Temperament

Background:

There is a fundamental problem that needs to be solved in tuning any fixed tone musical instrument: the notes cannot be made to fit into a "perfect" octave, when you attempt to tune all twelve of the chromatic tones according to perfect fifths. Western music customarily demands that the resulting octave needs to be tuned to exactly twice the frequency as the starting note (stretched tuning notwithstanding). If you were a tuner who was determined to maintain perfect fifths as you proceeded around the circle of fifths, the resulting octave tone would sound approximately 20+% of a semitone flatter than the original starting tone! As a corrective action to force a so-called "perfect" octave interval to occur across the keyboard's range, some of the note names' frequencies within the musical scale have to be compromised or mis-tuned slightly (or "tempered").

Aside:

When one speaks of intervals of so-called "perfect" thirds, fourths, fifths, octaves, etc., we mean to say that the two simultaneously sounding notes' overtones do not fight each other; you should hear no background undulation in the tones nor of their overtones caused by constructive- and destructive interference of the overlapping coincident harmonics' frequencies.

Let's explore this concept of temperament in greater detail, using a specific example. If you were to tune a keyboard instrument in the circle of purely perfect fifths, starting at the octave below middle C and ending at Middle C after 12 iterations ......

c - g - d - a - e - b - f# - c# - g# - d# - a# - f - c

.... the final "Middle C" note you would end up with is not exactly in tune with the original reference "c" note from which you started. In fact, the resulting "C" will be off by approximately 1/53rd of an octave, or flat by about 23% of a semitone (flattened towards B Natural); this mathematical peculiarity resulting in flattened pitch is called a "comma." To get an idea of how far off -23 cents sounds to the human ear, consider the normal note Middle C, which is often tuned to 261.6Hz (vibrations per second and annotated as "C261.6") in equal temperament when the "A" above Middle C is specified as being tuned to 440Hz. The frequency that corresponds to 23 cents flatter than Middle C261.6 is a noticeably flatter C sounding at 256.8Hz. Clearly, a frequency difference between C261.6 and C256.8 (if sounded simultaneously) would result in an out-of-tune undulation that varies just under 5 times per second, almost the speed of some organs' tremulants. Worse yet, at just one octave higher, the undulations are doubled in speed and sound noticeably more out of tune, and the undulation happens to warble at rates faster than what most organs' tremulants are set.

As we know, the pipe organ, the piano and practically all mainstream keyboard instruments made over the past 500 years are constructed on a basis of seven natural (usually white) and five sharp keys (usually black) to each octave; or as it is generally said, on a 12-to-the-octave basis (or 13 if you include the octave tone). From the above discussion of having to deliberately mis-tune or temper some of the tones within the musical scale to maintain perfect sounding octaves, we know that we cannot tune a fixed-tone instrument on a one-time basis in a way to obtain "pure" diatonic ratios for all of the twelve tonalities in major and minor keys at the same time.

This peculiarity of the "comma" resulting at the octave when attempting to tune perfect fifths has been known ever since fixed tone instruments have existed. If the demands for a musical composition calls for free modulation of tonalities among all the possible tonalities, then the fixed-tone instrument must be tuned according to some sort of compromise in the form of deliberate mis-tuning of at least certain intervals, and quite possibly all twelve intervals.

A given "temperament" is a system of mis-tuning the notes within the musical scale as defined by a list of frequencies specified for each distinct note, and often consists of tuning certain intervals as mandatory "perfect intervals" and modifying (deliberately mis-tuning upward or downward -- tempering) the remaining notes in a way that maintains perfect octaves in that musical scale. Think of it as a system of deliberately mis-tuning that is designed metaphorically "to rob Peter to pay Paul". A single instance of a temperament’s underlying pattern is a scale (e.g. do, re, mi, fa, so, la, ti, do) and the relative tuning of that scale is repeated throughout the instrument's keyboard range. Tuning also refers to the mapping of specific pitches' frequencies to the keys/notes played by individual musical instruments.

PLEASE NOTE THAT TEMPERAMENT RELATES PRIMARILY TO FIXED-TONE INSTRUMENTS SUCH AS PIPE ORGANS, PIANOS and HARPSICHORDS:

Instruments of the violin family, together with the slide trombone and the human voice, may be performed according to the perfect intervals with completely accurate intonation, simply because the performer (assuming one is an accomplished player or singer) can adjust the tuning on-the-fly by: (1) moving his or her finger on the string; (2) varying the vibrating length of the tube; or (3) modifying the position of one's vocal chords, as the case may be. Of course, when any such instrument is played along with one of the fixed-tone type, its intonation will invariably be modified by the performer (witting or unwittingly) to fit the compromised (tempered) intonation of the fixed-tone keyboard instrument during the course of a musical performance.

The system of keyboard tuning compromise (temperament) now universally employed in the tuning of fixed-tone instruments for Western music is what has been known as Equal Temperament. Other systems, such as meantone and just intonation have come into use and have passed out of existence -- except for some historical pipe organs throughout Europe that retain various well-tempered tuning methods.

Well-Temperament versus Equal Temperament

In so-called "equal temperament" of fixed tone instruments, all twelve semitones in the chromatic scale are equally proportioned in pitches' frequencies, to force the octave note to be precisely in tune with the starting note. Restated, the ratio of frequencies from C# to C, D to C#, D# to D, etc, all the way to the next octave c, is a constant number: the 12th root of 2 (or 1.059461, as calculated to seven significant figures). This means if you start out with the frequency of any C on the keyboard, and multiply its frequency by 1.059461, you will get the equal tempered frequency of C#. Then multiplying the frequency of C# by the same 1.059461, you will get the equal-tempered frequency of D, and so on, throughout the chromatic semitone tuning of the instrument. The important idea to realize is, after you have completed your mathematical tuning journey to the next higher C, the equal-tempered frequency of that higher C is exactly twice the frequency of the original C -- a so-called "perfect" octave.

In all other tuning schemes other than equal-temperament, specific notes of the musical scale are shifted upward or downward by differing amounts, giving each temperament a certain tonal character or flavor: some chords and intervals are very beautiful sounding, whereas other intervals or chords in the same temperament tend to sound harshly out of tune. These particular chords and intervals are generally avoided by the composer.

A few words on BACH and temperament:

Contrary to what you may have been told or taught, J.S. Bach did NOT at any time advocate the use of equal temperament! He wrote two sets of pieces called Das Wohltemperierte Klavier ('The Well-tempered Clavier' or WTC), avoiding the German term for equal temperament, which would have been gleich-schwebende temperatur. These forty-eight preludes and fugues were each designed to exploit the full range of key-color available from a circulating (or "well-tempered") temperament, and careful examination of the texts shows that Bach varied his compositional technique according to the key he in which he was writing.

Bach's motivation for composing the WTC was to demonstrate the feasibility of composing FOR KEYBOARD (hence, the name Well-Tempered CLAVIER) in well-temperament, and to demonstrate the varying key colors in well-tempered fixed-tone keyboard tuning as one progresses through the major and minor tonalities of each of the twelve chromatic notes. Bach repeated the WTC some 20 years later with Book II of the WTC. (12 individual notes of the chromatic keyboard x 2 for major and minor keys x two publications = 48 Preludes and Fugues.) The various types of well-temperament used in Bach's time are distinct from our system of equal temperament.

Well-temperament represented a departure from the various meantone keyboard tunings that were used in earlier music. In fact, Western fixed tone (e.g. keyboard) music from the time of Bach until the turn of the 20th century was not intended to be performed in equal temperament. Equal temperament is appropriate for some keyboard music of the 20th century, especially atonal music, and music based on the whole tone scale, but not for the works of the 18th and 19th centuries. (Interestingly, modern orchestral flutes, oboes and clarinets are far more complex in operation than their corresponding 18th century counterparts, because they are called upon to play the 20th Century repertoire of Schoenberg, Stockhausen and Stravinsky, as well as to be able to play well in all twelve major and minor tonalities.)

Equal temperament, the modern accepted system of keyboard tuning used in Western music, is based on twelve equally tuned semitone intervals. The ratio of frequencies, for each consecutively higher semitone pitch, is equal to the twelfth root of two. So, twelve semitones, played consecutively to complete one octave, lead to a doubling of frequency of the original starting pitch. The uniformity that one gets by having each semitone equally tuned allows a composer or keyboard performer to freely modulate among the different keys. One main drawback to equal temperament is that all major thirds are tuned quite a bit away from where they ought to be, roughly fourteen percent of a semitone. Fourths and fifths are all pretty close to perfect tuning. For example, when one plays the interval of a fourth, say, G3 and C4 with an 8' flute in equal temperament tuning, the overtones undulate at a speed of three cycles over a duration five seconds, a slightly faster than 1/2 a cycle per second. Unless this feature is pointed out to them, modern listeners are usually unaware that a 0.6Hz undulation is even present.

This very obvious solution of equal temperament has been known since approximately 350 BC(!) in Greece, but did not become widespread in Western music until the late 18th century. Theoreticians in Europe have described equal temperament from times earlier than the late 18th century (e.g. Bartolomé Ramis in 1482, Francisco Salinas in 1577), and occasional attempts to introduce it as a practical tuning started as early as the 16th century (e.g. Vincenzo Galilei in 1581, Galileo Galilei's father). These attempts became more serious in the late 18th century when the use of other well-temperaments had become common, but it did not really come into common use until later in the 19th century. Even then, it was probably adopted more slowly by organ builders than by any other group, because the effect of the purposefully mis-tuned intervals is more noticeable on the organ's capability to sustain notes indefinitely, than on any other common instrument.

In essence, for Equal Temperament keyboard tuning, you gain something, and you lose something: You gain the ability to modulate into any other key tonality; but, you give up the tonally rich sound palette of perfectly tuned thirds, fourths, and/or fifths in keyboard music as you play in different keys.

Most importantly, though, other than pitch, equal keyboard temperament DISTINGUISHES ABSOLUTELY NOTHING between the various tonal keys of do-re-mi-fa-sol-la-ti-do. Therefore, all of your organist and pianist friends using equal temperament tuning who claim that they enjoy, say, the key of D more than A Flat --- have delusions that they are actually hearing something that is not there. Wait until you are able to relay your newly acquired information about equal temperament to your delusional friends after they (supposedly) brag how they enjoy certain "keys" more than others.

The various keyboard well-temperaments used throughout the 18th and 19th centuries also allow one to modulate among different key tonalities. However, the octave is not divided into equal steps. Rather, some semitones are smaller and some are larger. Overall, intervals of fourths and fifths tend to be somewhat close to being tuned perfectly, while the quality of major thirds varies around the circle of fifths, with the more unstable major thirds tending to fall on the black keys, giving the various key tonalities different but distinctive sonic characteristics. Composers of the eighteenth and nineteenth centuries used this fact to gravitate towards certain fifths -- yet avoid other fifths -- in their musical compositions. When we listen to their music in our modern equal temperament, we are not hearing their harmonic intentions. Key color has been lost. However, it should be remarked that different musicians have described the colors of the various keys in often quite contradictory ways!

What should you do if you want to play the music of Bach, Mozart, Handel, Mendelssohn, etc. the way the authentic tuning was intended? Find a competent tuner who knows how to tune historic well temperaments! Fortunately for Hauptwerk users for example, many sample libraries contain tables of alternate tuning that are available with a few on-screen clicks of a mouse.

Pitch Perception and Frequency

The position occupied by any musical sound in a musical scale (such as do-re-mi-fa-sol-la-ti-do) is determined by its "pitch." The term relates to perception, not to physical measurement. The so-called "pitch" of a musical sound is the measure of its position, its "highness" or "lowness" relative to some standard sound as judged by the sense of human hearing.

This characteristic of "pitch" depends of course upon the frequency of the sound, i.e., upon the number of oscillations per unit of time made by a tuning-fork, vibrating string, column of air, or other body which gives rise to it. The relation between frequency and pitch is not indeed always linear, for very loud or very faint sounds may give rise to sensations of pitch not corresponding to what their frequency would indicate. (For example, if you place a vibrating tuning fork very close to a person's ear, that person may sing the tone noticeably flatter than when he or she hears it from arm's length.)

Usually, the terms "pitch" and "frequency" are used interchangeably, but it is perhaps best to speak of "frequency" when the question is of the actual physical generation of sound. The term "pitch" is perhaps best used in connection with the sensation of sound, such as sound that is heard. Stated another way, "pitch" is a sensory characteristic of a sound, arising out of, usually running parallel with, and always indicated by, its frequency.

Frequency is more precisely defined as a vibration number, recording the number of the pulsations of a tense string, a column of air, an electronic loudspeaker, a vocal chord or other vibrating source, in a unit of time, usually a second of time. The vibrations of the sound source interact with the air in the surroundings, as slightly higher, followed by slightly lower air pressure, occurring at more or less the same frequency as the sound source. These repetitive changes in ambient air pressure cause our eardrums to vibrate to and fro with the same frequency. This sympathetic vibration of the eardrum is perceived as sound in humans and animals. The pitch of a musical sound is defined in a hearing sense by its absolute position in the musical scale and by its relative position with regard to other musical sounds.

In Great Britain and the United States, complete vibration cycle of “to-and-fro” (swinging both ways of a pendulum) is taken as the unit; elsewhere, the vibration in one direction only (swing one way of the pendulum). The official French standard dates from 1859, preserved by a tuning-fork vibrating 870.9 (double the British and American method of 435.45) in a second, sampled at a temperature of 15° Centigrade (59° Fahrenheit) and standard barometric pressure of 760mm (or 29.92 inches) of mercury. This was the origin of Ariste Cavaille-Coll’s organ tuning of A = 435 cycles per second, or Hertz.

Humans perceive pitch in a manner such that higher frequencies correspond to higher pitch. Most people often perceive pitch in a relative sense, absolute hearing is a special ability of the few. The relative perception is based on the sensation of differences of frequencies. One of the most common frequency ratio is the octave, which corresponds to a 2:1 frequency ratio (or <math>\log_{2} f_{1}/f_{2}</math>). However some musical scales utilize an interesting property of hearing such that an octave is perceived as a different ratio, such as 2.02:1 (stretched octave). A stretched octave sounds in tune when the higher octave's fundamental frequency is in tune with the raised upper octave harmonics of the lower tone. Gamelans from Indonesia for example feature stretched octaves. When such frequency ratios are used to tune an instrument, the term octave is not used any more but Interval of Equivalence (IoE), Repeat Ratio or non-octave is used.

Mel scale

The scale of pitch perception is the mel scale (named after melody), which was found empirically using subjective tests in 1937 (J. Acoust. Soc. Am 8(3) 185--190). The mel scale can be converted to frequency by

<math>m = 1127.01048 \log_e(1+f/700) \ </math> also known as the Stevens-Volkman formula, or also by using another formula:

<math>m = (1000/\log_{10}2) (\log_{10}(1+f/1000)) \ </math>

The mel scale shows equal distances of pitch versus frequency, and it was found to be approachable with a logarithmic curve.

Tonality is a perception of the listener and includes both perceiving an established tonic note (key center/tonal center) at a specific pitch, and the scale that is anchored to that pitch.

Ratios, intervals and scales

Most interval scales are of logarithmic sizes. Before using logarithmic intervals, equal intervals were used, such as by Simon Stevin (1585). Notable first proposers of logarithmic intervals were Bonaventura Cavalieri (1639), Juan Caramel de Lobkowitz (1647), Lemme Rossi (1666), Christiaan Huygens and Isaac Newton (1665), who used 1/12 octave scale to express intervals. Alexander John Ellis (Sharpe) (1814-1890), considered as the founder of ethnomusicology, introduced the definition of cents in 1884. A cent is one hundredth part of the semitone in the 12-tone Equal Temperament (ET), centisemitone. The frequency ratio represented by one cent is the 1200th root of 2, or in other words, one octave is equal to 1200 cents. Today cents are most widely used and the historical scales are expressed with cent values. Since cents express intervals, they can be converted to exact frequency values using a reference frequency, such as a = 440 Hz.

A comma – originating from the Latin word coma (hair) - is not a fixed measure but often used as an interval unit. The syntonic (Didymic) and Pythagorean (ditonic) commas have almost the same size. Often 1/53 of an octave is named a comma. The chromatic and diatonic semitones are 4 and 5 commas. In regular meantone temperaments where the tempering of the fifth is expressed in a fraction of a comma (e.g. 1/8 comma meantone), the comma is the syntonic comma. In well-temperaments, like Werckmeister's, where the cycle of fifths closes in a circle, the temperings are usually expressed in fractions of a Pythagorean comma. The measure 1/55 of an octave is called a Sauveur comma and 1/50 of an octave could be called a Henfling comma. A logarithmic absolute frequency scale measured to the 64’ C note as a reference is called Ellis or El (absolute cent).

When an instrument is tuned by ear, the term of directions is called a bearing plan.

Just intonation

Intervals which are related by a frequency ratio in which both the numerator and denominator of the ratio are (usually small) integers (e.g. 2, 3, 5, 7 and 11) are called just intervals. For example 3/2 or 7/5 are just ratios. When a temperament is based on the ratios of two integers, it is called a Just Intonation (Just Intonation in C for the following example):

Interval: .......................... I ......... II ......... III ........ IV ....... V ........ VI ....... VII ...... VIII

Name of Note: ............... C ........ D ......... E ......... F ....... G ........ A ......... B ........ C

Frequency Ratio to C: ... 1 ....... 9/8 ....... 5/4 ...... 4/3 ..... 3/2 ...... 5/3 ..... 15/8 ..... 2/1

Ratio of integers: ....... 24/24 .. 27/24 .. 30/24 .. 32/24 .. 36/24 .. 40/24 .. 45/24 .. 48/24

Pythagorean tempering

An instrument that is tuned to have perfect fifths and octaves, often found in medieval music is tuned in a Pythagorean tempering. Pythagorean tuning seems to have been in use up to the end of the 16th century. Almost all the fourths and fifths are dead in tune, and the octaves are mandated to be exactly in tune -- such that the entire comma ("error") is 'dumped' on one specific interval (according to Arnaut de Zwolle between F and B flat), which is therefore rendered unusable by definition. (In contrast, various Well-Tempered tuning schemes dilute the comma's error throughout the entire scale, such that no single interval is hideously out of tune.) Pythagorean temperament is easy to explain and to execute, but it places many of the notes of the scale in rather compromising positions. It is quite satisfactory for music written in the old 'modes' that preceded the major and minor scales, provided there is no modulation, whatsoever, between different key signatures.

Meantone tempering

The aim of the meantone tempering is to provide as many almost pure major triads as possible. Roughly speaking, meantone temperament will render approximately eight of the twelve available semitones as passable for listening.

By the early 17th century, MEANTONE temperament was the norm. In this temperament, the major thirds are perfectly in tune and the fourths and fifths slightly compromised - except for one hideously catastrophic fifth, usually between G sharp and E flat, the famous term was called a "wolf". Said in another way, the particular interval of that fifth was so hideously out of tune, its irritating sound was metaphorically likened to the howling of a 'wolf'. However, this was a 'regular' temperament, because in major key signatures requiring less than four accidentals, the notes of the major scale were in the same relative positions, with the thirds remaining all pure. For the first time in Western musical history, meantone temperament allowed the composer freedom to include harmonic modulation in one direction or another, and to choose a key that mirrored his thoughts. However, during the course of modulated, passages there was often an audible 'shift' of tonality, perceived rather like changing gears in an automobile's transmission.

The deliberate appearance of a black note that was technically 'unavailable' in music of the 17th century (they are A flat, A sharp, D flat, D sharp and G flat), was a sure indication that a sudden clash was intended - rather like the deliberate use of false relations. The more extreme accidentals (C flat and onwards) hardly ever appeared in compositions tuned in Meantone temperament. The occasional appearance in the mean-tone era of keys such F minor (with four flats in the key signature) suggested the dawning of an awareness of the possibilities of key-color: with four flats it has a very strange minor third (G sharp, not A flat) and if the G flat is called for, there is further trouble in store. (Also known as QUARTER-COMMA MEANTONE)

The wolf in meantone tuning was so horrible and such an obstacle that, by the later seventeenth century, it was modified substantially in practice. MODIFIED MEANTONE (including FIFTH-COMMA MEANTONE and SIXTH-COMMA MEANTONE, the latter sometimes also known as SILBERMANN TEMPERAMENT) was probably the most appropriate temperament for most of the 'early' organ music we now hear - even though Buxtehude and Bach were clearly among those exploring new tuning systems, their compositional technique remained influenced by the meantone system. Simply put, the pure thirds of meantone are de-tuned slightly, in order to lessen the wolf. Modified meantone temperament was still being used by English organ builders, including Willis, as late as the 1850s. Naturally, the reduced wolf interval in modified meantone temperament allowed composers to modulate the key signatures more freely and frequently, perhaps permitting an occasional excursion into five sharps or flats before returning to a more reasonable home key.

Well tempering

Late in the seventeenth century, theorists started to experiment with various WELL-TEMPERED systems, or CIRCULATING TEMPERAMENTS. The object was to finally hide the wolf, making all keys usable. It is perfectly obvious that this could be done by distributing the intervals equally across the scale, but this was not the path they took (except as an academic exercise). Why? The answer lies in the fact that these circulating (i.e. no wolf) temperaments allow the widest exploration of key color.

In the 18th century, well-tempering became popular, which made all 12 keys usable. J.S. Bach wrote two independent sets of twelve preludes and fugues for each major and minor scale in the collection Das Wohltemperirte Clavier, for instruments using this temperament. The Well tempering is any unequal temperament that can be used in all keys; it is different from Equal tempering because the individual intervals that make up the musical scale ... are still unequal.

There is every indication that musicians of the 18th century were very happy with the expressive possibilities offered by writing in different keys, and sought to exploit the quite different character of each in their writing. Temperaments of this type include the various tunings by WERCKMEISTER (organ expert, 1691), KIRNBERGER (Bach pupil, early 18thC), NEIDHARDT (1724) and VALLOTTI (c1730). Of these systems, Werckmeister III is notable for its purity in the best keys and its suitability for organs with large quint mixtures (many of the fourths and fifths are in tune); but it is irregular and bumpy in the way it deals with modulation and key color. Vallotti is smooth and regular, but the key color is generally rather mild . In all these systems it is possible to play in any key, though the more remote keys may sound unpleasant, and enharmonic modulation is not always happy. Other circulating temperaments have been devised in modern times, almost all of them suffering from the grave defect that they are difficult to commit to memory and therefore difficult to use in practice (you can't tune an organ with a book in one hand).

Equal tempering

Equal tempering and well-tempering are different from one another. Since the early 20th century, equal temperament became common. The reader is referred to "Well-Temperament versus Equal Temperament" in the Temperament section.

This very obvious solution has been known since 350 BC (!), but did not become widespread until the late 18th century (50-100 years later in the English speaking world). The advantages are obvious - all keys are usable without fear or favor, and full enharmonic modulation is possible. The disadvantages are also clear: not one interval is dead in tune (indeed in any major scale the thirds and leading notes are extremely sharp), and there is no key-color whatever. In organs, reeds sound grittier and tierce mixtures begin to scream.

Scales

Hepatonic scales (using all 7 notes of the 12-scale: a,b,c,d,e,f,g)

As a convenient reference, the twelve ecclesiastical modes are listed here. For example: The 7th mode, known as mixolydian, corresponds to the scale of notes on the white keys of the organ keyboard from G to G with G as the "finalis" or "homebase" tonality:

Temperament generator for Hauptwerk

• Temperament file generator (calculator) for Hauptwerk created by another Hauptwerk's forum member, Pere Casulleras.

Click on the above link, and scroll to the name of the available tuning temperament -- the calculator will furnish the deviation in cents for each semitone required to achieve the desired temperament. These values of tuning deviation may be entered into Hauptwerk's Organ Settings.

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