Vallotti style arrangement of major thirds

Transcription

1 In Temperament: A Beginner s Guide, whilst hoping that we will be encouraged to listen out for the many positive features of unequal temperament, Stephen Bicknell (1997) notes that due to repertoire requirements most new organs will (and probably should) be tuned to equal temperament. In addition to these equally tuned organs others are designed in a specific historic manner and associated with a particular temperament. Between these groups we find organs where, in the right circumstances, a mild unequal temperament may be appropriate. To investigate this further, and find suitable tunings for two new organs, I decided to construct some temperaments using Vallotti as a model. Vallotti, like most temperaments from the past 300 years, falls into the category of temperaments that are irregular, circulating and well or good. This means that they have different sizes of fifth, the circle of fifths is closed and all keys may be used. The irregular sized fifths produce a variety of key colour with major thirds best in keys with few sharps or flats and worst in more remote keys. The usability of the worst thirds and the quality of the best thirds will be equally considered if the temperament is to be unrestrictive. A smooth progression from the best third (C major) to the worst (F# major) is just one ideal situation, and gives this arrangement of major keys: C G/F D/Bb A/Eb E/Ab B/Db F# Unfortunately, this grouping is only possible if the difference between each pair of thirds varies or if the keys are in different positions. A solution that will keep the order of keys and an equal spacing is to place the best and worst thirds in groups of three. This is the arrangement used by Vallotti: C/G/F D/Bb A/Eb E/Ab B/Db/F# Vallotti style arrangement of major thirds C, G and F D and Bb A and Eb E and Ab B, Db and F# Vallotti ET Some of the references at the end of this article give a good overview of temperaments and clear descriptions of the arithmetic used (Padgham 1986; Lloyd and Boyle 1978). It is sufficient here to know that the Pythagorean comma is the amount twelve pure fifths exceed seven octaves, and the syntonic comma is the amount four pure fifths exceed two octaves plus a major third. In any temperament where the circle of fifths is closed the Pythagorean comma must be completely used in the tempering of some, or all, the fifths. As most circulating temperaments are based on divisions of the Pythagorean rather than the syntonic comma, unless otherwise stated, any mention of comma will refer to that of Pythagoras. A minus sign indicates the value is tempered flat. The emergence of circulating temperaments at the beginning of the eighteenth century enabled musicians to modulate more freely. However, with it came problems. An increase in the number of keys available brought a reduction in the quality of important intervals such as the major third. For ensembles there were clashes between instruments with little flexibility in intonation, and the use of more than one pitch caused complications once modulation was involved. It is no wonder that so many circulating temperaments were devised to alleviate the problems in various ways. After the publication of Werckmeister s third temperament in 1691, many more were constructed in the next century including those by Neidhardt, Vallotti, Marpurg, Kirnberger and at the turn of the nineteenth century, Young.

2 Vallotti Francesco Antonio Vallotti ( ) spent his musical life at the Basilica di St Antonio in Padua first as organist and later the Maestro di Cappella. He claimed to have worked out his temperament by 1728 though the theoretical work in which it appeared, Trattato della scienza teorica e pratica della moderna musica, was only part published in 1779 with the rest in manuscript until 1950 (Duffin, 2000: 2). His temperament has six consecutive fifths tempered by -1/6 comma therefore similar to other temperaments being devised. For example, Neidhardt, and later Marpurg, give temperaments with six -1/6 comma fifths in other arrangements. Because of the number of -1/6 and -1/12 comma temperaments presented by Neidhardt alone it is plausible that many similar tunings could have been tried around this time. For this reason, and because Vallotti was active an musician, it may be unfair to reject his tuning for authentic baroque performance purely because of publication dates. Today, Vallotti is probably the most commonly used unequal temperament for both period and modern instrument ensembles and many highly respected groups use Vallotti exclusively. Apart from issues with publication dates, a criticism of Vallotti (or any irregular temperament) is that the variation of each interval size complicates the pitching of notes for singers and instruments with flexible intonation (Duffin 2000: 3). On this issue Duffin (2007) makes a case for a tuning such as 1/6 syntonic comma meantone but, while there may be fewer different interval sizes in a meantone tuning (two sizes of semitone in 1/6 syntonic comma meantone compared with six in Vallotti) there are only good and bad keys and no key colour. Also, while the meantone intervals may be good for harmony the flatter thirds and leading notes result in a deadening of melody. Irregular temperaments, on the other hand, have a graduation of key colour and a graduation from the harmonic to the melodic. The harmonic is found in the keys where the fifths are most tempered and the melodic in the keys where the fifths are tempered the least. Lincoln s Inn and St Mary le Bow For the new organ at Lincoln s Inn we wanted a temperament with similar key grouping to Vallotti but in a milder manner. The organ has an eclectic specification with a good selection of more romantic stops. As the chapel choir comprises professional singers who perform a wide repertoire a new temperament needed minimal restriction in the poorest keys. A simple variation on Vallotti s arrangement of thirds is achieved by reducing the tempering of the six fifths (F-C-G-D-A-E-B), and tempering the otherwise pure fifths by this amount to take up the comma. The six fifths (F-C-G-D-A-E-B) can be tempered any amount between -1/6 and -1/12 comma. Vallotti Vallotti variation 1 CG -1/6 CG X GD -1/6 GD X DA -1/6 DA X AE -1/6 AE X EB -1/6 EB X -1/6 X -1/12 BF# 0 BF# Y Y = -1/6 - X F#C# 0 F#C# Y C#G# 0 C#G# Y G#Eb 0 G#Eb Y EbBb 0 EbBb Y BbF 0 BbF Y FC -1/6 FC X Using this arrangement, any gradient of temperament is possible from Vallotti through to equal. Six fifths at -1/8 comma (X) and six at -1/24 comma (Y) places the value of the major thirds exactly between Vallotti and equal temperament (see chart of major thirds below). Another way to vary Vallotti is to keep the middle pair of fifths (G-D and D-A) at -1/6 comma and divide the tempering of the two outside pairs of the group (F-C- G and A-E-B) over four fifths instead of two. The four fifths F-C-G and A-E-B can be tempered by any amount of the comma between -1/6 and pure, and the remainder is spread equally over B-F#-C# and Eb-Bb-F. The two fifths at C#-G#-Eb are kept pure. There can be up to three sizes of tempered fifth including those fixed at -1/6 comma. Vallotti variation 2 CG X' GD -1/6 DA -1/6 AE X' EB X' -1/6 X' 0 BF# Y' Y' = -1/6 - X' F#C# Y' C#G# 0 G#Eb 0 EbBb Y' BbF Y' FC X'

3 At Lincoln s Inn the eight fifths (X' and Y') are all flattened by -1/12 comma, so all tempered fifths are -1/6 or -1/12 comma. The values of the major thirds are the same as the first method using six fifths at -1/8 and six at -1/24 comma. Varying X' and Y' can produce different key quality gradients from Vallotti through to the mildest as used at Lincoln s Inn. In the very average acoustic of Lincoln s Inn chapel it is likely that anything milder than this would be too subtle to be perceived. The livelier acoustic, and more classical organ, at St Mary le Bow affected the temperament choice in a different way. Though it had been decided early in the project that the organ would be tuned to Vallotti when we got on site and tuned through we found the temperament to be quite extreme in the building. The organ was retuned so that the temperament is between Vallotti and Lincoln s Inn. The arrangement of fifths is shown below. L Inn St Mary le Bow CG -1/12 CG -1/8 GD -1/6 GD -1/6 DA -1/6 DA -1/6 AE -1/12 AE -1/8 EB -1/12 EB -1/8 BF# -1/12 BF# -1/24 F#C# -1/12 F#C# -1/24 C#G# 0 C#G# 0 G#Eb 0 G#Eb 0 EbBb -1/12 EbBb -1/24 BbF -1/12 BbF -1/24 FC -1/12 FC -1/8 Major Thirds: Vallotti, St Mary le Bow and Lincoln's Inn C G F D Bb A Eb E G#/Ab B C#/Db F# Vallotti SMLB L Inn Neidhardt Though Vallotti seemed to be the ideal model for creating some new temperaments, it is possible one of the many constructed by Johann Georg Neidhardt could have been used either as the tuning or the model. Neidhardt (c ) was in the circle of musicians and theorists that included Werckmeister, Hildebrandt, Silbermann and Bach. His theories and temperaments were used during his lifetime and published in Sectio canonis harmonici (1724) and Gäntzlich erschöpfte, Mathematisches Abtheilungen (1732/4), both of which are available online. Here Neidhardt detailed many new temperaments, some as technical examples and others real temperaments for different situations. From these he offers four main temperaments that considered the musical needs of different places; the village, the small city, the big city and the court. The village temperament is

4 the most unequal, the cities less so, and for the court he proposes equal temperament. Neidhardt gives string lengths for these temperaments, shown as parts of the comma below. Neidhardt Big City 1724 Neidhardt Big City 1732 Neidhardt Sm City 1732 Neidhardt Village 1732 (Small City 1724) (Village 1724) CG -1/6 CG -1/6 CG -1/6 CG -1/12 GD -1/6 GD -1/6 GD -1/6 GD -1/6 DA -1/6 DA -1/6 DA -1/6 DA -1/4 AE -1/12 AE -1/12 AE -1/6 AE -1/4 EB 0 EB 0 EB -1/12 EB 0 BF# -1/12 BF# -1/12 BF# -1/12 BF# -1/12 F#C# -1/12 F#C# -1/12 F#C# 0 F#C# 0 C#G# -1/12 C#G# -1/12 C#G# 0 C#G# -1/12 G#Eb 0 G#Eb 0 G#Eb -1/12 G#Eb 0 EbBb -1/12 EbBb 0 EbBb -1/12 EbBb 0 BbF -1/12 BbF -1/12 BbF 0 BbF -1/12 FC 0 FC -1/12 FC 0 FC 0 The milder temperaments for the big cities enabled a wider range of music to be played, and equal temperament for the court resolved the complication of temperament combined with transposition. As a musician himself (he became Kapellmeister at Konigsberg in 1720) Neidhardt thoroughly considered the multifaceted problem of tuning; a desire for wide ranging modulation, transposition issues, combination with other instruments of fixed intonation, ease of tuning and also the organ sounding well on it s own. For the 1732 versions he shifted all the temperaments, apart from the court, to favour more unequal versions, i.e. the 1724 village becomes the 1732 small city etc. For his big city tunings Neidhardt favours a different arrangement of thirds than Vallotti so the 1724 and 1732 versions are presented alone. Major Thirds: Neidhardt Big Cities C F Bb G D Eb A E B C#/Db F# G#/Ab Big City Big City Neidhardt s temperaments have a good progression through the thirds of the most used keys, and the worst thirds are grouped together. This combines some of the best features of circulating and meantone temperaments in an ingenious way. It is particularly evident in the Small City temperament of 1732, recently used for the tuning of the new Richards, Fowkes & Co organ at St George s, Hanover Square. For this temperament a comparison with Vallotti and Lincoln s Inn can be made.

5 Major Thirds: Neidhardt Small City (1732) C G F D Bb A Eb E G#/Ab B C#/Db F# Vallotti L Inn Small City Young Any discussion of Vallotti s temperament will usually include a mention of Thomas Young s temperament of Like Vallotti, it has six consecutive fifths tempered by 1/6 comma and six pure fifths, but the group of tempered fifths is moved one step around the circle so that G major is the tonal centre rather than Vallotti s C major. Though usually considered to be his intended temperament, Young only suggested this as an approximation for a more detailed temperament he gave that centres on C major and has the arrangement of thirds we started with; C G/F D/Bb A/Eb E/Ab B/C# F# Young gives string lengths for all notes and logarithms for major thirds, minor thirds and the fifths. The information published in 1800 contains error in a string length and a logarithm for one of the groups of fifths, but these values are corrected in the 1807 publication and are very exact. In both versions Young describes the process: It appears to me, that every purpose may be answered, by making the third C : E too sharp by a quarter of a comma, which will not offend the nicest ear; E : G#, and Ab : C, equal [sic]; F# : A# too sharp by a comma; and the major thirds of all the intermediate keys more or less perfect, as they approach more or less to C in the order of modulation. The fifths are perfect enough in every system In practice, nearly the same effect may be very simply produced, by tuning from C, to F, Bb, Eb, G#, C#, F#, six perfect fourths; and C, G, D, A, E, B, F#, six equally imperfect fifths. If the unavoidable imperfections of the fourths be such as to incline them to sharpness, the temperament will approach more nearly to equality, which is preferable to an inaccuracy on the other side (Young 1807: 551-2). From the description, logarithms and string lengths, the values Young gives for the fifths are; four pure, four -3/16 Syntonic comma, and the remainder of the Pythagorean comma distributed over the last four. Young S comma CG - -3/16 GD - -3/16 DA - -3/16 AE - -3/16 EB BF# F#C# 0 0 C#G# 0 0 G#Eb 0 0 EbBb 0 0 BbF FC

6 Major Thirds: Young C G F D Bb A Eb E G#/Ab B C#/Db F# Vallotti SMLB Young We have found the two temperaments based on Vallotti to be successful. The St Mary le Bow temperament has been used again on the organ for St John s College, The University of Queensland, and it is likely that the new organ for All Saints Chapel, Rikkyo University, will be tuned in the Lincoln s Inn temperament. The chapel musicians at Rikkyo are very keen to have an unequal temperament but, as the main purpose of the organ is to accompany the chapel choir in traditional Anglican services, we feel a deviation from equal should be small. This short account mentions only a few circulating temperaments from the wide and varied range available. For any objectives, such as a particular progression through keys, usability in a wide repertoire, the acoustic, or historical accuracy, there are many compatible temperaments. In addition to those already known, new temperaments for authentic performances are still being created and existing ones tweaked. For any organ where an unequal temperament may be desired there is certainly plenty of choice. References Bicknell, S. (1997) A beginner's guide to temperament, [Online], Available: [30 Mar 2013]. Duffin, R. W. (2000) Why I Hate Vallotti (or is it Young?), [Online], Available: [30 Mar 2013] Duffin, R. W. (2007) How equal temperament ruined harmony (and why you should care), New York: W. W. Norton and Co. Lloyd, L. S. and Boyle, H (1978) Intervals, scales and temperaments, London: Macdonald and Janeʼs Padgham, C. A. (1986) The Well-Tempered Organ, Oxford: Positif Press Young, T (1807) A Course of Lectures on Natural Philosophy and the Mechanical Arts. Vol 2, [Online], Available: [31 Mar 2013]