diatonic and chromatic are terms in music theory that are most used to characterize scales, are applied to musical instruments, chords, musical styles, kinds of harmony. They are often used as a pair when applied to contrasting features of the common practice music of the period 1600–1900; these terms may mean different things in different contexts. Diatonic refers to musical elements derived from the modes and transpositions of the "white note scale" C–D–E–F–G–A–B. In some usages it includes all forms of heptatonic scale. Chromatic most refers to structures derived from the twelve-note chromatic scale, which consists of all semitones. However, it had other senses, referring in Ancient Greek music theory to a particular tuning of the tetrachord, to a rhythmic notational convention in mensural music of the 14th through 16th centuries. In ancient Greece there were three standard tunings of a lyre; these three tunings were called diatonic and enharmonic, the sequences of four notes that they produced were called tetrachords.
A diatonic tetrachord comprised, in descending order, two whole tones and a semitone, such as A G F E. In the chromatic tetrachord the second string of the lyre was lowered from G to G♭, so that the two lower intervals in the tetrachord were semitones, making the pitches A G♭ F E. In the enharmonic tetrachord the tuning had two quarter tone intervals at the bottom: A G F E. For all three tetrachords, only the middle two strings varied in their pitch; the term cromatico was used in the Medieval and Renaissance periods to refer to the coloration of certain notes. The details vary by period and place, but the addition of a colour to an empty or filled head of a note, or the "colouring in" of an otherwise empty head of a note, shortens the duration of the note. In works of the Ars Nova from the 14th century, this was used to indicate a temporary change in metre from triple to duple, or vice versa; this usage became less common in the 15th century as open white noteheads became the standard notational form for minims and longer notes called white mensural notation.
In the 16th century, a form of notating secular music madrigals in was referred to as "chromatic" because of its abundance of "coloured in" black notes, semiminims and shorter notes, as opposed to the open white notes in used for the notation of sacred music. These uses for the word have no relationship to the modern meaning of chromatic, but the sense survives in the current term coloratura; the term chromatic began to approach its modern usage in the 16th century. For instance Orlando Lasso's Prophetiae Sibyllarum opens with a prologue proclaiming, "these chromatic songs, heard in modulation, are those in which the mysteries of the Sibyls are sung, intrepidly," which here takes its modern meaning referring to the frequent change of key and use of chromatic intervals in the work.. This usage comes from a renewed interest in the Greek genera its chromatic tetrachord, notably by the influential theorist Nicola Vicentino in his treatise on ancient and modern practice, 1555. Diatonic scale on C equal just.
Medieval theorists defined scales in terms of the Greek tetrachords. The gamut was the series of pitches from which all the Medieval "scales" notionally derive, it may be thought of as constructed in a certain way from diatonic tetrachords; the origin of the word gamut is explained at the article Guidonian hand. The intervals from one note to the next in this Medieval gamut are all tones or semitones, recurring in a certain pattern with five tones and two semitones in any given octave; the semitones are separated as much as they can be, between alternating groups of three tones and two tones. Here are the intervals for a string of ascending notes from the gamut:... –T–T–T–S–T–T–S–T–T–T–S–T–... And here are the intervals for an ascending octave from the gamut: T–S–T–T–S–T–T In its most strict definition, therefore, a diatonic scale is one that may be derived from the pitches represented in successive white keys of the piano: the modern equivalent of the gamut; this would include the major scale, the natural minor scale, but not the old ecclesiastical church modes, most of which included both versions of the "variable" note B♮/B♭.
There are specific applications in the music of the Common Practice Period, music that shares its core features. Most, but not all writers, accept the natural minor as diatonic; as for other forms of the minor: "Exclusive" usageSome writers classify the other variants of the minor scale – the melodic minor and the harmonic minor – as non-diatonic, since they are not transpositions of the white-note pitches of the piano. Among such theorists there is no agreed general term that encompasses the major and all forms of the minor scale."Inclusive" usageSome writers i
Arnolt Schlick
Arnolt Schlick was a German organist and composer of the Renaissance. He is grouped among the composers known as the Colorists, he was most born in Heidelberg and by 1482 established himself as court organist for the Electorate of the Palatinate. Regarded by his superiors and colleagues alike, Schlick played at important historical events, such as the election of Maximilian I as King of the Romans, was sought after as organ consultant throughout his career; the last known references to him are from 1521. Schlick was blind for much of his life from birth. However, that did not stop him from publishing his work, he is best known for Spiegel der Orgelmacher und Organisten, the first German treatise on building and playing organs. This work influential during the 16th century, was republished in 1869 and is regarded today as one of the most important books of its kind. Schlick's surviving compositions include Tabulaturen etlicher lobgesang, a collection of organ and lute music, a few pieces in manuscript.
The lute pieces—mostly settings of popular songs—are among the earliest published. It features sophisticated cantus firmus techniques, multiple independent lines, extensive use of imitation. Thus, it predates the advances of Baroque music by about a hundred years, making Schlick one of the most important composers in the history of keyboard music. Records of Schlick's early life are sparse: he lived and worked at Heidelberg, completely destroyed during the War of the Grand Alliance, so no records survive from the time Schlick was born. Linguistic analysis of his writings has shown that Schlick was most from the area around Heidelberg, recent research showed that Schlick was most born into a family of a Heidelberg butcher, whose family name may have been Slicksupp. If Schlick's parents followed the contemporary German custom to name children after the saint on whose day they were born, Schlick must have been born on July 18, St. Arnold's day; as for the year of birth, since Schlick married in 1482 and described himself as "an old man" by 1520, he was born in 1455–60.
Schlick was blind for much of his life, may have been born blind. No documents survive concerning Schlick's apprenticeship. Johannes von Soest and an otherwise unknown "Petrus Organista de Oppenheim" could be his teachers, as could Conrad Paumann, if only for a brief time when he visited Heidelberg in 1472; the earliest mention of Schlick's place of employment is in his marriage contract: in 1482 he married Barbara Struplerin, a servant of Elector Philip's sons, the contract lists him as a court organist. Schlick and his family lived in a house on a path that led to the Heidelberg Castle. Schlick was held in great regard by his superiors. By 1509 he was the highest-paid musician at the court with a salary twice as high as that of the next-best-paid musician, comparable to the salary of the court treasurer. Evidently, this position was established by 1486, when Schlick performed at the election of Archduke Maximilian as King of the Romans at Frankfurt, on February 16 of that year, it was at this election.
In either 1489 or 1490, Schlick travelled to the Netherlands: he alludes to the journey in his preface to Tabulaturen etlicher lobgesang, but his reasons remain obscure. Recent scholarship unearthed evidence of payments to other Electorate of the Palatinate musicians, made by Utrecht authorities, although no mention of the court travelling to Utrecht in 1489–1490 has been found, it is possible that such a journey did happen. An older version of Schlick's motives was that he went to the Low Countries to escape from the plague, ravaging the Heidelberg area. In October 1503 King Philip I of Castile visited Heidelberg, bringing with him a large enoutrage that included the composers Pierre de la Rue and Alexander Agricola, organist Henry Bredemers. Schlick certainly met these musicians, played the organ at the performance of the Mass that took place during Philip's visit; the next known contemporary report that mentions Schlick is from February 23, 1511, when he played at the wedding of Louis V, Elector Palatine and Sibylle of Bavaria.
Nothing certain is known about Schlick's other performances. We know that he was present at one the diets at Worms, either in 1509 or at the famous diet of 1495; the presence of an unnamed Heidelberg court lutenist in Basel in 1509 is documented, as Schlick was an accomplished lutenist, it might have been him. In 1516, Schlick visited Torgau for unknown reasons; the year 1511 saw publication of Spiegel der Orgelmacher und Organisten. The book was published in Speyer. In 1511, Schlick's son Arnolt the Younger pleaded to his father to publish at least some of his music. A few of the biographical
Andreas Werckmeister
Andreas Werckmeister was a German organist, music theorist, composer of the Baroque era. Born in Benneckenstein, Werckmeister attended schools in Quedlinburg, he received his musical training from his uncles Heinrich Christian Werckmeister and Heinrich Victor Werckmeister. In 1664 he became an organist in Hasselfelde. Of his compositions only a booklet remains: pieces for violin with basso continuo, with the title Musikalische Privatlust. Werckmeister is best known today as a theorist, in particular through his writings Musicae mathematicae hodegus curiosus... and Musikalische Temperatur, in which he described a system of what we would now refer to as well temperament now known as Werckmeister temperament. Werckmeister's writings were well known to Johann Sebastian Bach, in particular his writings on counterpoint. Werckmeister believed that well-crafted counterpoint, in particular invertible counterpoint, was tied to the orderly movements of the planets, reminiscent of Kepler's view in Harmonice Mundi.
According to George Buelow, "No other writer of the period regarded music so unequivocally as the end result of God’s work,". Yet in spite of his focus on counterpoint, Werckmeister's work emphasized underlying harmonic principles. Musicae mathematicae hodegus curiosus... Musikalische Temperatur, oder... Der Edlen Music-Kunst... Hypomnemata musica Erweierte und verbesserte Orgel-Probe Die nothwendigsten Anmerckungen und Reglen, wie der Bassus continuus... Cribrum musicum Harmonologia musica Musikalische Paradoxal-Discourse Werckmeister Harmonies ^ George B. Stauffer, Journal of the American Musicological Society, Fall 2005, p. 711. ^ George J. Buelow: "Andreas Werckmeister", Grove Music Online, ed. L. Macy. David Yearsley and the Meanings of Counterpoint. New Perspectives in Music History and Criticism. Cambridge and New York: Cambridge University Press, 2002. Free scores by Andreas Werckmeister at the International Music Score Library Project "Well Temperaments based on the Werckmeister Definition" Bio which contains details of locations of surviving copies of Werckmeister's publications Well Tempered based on Werckmeisters last book Musikalische Paradoxal-Discourse is Equal Temperament see: https://www.academia.edu/5210832/18th_Century_Quotes_on_J.
S._Bachs_Temperament
