# Introduction to European historical tunings and temperaments

By Marc SabatWhen organs began to be used across Europe, they were tuned in pure "PythagoreanPythagorean tuning is an approach to tuning based on stacking pure fifths (i.e. fifths with an exact frequency ratio of 2:3).Visit the link to learn more" perfect fifthsPitches a fifth apart have a 2:3 frequency ratio (in just intonation) or a difference of 700 cents (in 12 tone equal temperament).Visit the link to learn more and fourthsPitches a fourth apart have a 3:4 frequency ratio (in just intonation) or a difference of 500 cents (in 12 tone equal temperament).Visit the link to learn more (frequency proportions 2:3 and 3:4). The first keyboards had only eight notes, B♭-F-C-G-D-A-E-B, following the classical division of a single-string monochord into whole number ratios. By the 13th century, ensembles of singers were commonly expanding the repertoire of pitches, including additional flats and sharps. Organ keyboards were adapted accordingly by adding semitones, extending the chain of fifths until all the wholetones were divided. The presets in Live 12 (and on this site) include two such tunings under the name "12 Tertial...".

A tertialIn Just/Rational Intonation tunings, ratios are often categorized by the prime numbers that constitute them. The Latin-derived terms "tertial" and "quintal," initiated by composer Catherine Lamb, are used to semantically describe the color and function qualities of the specific primes referenced (3 and 5), as well as the chordal and scale structures these primes generate. Higher primes have traditionally been named in similar manner, i.e. "septimal" (7), "undecimal" (11), "tridecimal" (13), etc.Visit the link to learn more tuning with 12 notes does not make a "circle" of fifths. There are 11 pure fifths, but the interval left over, between the 12th pitch in a chain of fifths and the starting note, is smaller than a fifth by an interval called the tertial (Pythagorean) commaA comma is a small difference in pitch between the same interval in different tuning systems.Visit the link to learn more, about 1/8 of a whole-tone. This dissonant sound is sometimes called a "wolfA wolf refers to a noticeably out-of-tune interval in a given tuning system that results from adjusting or "tempering" that interval to fit within a fixed number of notes.Visit the link to learn more fifth." Also, in a tertial tuning major thirds are tuned by a chain of five notes spanning four perfect fifths, for example: C-G-D-A-E. The very brightly beating interval C-E is larger than the harmonic major third (frequency proportion 4:5) by one quintalIn Just/Rational Intonation tunings, ratios are often categorized by the prime numbers that constitute them. The Latin-derived terms "tertial" and "quintal," initiated by composer Catherine Lamb, are used to semantically describe the color and function qualities of the specific primes referenced (3 and 5), as well as the chordal and scale structures these primes generate. Higher primes have traditionally been named in similar manner, i.e. "septimal" (7), "undecimal" (11), "tridecimal" (13), etc.Visit the link to learn more (syntonic) comma, about 1/9 of a whole-tone. Tertial tunings also have four enharmonically spelled "diminished fourths," which happen to be very close to harmonic thirds. Trying to find a tuning compromise that softens the wolf fifth and includes some harmonious thirds inspired many tuning experiments.

The presets in Live 12 includes two early tunings that depart from a strict tertial approach. In the "Erlanger Traktat", the four enharmonic diminished fourths are slightly corrected to make harmonic thirds (so the tuning becomes a quintal-tertial system). This produces two slightly smaller perfect fifths, both very close to the modern 12edo equal tempered scale. Also, the wolf fifth is made slightly less dissonant, but it still howls. The "Henricus Grammateus" tuning goes a bit further to mitigate the wolf, but does not have any pure thirds. The original eight notes (the chain of fifths from B♭ through B) are tuned in pure fifths. Then, by comparing D-F♯ and F♯-B in the tenor register, F♯ is tuned down until the two intervals beat at an equal speed. In effect, F♯ is lowered by about 1/2 a comma from tertial tuning. The remaining notes F♯-C♯-G♯-D♯ are tuned in pure fifths. Therefore, B-F♯ and D♯-B♭ each produce a tempered fifth about 1/2 comma too small: there are two "half-wolves".

As tempered fifths became more commonly accepted, musicians experimented with narrowing the four fifths that span a major third. If each fifth is made 1/4 comma smaller, for example, the resulting major third will be the harmonic ratio 4:5. Variations on this approach came to be known as Meantone tuning, and the presets in Live 12 include many examples. Each version balances the beating and purity of fifths/fourths and thirds/sixths in a slightly different way. In a Meantone system, sharps and flats end up being tuned differently from each other: they can be as much as a quarter-tone apart. The usual tuning included three sharps (F♯-C♯-G♯) and two flats (E♭-B♭). Thus, notes needed for certain keys and chords are not available (for example, there is no A♭ major triad). One of the highlights of Meantone tuning is that it not only produces the harmonic sounds of partial 5°, but also some beautiful septimalIn Just/Rational Intonation tunings, ratios are often categorized by the prime numbers that constitute them. The Latin-derived terms "tertial" and "quintal," initiated by composer Catherine Lamb, are used to semantically describe the color and function qualities of the specific primes referenced (3 and 5), as well as the chordal and scale structures these primes generate. Higher primes have traditionally been named in similar manner, i.e. "septimal" (7), "undecimal" (11), "tridecimal" (13), etc.Visit the link to learn more sounds as well. In the 1500s the flowering of music in Meantone tuning also inspired composers to explore microtonal tunings with subsemitones and more than 12 notes in the octave.

In the late 1600s, the limited repertoire of keys in Meantone led to the emergence of a new wave of tunings that tried to combine "the best of both worlds." By mixing pure fifths and narrow fifths in various ways, it was possible to avoid a wolf fifth and at the same time preserve some of the harmoniousness of Meantone. These tunings, called "Well Tempered", are generally not symmetric or equal in construction. Instead, they seek to color every key in a distinct way. Some keys become harsher and more dissonant, others are more concordant.

In the 19th century, motivated by the writings of Helmholtz, various attempts were made to revisit the approach of the Erlanger Traktat, combining pure fifths and pure thirds. To allow modulation into various keys, more notes were introduced. One of the most interesting theoretical tunings was proposed by Helmholtz himself: he used it on his own pump organ across 24 notes available on two keyboard manuals. It is included in the Live 12 presets as "24 Helmholtz temperament". Each fifth is narrowed by a tiny amount so that eight fifths produce a pure major third, allowing nearly-pure major and minor triads in many keys.

Finally, with the rise of mass-produced concert pianos in the late 19th century and later, the introduction of MIDI 1.0, 12edo equal temperament gradually became an ubiquitous part of concert life and music production. Now, as the new era of MIDI 2.0 approaches, and gestural controllers that can play any pitch become widespread, we are again seeing active exploration across a diversity of tunings with many different notes: EDOs and JI systems, as well as traditional scales from across the world.

## Some musical examples:

Overview of tuning and temperament basics

### Pythagorean/tertial:

- Guillaume de Machaut,
*Messe de Notre Dame*(Ensemble Organum) *Codex Faenza*, performed by Mala Punica

### Meantone:

- Frescobaldi,
*Cento Partite Sopra Passacaglia*(Yoann Moulin) - Studio31 Basel (microtonal music from the 1500s)