Tuning and Temperament

Einleitung
The temperament is a vital part of the intonation procedures of the pipes and the wind influence and belongs to the most relevant features of an organ. It is a genuine part of the sound concept. As far as ever possible it needs to be examined and documented during any restoration of an historical instrument establishing a fundament for a reconstructed temperament.

Here 's to explain some simple features of temperament issue. Interval ratios are given in „cent“ as defined by Alexander Ellis (1 cent = 1/100 equal temperament semitone. 1 octave = 12 semitomes = 1200 cents).

Basics
Tuning (as opposed to temperament) implies the tuning of physically pure intervals - insofar as possible under the following limitations. Details of specific temperament are beyond the epoch framework described here. This applies to the pythagorean temperament, based on eleven pure fifths (1 pythagorean pure fifth, frequency ratio 2:3 = 702 cents), disappearing from practical use during the 16th century.

Nonetheless to avoid any misundertandings the term "well tempered tuning" oder "well temperament" is to be avoided!

Fundament of temperatures
Every keyboard temperature (with 12 semitones per octave) must deal with the surplus "pythagorean comma" resulting from the sum of 12 pure fifths (12x2 cents =24 cents; 1224 cents vs. 1200 cents) by reducing some interval somewhere in the circle of fifths; since a sum of 12 fifths „overshoots“ a sum of octaves by approximately a quarter of a semitone.
The pythagorean comma could eg be reduced from a single fifth, or in equal or unequal distribution from more than one fifth; but this prodecure is in any way unavoidable!

Another important fundament is defined by the formation of a major third from a sequence of four fifths (eg: c-g-d-a-e, or two larger pure major seconds „ditonus“ 2 x (8:9)) . This results in a third too sharp with a defined surplus the"syntonic comma", (408 cents vs. pure third, frequency ratio 4:5 = 386 cents)

The syntonic comma results in a third audibly sharp by about 1/5 semitone (24 cents). Although these differences look rather tiny, they can result in audible differences particularly in addition.

To add to the confusion this too sharp major third is commonly called"pythagorean" major third. It is considered a limit of tolerable intervals. „Well-tempered" temperaments might approach that limit but won't overstretch it. To achieve a pure major third the four fifths defining it must be reduced by the surplus of the syntonic comma in som (way.

The procedures of dividing and subtracting these comma surplusses are unavoidable. But there is not the one and only solution of these dilemmas so in theory there are manifold and innumerable temperament variations.

History
in history three completely varied temperament system have evolved for different achievements.

Mean-tone temperament
Mean-tone temperaments try to include so many pure or almost-pure thirds as possible. This achievement of pure thirds in harmony is rated so high that heavily flattened fifths are tolerated and the big resulting dilemma of heavily reduceded fifths shall be accepted and the relativily strong reduction is shifted to one or two fifths heavily out of tune called „wolves“. The "wolf" howls in the circle of fifths commonly on the fifth g#-e flat, or nearby. This wolf placement results in good and fairly equal sounding chords on the opposing side of the circle with the opposing fifth d-a as central fifth.

As a central note in the old modal system ("Primus Tonus") d has an important position Already the most common pythagorean temperament put the first almost pure third over d.
The tone d, and its fifth d-a mark the axis of the most common mean tone temperament (from d-a it is five mean tone fifths to the „wolf“ g#-e flat. The d' also marks the symmetry axis of the long prevalent manual compass with short octave: CDEFGA-c'''. this visual symmetry was rated so highly that the additional basses F# and G# were added even after 1700 als split keys in new keyboards to maintain the visual symmetry. Only the further additions of D# (E flat) and C#, usually in connection with abandoning mean tone temperaments required the fully expanded bass octave.

Those „wolf“ intervals containing those as a whole or in in smaller parts are considered musically unsuitable while all the other may be very useful or even pure. Pitches are clearly defined, "enharmonic changes" are impossible.

For example an "e flat" can not be used as a "d sharp" asf.

The most familiar mean-tone temperature is based on eight pure major thirds, to achieve this the synotnic comma (s.above) is quartered, every fifth in the chain of fifths leading to a pure major third is reduced by a quarter syntonic comma.This terperature is also commonly though not quite correctly called „pure thirds“ or „pure“ mean-tone. bezeichnet. In German speaking lands it is also called Praetorian after Michael Praetorius (1571–1621) or „Printzian“ after Wolfgang Caspar Printz (1641–1717) or c. 1700 following Andreas Werckmeister (1645–1706) as „old“ or „common“ temperature.

The limitations of usable intervals and chords in mean tone temperature were familiar knowledge But when in the baroque conflicts arose from the different pitch standards existing (choir- bzw. cornett-pitch and chamber pitch asf.) organ players needed to be able to transpose in different keys. Since the framework of mean tone temperature woud be often transgressed , there were early experiments to enhance the chordal options. This led to eg the technical solution of split keys (so called subsemitones, rare after 1700). Another solution occasionally used were smaller modifications near the „edges“ of mean tone temparature to gain one or two more useable chord, though of inferior quality compareed to the nucleus of good chords (some of those diminished by the modifications).

The charakteristics ot those modified mean tone temperatures are:

a sequence of at least six to seven remarkably lowered fifth in the central range.
at least three or four similarly improved major thirds within this nucleus resulting from the previous: often eg f-a, c-e, g-b natural, d-f#.
most other thirds are good or usable , some directly adjacent to these better ones. The just useful are usually rather at a distance( often e flat-g und b-d#).
One sharpened „wolf“ fifth, mostly g#-e flat, maybe „regulated“ by one or two other fifths (similarly sharpened) nearby, maybe g#-e flat plus another fifth in the vicinity.
two or three more or less unusable thirds resulting from those altered fifths - often eg f#-b flat, c#-f, g#-c.

The most important features of "pure thirds" mean tone could be kept thus , but the greater the modifications the purity of the best thirds would be weakened for instance those in the nucleus of those mean tone temperature in 1/6 pyth. comma division.
Mean tone temperatures – mostly the „pure third“ mean tone – were common in Europe from mid-15th c. to after 1800. The shift to „well-temperered“ temperatures or often directly to equal temperament went on for often more than a century since this often was a very laborous and expensive task.

Well-tempered tuning
Typical for all well-tempered tunings is the absence of not-useful intervals - there are no „wolves". All intervalle can be used although useful differently. The fifths are commonly reduced any parts of a pythagorean comma and so dispered within the circle of fifths that purer third are perceivable in keys with fewer accidentals and with the further addition of accidentals the purity of thirds is reduced step by step.

Since a fifth contains a major and a minor third any alterations of major third and fifths also affects the minor thirds within the circle of fifths and so all the minor chords.

In well-tempered tuning modulating in all keys is possible without limits and changed enharmonically. There are notable small differencesof interval sizes between keys. This led to different characteristics of different keys of an individual character and subjective interpretation.

All those novelties had a positive appeal to an increasing number of theoreticians and musicians of the 18th c. compared to mean-tone with its limits in modulation. To achieve this the well-tempered tunings had to abandon mean-tone pure thirds flourishing from late 17th to early 19th centuries.

There was a certain resistance to general introduction of well-tempered tunings in orgen-making: Re-tuning of any organ of considerable size resulted in a major effort. Pipes had to be shortened (less often lengthened), and access to the bigger pipes was always an issue. Any altering of pipe lengths affected intonation and sound – any shortening of a pipes widens bore relation and widens cut relation. This could lead to further ingressions in pipes and alterations in wind control (pressure) . Re-tuning some medium or bigger instruments could take several months.

But it ran parallel with a strong tradition of mean- tone and often well-tempered designs were theoretical concepts followed after long years' distance!

Equal temperament
Equal temperament was established in renaissance times but spread very slowly since the mid 18th c .but after c. 1850 abolished all other temperaments almost completely and is widespread today. It implies reducing every fifth equally by a twelfth of the pythagorean comma (2 cents). This reduction is so small that the fifth appear almost pure but every major third is equally wider than pure and very perceivably too wide (by 14 cents).

Equal temperament has lost those many pure intervals of mean-tone but also the acoustically based tonality characteristics of well-tempered temperaments.