Temperaments - Historical and Technical Overview

FOOTNOTES:

1 [00:23] "Tuning and temperaments in the Renaissance" (two parts) [link], "Just intonation in the Renaissance [link], and "Emilio de' Cavalieri's mysterious enharmonic passage" [link].

2 [05:25] One of the definitions of a “tempered interval” is that it cannot be expressed with a ratio. In Pythagorean tuning the fifths are pure - not tempered (ratio 3:2), and the thirds are not pure, but can still be expressed with a ratio (81:64). The tempered fifths of the equal or meantone temperament for example cannot be expressed with ratios and therefore referred to as “tempered”.

3 [05:53] According to Mark Lindley, in late 14th century and early 15th century, the “wolf” was put between B and F#, and thus creating several thirds that are closer to pure. See Grove music online, “Temperaments”, no. 2.

4 [06:38] The main source that was a starting point for most theorists in the medieval and renaissance period was Boethius. He translated Greek texts about music theory and made them accessible for later generations. In his text, the 'ditonus' as a double-tone is assigned to the ratio 81:64. See Boethius, De institutione musica. Also Franchino Gaffurio, Practica musice and the treatises mentioned in footnote 8.

5 [07:02] If you are not familiar with the term mi and fa notes, see our episode about Solmization.

6 [08:25] See for example Jan Herlinger’s article “Marchetto's Division of the Whole Tone”, Journal of the American Musicological Society, Vol. 34, No. 2 (Summer, 1981), pp. 193-216 [Jstor].

7 [09:25] See our video "Just intonation in the Renaissance [link].

8 [09:29] This approximation needs further explanation, because it goes much further than fifths and thirds. The 'utopia' of using pure intervals for polyphony is visible in most of the main treatises of the 16^th century: Vicente Lusitano, Introduttione facilissima, et novissima, di canto fermo, figurato, contraponto semplice, et in concerto, ... Venice 1553, Gioseffo Zarlino, Dimostrationi harmoniche, Venice 1571, Nicola Vicentino, L'antica musica ridotta alla prattica moderna, Rome 1555, Francisco de Salinas, De Musica libri septem, Salamanca 1577. Usually, the consonant intervals (octave, fifth and thirds) are expressed as ratios, see pure intervals in the glossary. Using these intervals in a polyphonic score leads to the melodically relevant intervals, such as whole tones and semitones. But these intervals (see table in article pure intervals in the glossary) describe only moments in a score, you can apply them between two written notes, not to the whole score at once. Using all these intervals leads to many many different pitches within a polyphonic score, many more than what we can notate. Meantone temperament is a very efficient solution to reduce the overall number of pitches to just the ones we can actually write. Most importantly, we neutralise the two whole tones: there is just one whole tone in meantone world, and it's exactly between 9:8 and 10:9, which is exactly half a major third (5:4). The two semitones are almost the same as in Just Intonation (16:15 and 25:24), only narrowed by a quarter of the syntonic comma. The table belows shows a comparison of pure intervals and meantone intervals (described as difference to pure intervals in fractions of a syntonic comma, 81:80, which is ca. 21.5 cent). This makes meantone temperament an extremely powerful 'replacement' for Just Intonation, which is a system that can't be transformed into a stable and pragmatic keyboard tuning. In fact, our musical notation and its note naming convention is at its heart a meantone concept: we have 7 letters, and each one and be altered upwards (♯) or downwards (♭).

9 [11:05] The meantone system has conceptually an infinite number of pitches. The chain of fifths can be prolonged 'flat-wards' as well as 'sharp-wards'. Going into the flats, we will hit the pitch called f♭ at some point (which is not the same pitch as e♮ at all!), and we would go on to b♭♭. Going into the sharps, we would continue with f♯♯ after having arrived on b♯. A sensible range is g♭♭ to a♯♯. This range consists of 31 pitches, defined by a chain of 30 fifths. Interestingly, the fifth above a♯♯ (a pitch called e♯♯) sounds almost exactly like g♭♭. In a sense, this line of 30 pitches does form a circle of fifths with 31 pitches. Therefore, the division of the octave in 31 equal parts is a plausible meantone system. You could say that a usual meantone temperament on 12 keys is a subset of these 31 pitches, also called 31ed2 (31 equal division of 2:1, which is the octave).

10 [12:27] Instruments such as trombones, unfretted string instruments and the human voice are capable of being very flexible with their intonation and can create many different pitches. Instruments such as cornettos, recorders, traversos, and some harps are less flexible but still may produce more notes than a 12-notes-per-octave keyboard instrument.

11 [12:45] For meantone systems, this means: you can have a 'flat'-fret or a 'sharp'-fret, but not both at the same time. But usually you would like to have a mix of both alterations on the same fret. It is therefore much more obvious to place the frets regularly, creating some sort of equal temperament. There are many sources out there describing this. It's also well documented on period paintings that are detailed enough to see the fret positions.

12 [12:50] A very detailed source talking about the problem of colliding systems within instrument groups is Ercole Bottrigari, Il desiderio, overo de' concerti di varii strumenti musicali... Venice 1594.

13 [13:20] One of the ways that this transition was manifested is by the use of major thirds on the final chords of pieces. When Pythagorean tuning is used, ending a piece with a sounding major third would be very unsettling, but if meantone tunings are considered it makes more sense. Towards 1500 more and more pieces are found to have major thirds in their final chords. The following summary was made by Baptiste Romain (Le Miroir de Musique) and placed at my disposal during an email conversation: Before 1470: thirds in final chords are anecdotal or found in keyboard sources (statistically, very few occurrences); From 1470 onward: rare occurrences in the 4-part French song, generally with light-hearted texts, rare occurrences in 'si placet' voices; From around 1490: final thirds became common in Italian four-part songs (frottole, laude) and very common from around 1500; In Franco-Flemish counterpoint (motets, chansons, mass music) final cadences with octave and fifth remain the rule until 1520-30; Franco-Flemish composers born in the 16th century use final thirds as a possibility. For instance, of Manchicourt's 29 chansons (printed in 1554), only half of them have a final third. He was born around 1510.

14 [14:00] See our translated edition, p. 8 [link] and our episode about that treatise [link, 19:20]

15 [14:54] The terminology might be a bit confusing: some regard meantone temperaments as “irregular” due to their small and big semitones. When we use the term “irregular temperaments” we mean that the fifths are tempered irregularly, as opposed to globally as in the case of Phythagorean and meantone tunings.

16 [16:58] In Rousseau's description of the temperament ordinaire (some sort of well-tempered temperament), he sees the necessity for the irregularity of interval qualities within the circle of fifths as an advantage. Because it offers different grades of expressivity to the player, it is superior to any other system. See Denis Diderot, Encyclopédie, ou dictionnaire raisonné des sciences des arts et des métiers..., 1761-1766, article "Tempérament" (volume XVI, p. 56) by Jean-Jacques Rousseau.

17 [17:42] Werckmeister, title page of "Orgelprobe" 1681: “Unterricht, Wie durch Anweiß und Hülffe des Monochordi ein Clavier wohl zu temperiren und zu stimmen sei”.

18 [17:58] Many examples are brought in the book “How Equal Temperament Ruined Harmony” by Ross Duffin (2006).

19 [18:50] Friedrich Wilhelm Marpurg, “Versuch über die musikalische Temperatur” (1776) [link]. Quoted in Grove, “Temperaments”, no. 8.

ERRATA:

• In minute 06:00 we say that the "wolf" in Phythagorean tuning is larger than pure, but it is in fact the opposite. However, the "wolf" in Meantone tunings is indeed larger than pure.

TEMPERAMENTS GLOSSARY - by Johannes Keller

Temperament

A definition of interval sizes that are used to tune an instrument. Usually a temperament is described with the help of a circle of fifths, describing the exact sizes or musical qualities of the fifths. Another useful model to describe temperaments are linear systems, for the contexts where a closed circle is not required or appropriate.

Pure Intervals

Musical intervals can be described as string length ratios. If you pluck two identical strings (as it is the case on a monochord), they will produce identical pitch. If you change the length of one string, the pitch changes accordingly. The pitch difference can be expressed in the ratio of the two string lengths.

The fundamental intervals in Western music can be expressed with simple ratios or fractions.

This table shows some differences between our modern interval naming conventions and approaches from the 16^th century. You might find any of these interval names in historical contexts. They usually refer to pure intervals.

The diesis is the difference between the diatonic and the chromatic semitone (semitono maggiore and semitono minore). The syntonic comma is the difference between the two whole tones 9:8 and 10:9. Also, it is the difference between a ditono (a double tone, twice 9:8, also called a Pythagorean third, 81:64) and a pure third, 5:4.

The pure consonances consist of the numbers 6:5:4:3:2:1. These numbers can be generated by combining prime numbers up to 5 (2, 3, 5). This limitation these prime numbers is also called limit-5. The Pythagorean tuning is a limit-3-system, because it is built with only the numbers 2 and 3. A limit-7-system includes the consonant interval 7:4 (the natural seventh), which is practically absent in Western music, but very important in other cultures.

Tempered Intervals

See also Pure Intervals. Tempered intervals share their musical identity with pure intervals, but they are slightly different. They can't be expressed with a ratio, but only with an irrational number. Example: the Pythagorean third is not a tempered interval, because it can be expressed as a ratio, 81:64 (the product of two whole tones 9:8, or four fifths 3:2). The meantone whole tone can't be expressed with a ratio, because it is the equal division of the major third 5:4, which can only be expressed as square root of 5:4. Therefore, the meantone tone is considered a tempered interval.

There is a threshold when a tempered interval turns into something 'out of tune'. This threshold is flexible and can't be defined objectively because it depends on cultural context, training, practical environment and individual preferences.

Comma

A comma is usually considered a very small interval. There are many different commas that were defined at some points in history. The most important one in the context of tuning and temperaments are:

• the syntonic comma, 81:80. It is the difference between a Pythagorean third (81:64) and a pure third (5:4) [in the video this difference is demonstrated on 06:30]. It can also be found in other places, for example the difference between the large and the small whole tone in Just Intonation (9:8 and 10:9).
• the Pythagorean comma, 531441:524288. It is the difference between 12 fifths and 7 octaves [on the video this is demonstrated on 02:35]. It is the 'enharmonic interval' in Pythagorean tuning, for example the interval between d♯ and e♭.

The diesis is not considered to be a comma, but a 'proper interval'. It is the difference between three major thirds (5:4) and an octave (2:1), which is 128:125. This difference is the 'enharmonic interval' in Just Intonation and in meantone temperament, where it can be found between d♯ and e♭ for example.

Throughout music history, the term comma has caused many misunderstandings and errors. It's always a good idea to treat texts using comma with caution, and to check the specific context.

Equal Temperament

There is no clear definition of this term. Usually it's used to describe the modern standard temperament that divides the octave into 12 equal parts (semitones). In a more literal understanding, equal temperament describes a temperament, in which all the intervals with identical names have identical sizes. In that sense, also meantone and Pythagorean tuning could be understood as equal temperaments. But this is not how the term is used usually.

Unequal temperaments are temperaments that use a closed circle of fifths with slightly different sizes. These temperaments are also called well-tempered. They are explicitly not equal. Although in modern English, well-tempered and equal temperament are often used as synonyms, it is not correct, they refer to distinctly different concepts. It would be plausible to see equal temperament as a special case within the family of well-tempered systems.

Pythagorean Tuning

A tuning system that is based on pure fifths. All the other intervals are defined by the combination of fifths. This can be modeled with a linear system, which is basically a chain of fifths, unfolding in two directions. Whatever leaves the octave (the identity interval of this system) will be moved back into the range of an octave. For example: the interval of a whole tone can be constructed by combining two fifths. This produces the interval of a ninth, which is larger than an octave. By reducing the newly found interval by an octave, we get the tone as a result (9:8). The minor third can be constructed with three pure fifths accordingly. Four fifths produce the major third. Any number of fifths generates a unique interval that is added to the system.

This principle was used to set up instruments throughout music history. It was particularly important until the 16^th century. As a keyboard tuning system it is rarely found today, but it is still present. You could argue that the open strings in a string quartet (c, g, d, a, e) are a Pythagorean system, if the players tune their strings in pure fifths. The major third c e (between the viola and the violin for example) will then be a Pythagorean third.

In contemporary intonation practice, the expression 'Pythagorean leading note' is sometimes used. In a Pythagorean system, the diatonic semitone (which is functionally a leading note) is significantly smaller than a semitone in equal temperament. 'Pythagorean leading notes' are borrowed from Pythagorean tuning, but of course used in a context that is much closer to equal temperament than an actual Pythagorean system.

Meantone temperament

Meantone temperament is a concept to organise all notes that are part of our notation system in a way that they have unique pitches. As opposed to equal temperament or well-tempered systems, where all the black keys have ambiguous notation (they can be either a sharp or a flat alteration of a white key), each note has its very own pitch.

Meantone intervals are close to Just Intonation intervals. Historically, the aesthetic context of meantone are pure intervals that are compromised due to mathematical problems caused by the nature of polyphonic structures. See pure intervals.

In practice, the nucleus of a meantone temperament on a keyboard instrument is usually a major third constructed by four fifths. These fifths are narrowed in a way that they have a similar quality and that the third is consonant (could be pure or close to pure). All the other notes are tuned by using the same fifths, creating more consonant thirds.

One of the earliest treatises describing something we could call meantone today is in Pietro Aron's Il Toscanello (Venice 1523) [link]. It is practical, doesn't include any numbers or formulas, but describes what a harpsichord player needs to do in order to prepare his instrument.

Irregular Temperaments

Irregular temperament is usually used as a synonym for well-tempered temperament or unequal temperaments. They are based on a closed circle of fifths, but intervals with the same name can have slightly different sizes. Today, some very specific descriptions of irregular temperaments are used in the context of historical performance practice. Usually, these temperaments are referred to by the name of the author of a treatise where the description can be found. In some cases, these descriptions are exact in a mathematical sense, making it easy to translate to a mathematical model that can be used to program a tuning machine. Others are a bit more vague and need interpretation.

Here is a list of temperaments that are often used in today's performances of Baroque music:

• "Vallotti": based on Francesco Antonio Vallotti, Della scienza teorica e practica della moderna musica. The description of the temperament is located in book 2, which was never published in the 18^th century, but only much later in this edition: G. Zanon, Trattato della moderna musica, Padova 1950. There are different interpretations of the original descriptions in use today.
• "Young": this is a very similar outcome as "Vallotti", but in a transposed form, and with a very different reasoning. The temperament is found in a text about physical experiments with sound and light: "VII. Outlines and Experiments and Inquiries respecting Sound and Light." in Philosophical Transactions, The Royal Society, Vol. 90, p. 106-150. The description of the temperament is on p. 145. See also a graphical comparison to other systems on plate VII (fig. 53), with an explanation on p. 150 [link].
• "Werckmeister": Andreas Werckmeister described many ways of tuning organs, in different treatises. The "Werckmeister Temperaments" that are well known today are mainly based on a monochord scale printed in Musicalische Temperatur, Quedlinburg 1691 (link). There is a comparative scale with six systems labelled I, II, III, IV, V, Today, a different numbering system is usually used, labeling III as I, IV as II and V as III. In Musicalische Paradoxal-Discourse, Werckmeister describes something like equal temperament in words, see p. 111 [link].
• "Neidhardt": This family of temperaments are described in detail by Johann Georg Neidhardt in the treatise Sectio Canonis Harmonici (Königsberg 1724) [link]. Starting on p. 12 he described the various temperaments as fitting to a "Hof" (court), "grosse Stadt" (large town), "kleine Stadt" (small town) and "Dorf" (village). In his treatise Gänzlich erschöpfte Mathematische Abtheilungen des Diatonisch-chromatischen temperierten Canonis monochordi (Königsberg and Leipzig, 1734) [link] he adds more temperaments and comparisons and gives slightly different definitions.
• "Kirnberger": This group of temperaments are found in a treatise by Johann Philipp Kirnberger, Die Kunst des reinen Satzes in der Musik (Berlin 1774) [link] and other treatises.

Credits:

Created by Elam Rotem and Johannes Keller, May 2020

Many thanks to Marc Lewon, Corina Marti, and Anne Smith.