Author Topic: Useful insight into 17th 18th century temperament - major and minor semitones  (Read 2381 times)

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David Pinnegar

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Beyond Temperament

Non-keyboard intonation in the 17th and 18th centuries

Bruce Haynes

İ Bruce Haynes, 2006

My system is not based on any keyboard temperament; rather, it displays the sounds found on unre­stricted in­struments like the cello, violin, etc., that can play purely in tune...

(G.P. Telemann, "Neues musicalisches System," 1743/44)

"Temperaments" are closed systems designed to help make the intonation of instruments with immovable pitch (like the or­gan and harpsichord) convincing. But singers and players of stringed and wind instruments have no such limitations -- "temperament" is too rigid a concept to apply to them.

Since keyboard temperaments have been studied and dis­cussed for some time,1 it seems odd that the intonation of singers and orchestral in­struments has had very little at­tention.2 It is a subject that is much harder to treat quantita­tively, as it depends so much on context. Play­ing "in tune" is a relative and very personal af­fair, and no set of rules or abstractions from prac­tice can possi­bly encom­pass its complexities, or sub­stitute for an alert ear and a will­ing spirit. But certain basic as­sumptions of a singer or vio­linist in the 17th and 18th cen­turies concern­ing intona­tion were quite different from ours, and an understanding of them is not only useful in everyday ensemble work, but adds an unexplored expressive element to baroque and classi­cal performances. Ultimately, using the available historical information, early musicians must work out this question for themselves.3 The second part of this article there­fore pre­sents ex­tensive extracts from original sources on non-keyboard tuning.4

Historic expedients to the tuning problem

It is a troublesome physical fact that it is not possible, either in the­ory or practice, to com­bine both pure fifths and pure major thirds in the same tuning system. A series of four pure fifths placed above each other (for instance, C-G, G-D, D-A, A-E) will produce a ma­jor third (C-E) considerably wider than pure. This is called Pythagorean tuning, a tuning commonly used in the Mid­dle Ages; the fifths are pure, which means the thirds are large --- larger even than in equal tempera­ment.5 A differ­ent system, meantone temperament, be­came com­mon by the mid­dle of the 15th cen­tury, in response to the need for better thirds. Meantone favors thirds: in order to get them low enough, the fifths must suffer by be­ing tuned small.6

Because of its one great advantage, practicality, equal tem­perament had some adherents even in the 18th century and be­fore, but the attitude of one writer of the time was proba­bly typical: it produced, he wrote, a "harmony extremely coarse and dis­agreeable."7 Sauveur in 1707 said equal tem­perament " used [only] among the least able instrumen­talists, because it is simple and easy."8

By contrast, the most common tuning of the time was de­scribed by a number of writers, including Telemann and Quantz, and was engagingly summarized by the singer and musi­cal theorist Pier Francesco Tosi, who wrote in 1723:

Everyone knows that there is a Semitone Major and Mi­nor, because the Difference cannot be known [ie. played] by an Organ or Harpsichord, if the Keys of the Instrument are not split. A Tone, that gradually passes to another, is di­vided into nine almost imperceptible Intervals, which are called Comma's, five of which con­stitute the Semi­tone Major, and four the Minor....If one were continually to sing only to those above-men­tion'd Instruments [the organ and harpsichord], this Knowledge might be unnecessary; but since the time that Composers introduced the Custom of crowding the Opera's with a vast Number of Songs accompanied with Bow In­struments, it becomes so necessary, that if a So­prano was to sing D-sharp, like E-flat, a nice Ear will find he is out of Tune, because this last rises. Who­ever is not satisfied in this, let him read those Au­thors who treat of it, and let him consult the best Performers on the Violin.9

Among Quantz's many comments on tuning, he explained that

What led me to add another key not previously used on the flute was the dif­ference between major and minor semitones.... The ma­jor semitone has five com­mas, the minor only four. For this reason, Eb must be a comma higher than D#.

From our perspective in the late 20th century, we are intro­duced here to two rather startling concepts:

1) the existence of major and minor semitones (a D# dif­ferent from an Eb, for in­stance); and

2) the possibility, therefore, that on some notes the harpsichord or or­gan might be tuned dif­ferently than the other members of an instrumen­tal ensem­ble.

A system that differentiates be­tween half-steps, according to their harmonic function, sug­gests refinements unknown to our ears, which have grown accustomed to a mere twelve notes to the octave. But as far as Quantz was con­cerned in 1752,

Appreciation of [this difference between flats and sharps] is needed by anyone who wants to develop a re­fined, exact and accu­rate ear in music.

Modern players usually raise sharps and lower flats to en­hance their melodic function as leading, or "tendency" tones. This practice has its roots at the be­ginning of the romantic period with the rise of equal tem­perament,10 and is the reverse of the normal practice of 17th- and 18th-century musicians, for whom leading tones were low. Our contemporary preoccupation with melody is apparently recent; a stronger harmonic ori­entation and more "vertical" awareness naturally tended to favor the pure ma­jor third (which is much smaller than the beat­ing, unresonant equal-tempered one).

The pure third is an interval that is both natural and very satisfying to play, and indeed most modern musicians seem to gravitate towards it, especially string players tuning to their open strings. But pure fifths are even easier and yet more tempting to tune on a stringed instrument. Since the end of the 18th century, therefore, fifths have usually won out over thirds in string intonation (cf. the Pythagorean sys­tem, with its perfect fifths and high thirds).11 Rameau in 1737, Quantz in 1752 (17/vii/4) and Sorge (1744:53) indi­cated that some violinists in their day were also inclined to pure fifths, but they considered this a mistake and asso­ciated it with poorer players.12 They rea­soned that a violin tuned to per­fect fifths would be out of tune with the harp­sichord or or­gan, but the deeper implica­tion was that it would also be unsuited to the general into­nation system of the period. As John Hind Chesnut wrote (page 271):

Modern intonation prac­ not appropriate if our goal is to play Mozart's music as he himself wanted it played. The quasi-Pythagorean "expressive" or "func­tion­al" intonation of nineteenth- and twentieth-cen­tury non-keyboard instruments is particularly for­eign to the tradition in which Mozart stood.

Tempering and "intoning justly"

We are not dealing here with a closed tuning system based on a cir­cle of fifths like a keyboard temperament. This says nothing about the naturals; it implies a general system but does not indicate any specific tempera­ment.

Quantz wrote

...the other instruments play [the notes] in their cor­rect ratios, whereas on the harpsichord they are merely tempered.

"Merely tempered" is the key phrase here. If we use both D# and Eb, G# and Ab, etc.), we will need more than twelve notes in an oc­tave. These different enhar­monics are avail­able for the singer or vi­olinist, who is able to adjust intona­tion while performing, but key­board players (unless they have in­struments with split keys) are forced to resort to compli­cated sys­tems of tempera­ment.

"Temperament" in this sense means "compromise," an expedient that attempts to make the best of the fact that only one note can be played when two are needed.13 It is an artifice that gives the illusion that a keyboard instrument is as well in tune as the other instru­ments when played by musi­cians with the "refined, exact and accu­rate ear" of Quantz's time.

For non-keyboard instruments, in fact, "temperament" is not even possible. Without a fixed tuning, intonation is influ­enced by technical situations, subjective percep­tions, even differences in dynamics.14 Players of such in­struments are incapable (even if they wanted it) of the level of consis­tency in into­nation implied by a temperament.15

But although they are not bound by any closed system, it would still be useful to see how original de­scriptions of their tuning might be roughly fitted into a keyboard system, since they normally per­form with harpsichords or organs. A key­board temperament can also operate as a frame of refer­ence or model, from which singers and players of instruments with flexible intonation can occasion­ally depart in the context of the moment. Ideally, a "syner­getic" rela­tionship will exist, in which the keyboard is first tem­pered as closely as possible to the physical and musical needs of the other instruments, who in their turn refer back to it for guidance.

By definition, we can deduce that a tuning that distin­guishes between enhar­monic pairs, with sharps being a comma lower than flats, does not resemble either equal temperament or the Pythagorean system (in which sharps are higher than flats). If it is a system at all, it must be closer to ei­ther just into­nation or some form of meantone.

Just intonation "has always had a kind of fatal fascina­tion for musicians because of the purity within the basic scale of the tonic, subdominant, and dominant chords, and of cer­tain melodic in­tervals"16 that can be easily tuned to the open strings. Some early violin tutors indicate the use of a kind of just into­nation, flexibly applied in a limited way (see Rameau 1726 and Tartini 1754:100-101).17 But just into­nation is a kind of "holy grail" that is impossible to apply con­tinuously,18 although ingenious attempts at it have been made.19 As Barbour put it,20

The bulk of the vio­linists [in c1730] were probably still accustomed to the just thirds and greatly flat­tened fifths of meantone temper­ament.

The line between just and meantone need not, of course, be strictly drawn on instruments whose tuning is not fixed.21 Some string players be­gin with open strings tuned to some­what narrow fifths and tune in­tervals purely to the open strings. Wind players, too, tend to ad­just long notes purely. Of any consistent system, this tun­ing most resem­bles "1/4-comma" meantone ("meantone" in its strictest sense), in which thirds are pure (as in just intonation) and fifths are smaller than pure by 1/4 of the syntonic comma.

But the difference between enhar­monic pairs in 1/4-comma meantone is much greater than that specified by early sources (41 cents as op­posed to 22).22 The consistent use of 1/4-comma meantone is not, therefore, what they de­scribe. Georg Muffat (1698) even warned violinists to resist the temptation to play leading notes too low (sic).

Tosi said that "A di­vided into nine...Intervals, which are called Comma's, five of which constitute the Semi­tone Major, and four the Minor." (The "comma" referred to here is just un­der 22 cents wide.)23 An example of a major semitone would be C-Db, a minor would be C-C#. Since the first is five commas and the second four, the difference be­tween them is one comma.

An octave, as Francesco Geminiani wrote in 1751, can be di­vided "...into 12 Semitones, that is, 7 of the greater and 5 of the lesser." Since the seven "greater" or major semitones each contain five com­mas and the five "lesser" have four, the octave will consist of a total of 55 commas, or parts. The 55-part oc­tave, as the sources quoted in Part 2 show, was a familiar concept in the 17th and 18th centuries.24 It corresponds to a tem­perament known now as "1/6-comma mean­tone."25

David Pinnegar, BSc ARCS

David Pinnegar

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written sources
« Reply #1 on: September 05, 2011, 10:23:23 PM »
Written Sources

The term "meantone" was not used in the 18th century; in fact, like many com­monly ac­cepted assumptions, musicians were so uncon­scious of alter­natives to a system that in­cluded major and minor semitones, that it had no name at all.26

Among the more interesting descriptions of non-keyboard tun­ing are those by Telemann and Quantz. Sorge (1748:61) said that Telemann's tuning system "cannot be applied to a key­board instru­ment, but it may be rather convenient for the fiddle and certain wind instruments, and is the easiest for singers." Chesnut has pointed out that Mozart also appar­ently distinguished the small and large half steps of a meantone temperament similar to 1/6-comma.27 Major and minor semitones were discussed as late as 1813.28

In his 1707 Méth­ode (206), Sauveur classes instruments ac­cording to their ability to alter their intonation: the voice and vio­lin are in a class in which accurate intonation depends en­tirely on the ear, while the keyboards are in one where no control is pos­sible during playing. The woodwinds fall in an intermedi­ate class, and are among instruments

...on which the pitch is governed by projections, tone-holes or touchpieces, but that can be neverthe­less cor­rected by a sensi­tive ear.29

A number of woodwind fin­gering charts from the end of the 17th to the end of the 18th century confirm the use of higher pitches for flats and lower for synonymous sharps, although the exact difference is not specified. Recorder charts are the most informative, since that instrument's in­flexible blowing tech­nique requires alternate fingerings for correcting intonation. Among the many fingering charts that appeared for the recor­der from 1630 to 1795, the earliest often choose only one of the two enharmonic pairs.30 By 1700, complete chro­matic charts began to appear that distin­guished most pairs, espe­cially the d#/eb1. The most inter­esting charts were those by Johann Christian Schickhardt (c1720), which distinguished g#/ab2,31 and Thomas Stanesby Jr. (c1732), that distin­guished every chromatic note.32

To a lesser extent, traverso charts also offer useful infor­mation; Quantz's additional key indicates that tuning cor­rections were more limited on the traverso than on the dou­ble-reed instruments (to which such keys were never added).33

Although embouchure adjustments make the oboe's intonation relatively flexible, most oboe charts indicate al­ternate fingerings for some sharps and flats, from the ear­liest ex­isting chart (Bismantova, 1688)34 to at least 1816 (Whitely).35 The synonymous pairs that are given the most alternate fingerings are the "left-hand" notes G#/Ab and A#/Bb. The development of double holes on the oboe and recorder has an obvious application for "intoning" enhar­monic pairs. On both instruments they affect the most am­biguous pair, G#-Ab.36

Bassoon fingering charts also distinguished enharmonic pairs.37 Towards the end of the century, however, keys began to be added whose purpose may have partially been to obscure these distinctions.38

Regular vs. irregular temperaments

As Telemann wrote of his tuning system (1743/44), "It estab­lishes a con­tinuous proportional equality between inter­vals..." This implies something similar to a standard "regular" meantone temperament, de­fined by Bar­bour as one "in which all the fifths save one are the same size."39

An in­teresting attribute of "regular" meantones is the ease with which standard transposi­tions can be made, since inter­vals are identical in strategic keys. This would ex­plain how German composers like Bach and Telemann were able to func­tion in meantone while using Chorton and Cammerton si­multaneously.40 "Transposing" instruments were a part of life for German musi­cians at this time. Parts for transpos­ing instruments were notated in different keys than the ma­jority of the parts, because they were "pitched" differently (being tuned to Chorton/Cammerton). The "d'amore" instru­ments and the violino piccolo also had transposed parts.41

Obviously, however notes are notated or fingered, they should be at the same frequency for all the instruments of an ensemble. But the dif­ferences in key among transposing instruments were always either a major second or a minor third. Since in a regular meantone, parallel scales a major second or minor third apart would normally be in­flected identically,42 their notes would have corre­sponded closely.43 Meantone tuning will therefore work with trans­pos­ing instruments, as long as the keyboard instruments in such mu­sic are tuned in reg­ular (rather than ir­regular) temper­aments.44

A model based on a regular temperament is relatively simple and easy to re­member.45 Let us take 1/6-comma meantone as an example. Since most mu­sicians nowadays use a Korg or simi­lar tuning ma­chine, the following table shows where its notes are placed in relation to equal temperament.46

C +5 cents

C# -8 Db +14

D +1

D# -11 Eb +10

E -2

F +7

F# -6 Gb +16

G +3

G# -10 Ab +12

A 0

A# -13 Bb +9

B -4

C +5

As flattened notes become more distant from C, they become gradually higher, whereas sharpened notes become lower. The note Bb, for instance, is 9c higher than in equal tempera­ment, Eb 10c, Ab 12c, etc. Going in the other direction, F# is 6c low, C# 8c, G# 10c, D# 11c, etc.47

Although a regular temperament might have been useful for the keyboard instruments, it is unlikely that other instru­mentalists and singers adhered strictly to it, since the thirds and fifths would not have been completely pure. Ir­regular meantone systems, which favor selected keys at the expense of others, were no doubt also used together with non-keyboard instruments.48 There are clear expressive ad­vantages to these tunings, in which modulations are more colorful.

But no system, regular or irregular, could possibly have been applied rigidly on the flex­ibly-pitched instruments. The regular 55-part octave was no more than a convenient theoretical framework, and it can be used to advantage by present-day musi­cians with either a simi­larly tuned keyboard instrument or one tuned in an irregu­lar temperament such as the well-known "Werckmeister III" or "Tempérament ordi­naire."

Reconciling the keyboard to the other instruments

Discussing intonation, Hubert LeBlanc (p.55) com­mented that

The divine artistry of Mr Blavet consists in adjust­ing [the tuning of his] flute by his manner of blowing. But students of the harpsichord praise the instrument for its into­nation, not perceiving that it is in fact never truly in tune.

It is nat­ural to refer to the keyboard instrument when into­nation ques­tions arise in an ensemble, since it is the only in­strument with a fixed pitch. But fixed pitch has the de­fect of its virtue: when the music changes and demands tun­ing modifications, the key­board can­not adapt as the other in­struments can. It is a case of the immovable object and the irresistible force. There isn't much sense, for in­stance, in tuning the G# of a flute to a harpsichord with an Ab.

A number of sources (among them Sauveur, Tosi, Quantz, Tele­mann, Tartini, Sorge, and Mozart) accepted the fact that key­boards used different systems of tuning than other instru­ments.49 There are suggestions as to how the problem was solved. Huygens, Rameau (1726) and Sorge (1744:53, 1758), all assumed that the melody in­struments should con­form to the keyboard. On the other hand, Rameau (1737), Rousseau (1743) and de Béthizey considered it self-evident that (except for unison notes and final tonics) singers pur­posely ignored the tem­perament of the accompany­ing instru­ments. Quantz (17/vi/20) pro­posed a more diplomatic solu­tion in which the fixed-pitch in­strument also adap­ted to the other in­struments.

In larger settings such as orchestras, a keyboard instrument is consid­erably less audible than the treble melody instru­ments. In the case of the harpsichord, the sound dies away quickly, while pure intervals are sustained by the other treble and bass instruments. De Béthizey and Quantz [16/7] suggest that singers and other play­ers would thus do better to adjust to the violins and oboes rather than the harpsi­chord (cf. also Tosi).50 The problem is more acute for the other bass instruments, since they usually play in unison with a harpsichord or organ.51 There are a number of possi­ble solu­tions.

The idea of a harpsichord or organ with split keys was men­tioned by Tosi and Quantz, and evidently used by Han­del.52 With both D#/Eb and G#/Ab, the keyboard would have good ma­jor triads as far as B and Ab major, making it possi­ble to venture into tonalities with as many as four sharps or flats and still keep the thirds relatively pure.53 For continuo playing, therefore, split keys clearly have a use.54

Barbour (1951:191) suggests that, when key changes were lim­ited, it was a historic practice to retune unsplit keyboard accidentals during a program. It takes about as long to change a D# to an Eb on a harpsichord as to tune a section of violins.55

Another solution is to use two harpsichords, one tuned (for instance) to sharps and the other to flats. Al­ternately, one two-manual harpsichord can be used in this way.56

Where frequent choices between enharmonics are necessary (ie., when a wide range of keys cannot be avoided), another approach is suggested by several sources. Quantz's "good temperament which al­lows either [synonymous flat/sharp] to be endurable" and Telemann's enhar­monic pairs that are "blended together" on keyboard instruments (1767) im­ply ei­ther the use of an irregular meantone or the splitting of the difference between the two or three trouble­making acci­dentals within the framework of a regular meantone sys­tem.57 The latter compromise (which is neces­sarily rather colorless in character) might look on a Korg tuner like this:

C +5 cents

C# -8

D +1

D#/Eb 0

E -2

F +7

F# -6

G +3

G#/Ab +1

A 0

Bb +9

B -4

C +5

This scale is based on 1/6th-comma meantone; C#, F# and Bb plus all the diatonic notes are left in their normal places (see previous table), and the difference be­tween the two am­biguous flat/sharps is split.

Some practical considerations

Quantz gave some advice on practicing intonation in 17/vii/8. He advised (as did Leopold Mozart) the use of a monochord to play­ers of melodic instruments.58

The best manner of escape from [poor intonation] is the monochord, on which one can clearly learn the inter­vals. Every singer and in­strumentalist should become familiar with its use. They would thereby learn to rec­ognize minor semitones much earlier as well as the fact that notes marked with a flat must be a comma higher than those with a sharp in front of them. Without these in­sights one is obliged to depend en­tirely on the ear, which can however deceive one at times. Knowl­edge of the monochord is re­quired especially of play­ers of the violin and other stringed instruments, on which one cannot use the placement of the fingers as an exact guide, as one can on wind instruments.

In our time, we can add that we have all grown up in a pre­vailing atmosphere of approximate equal temperament, making the help of a reference beyond our ears even more necessary. There is a "black box" on the mar­ket that func­tions much like a mono­chord; it is designed to play in any temperament the user wishes.59

A player using meantone as a model is theoretically expected to have alternate flats and sharps available for every note, but in practice, some accidentals are rarely used, since 18th-century music usually stays within the bounds of keys with four flats and sharps. One seldom has to play the notes E#, Fb, Gb, B#, Cb, etc. There are, then, three sets of enhar­monic pairs that are usually am­biguous and need at­tention: Ab/G#, Eb/D#, and Db/C#.60 The other notes (C, D, E, F, F#, G, A, Bb, B) are normally al­ways in the same place.

The less adaptable to different tonalities a temperament needs to be, the purer and richer it can be. Just intona­tion, the theoretical ideal, is practical in only one key; equal temperament works in all of them. When plan­ning con­cert programs, therefore, the choice of tonalities relates directly to the choice of keyboard tem­perament, and vice-versa.


It is hopefully clear by now why the concept of major and minor semitones is fundamental to 18th-century tuning prac­tice, why it can cause problems between the keyboard and the other instruments, and how it logically leads to intonation models that re­semble various temperaments known nowadays as "meantone." A closed system is artificial when applied to strings, winds and voices, but it can help players and singers understand how to work with the "immovable object," a keyboard instru­ment with its fixed pitch, as well as pro­vide them with a frame of reference with which to build a more expressive and "harmonious" structure of intervals.61

David Pinnegar, BSc ARCS


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