Pitch ::
We reproduce below Ellis' famous table entitled History of Musical Pitch which demonstrates the various pitches used at different times in different places.
Why did pitch vary so much even at the same period in history?
One obvious answer is that there was no universal pitch standard.
Before the widespread use of keyboard instruments, most serious music
in the Middle Ages, both sacred and secular, was song. The monochord,
used to check intervals, was too rudimentary a device to be of use as a
pitch reference. Margaret Bent (Diatonic ficta revisited: Josquin's Ave Maria in Context)
discusses what she calls 'her free-standing, vocally-conceived,
Pythagorean, pre-keyboard "medieval" view of pitch, tuning and vocal
counterpoint.'
"Our
musical culture has raised the definition of frequency and pitch-class
to a high status, for analysis, editing and performance. My reading of
a range of early theorists leads me to posit a slightly fuzzier status
both for what we would call pitch-class and for frequency, a status
that places pitch closer to the more flexible view of durations and
tempo that we still have. This reading rests partly on conspicuous
circumlocutions and the late arrival of precise language, notation and
measurement, partly on a pervasive Pythagorean mentality expressed in
the tuning system, partly on my understanding of counterpoint and the
internal evidence of some paradigmatic pieces. We routinely make
rhythmic and durational analyses on the basis of notated values even
though we know that performance fluctuations, some necessary, some
elective, expected but elusive to precise definition, are ignored by
the analyst. We are not necessarily shocked if an analysis disregards
the fact that a piece, any piece, may end slower than it began. A
terminal ritardando needn't affect certain kinds of analysis; nor need the ritardando of pitch caused by a logical downward sequential spiral (Obrecht Libenter gloriabor Kyrie) shock us."
"A rigid frequency stability which, however well established it became
in the keyboard-reference era, did not, I believe, govern earlier
music. Without cumbersome advance planning, I maintain that it is
virtually impossible to sing the Obrecht Libenter gloriabor Kyrie
(and about 30 other pieces) from original notation in any other artful
way than to let the sequence, indeed, wind smoothly down in its
contrapuntal operation (irrespective of the tuning used, even if that
were equal temperament). This happened in one of our singing sessions
when someone innocent of its notoriety brought a facsimile along. We
read it, it descended, as everyone was (and always would have been)
well aware as it was happening. The sequence of descending fifths and
rising fourths F B E A D G C F is notated only with a few encouraging B
and E flats, but its smooth counterpoint locks it into -- in our terms,
F Bb Eb Ab Db Gb Cb Fb. Another of us, who had previously been
sceptical of "my" solution on paper, exclaimed with surprise that it
sounded fine. That is precisely the point. Try it!" |
Even into the 16th century the pitch for a cappella performance was set not by the notated parts but rather, as Ludovico Zacconi writes in his Prattica di Musica,
pub. Venice (1596), "to have regard for those who are to sing, that
they be at ease with the pitch, neither too high nor too low."
Once we are in an era and a situation where the pitch was, in effect,
fixed by the presence of a keyboard instrument, at which pitch
musicians played would have varied according to where they were
employed. If they performed in a band, an orchestra, at court, in the
opera house or in a church they would have experienced several
different working pitches. For stringed and keyboard instruments the
solution was to retune the instrument. Wind and brass players, however,
were faced with very real difficulties which could only be overcome
either by purchasing completely new instruments when moving from place
to place, venue to venue, or by working from parts specifically
transposed to take account of the difference in pitch. Once the
Hotteterre family had redesigned woodwind instruments to be made in
sections rather than in a single piece, transverse flutes could be made
with extra sections which, if longer, lowered or, if shorter, raised
the pitch of the instrument. An adjustable plug in the head section was
used to correct the tuning and speaking properties of the flute as the
middle sections were exchanged. Brass instruments also had extra
crooks, small lengths of tubing called corps de réchange, which could be applied to the instruments to change their pitch. Quite
apart from the problems of starting at the same pitch, there was also
the reality of playing together as the ambient temperature changed. If
the ambient temperature rises, the pitch of stringed instruments, like
harpsichords, lutes and violins, drops, while that of wind and brass
instruments rises. Played together, the two groups move in opposite
directions and what might start out well enough would soon become
increasingly strained particularly if the instruments were being played
in small concert halls, theatres or opera houses. Churches were less of
a problem because they tended to remain cool whatever the weather
outside. Wind and brass players may have suffered a lower status than
stringed and keyboard instrumentalist because of their constant
struggle to remain in tune with their fellow musicians and the
difficulty they might face when moving from one employer to another,
that of differing pitches. For similar reasons, the instruments of even
the finest wind-instrument makers tended to travel less widely than
those of the finest stringed or keyboard instrument builders. Sir
John Hawkins, writing in 1776, tells us that the tuning fork,
originally called the 'pitch-fork', was invented in 1711, by John
Shore, a trumpeter in the band of Queen Anne. It provided the first
and, until the advent of electronic meters, the most trustworthy
pitch-carrier, and was in every way superior to the 'pitch-pipe' about
which the French philosopher Jean-Jacques Rousseau (1712-1778), writing
in 1764, noted "the impossibility of being certain of the same sound in
two places at the same time".
As an interesting aside, in Korea, pitch was set using resonant stones, called kyong-sok, which whatever the temperature or the humidity would, when struck, produce a reliable pitch reference. Until
comparatively recently, most musicians and scientists, set the note C
rather than A. Today, we tune our instruments to internationally agreed
pitch standards set for A (actually a') although it should be pointed
out that in a world where equal temperament is widely used, setting A
also uniquely sets every other note of the chromatic scale, including
C.
The most widely used standard, first proposed at the Stuttgart
conference of 1838, but not properly established until 1938 in Britain
and in 1939 by the International Organization for Standardization (ISO),
is a'=440 Hz. Hz. is an abbreviation of the name of the German
physicist, Heinrich Hertz, and is a unit of frequency equivalent to one
cycle per second. Neither the Stuttgart (1838), the ISO conference
(1939) nor its successor held in London in October 1953, was successful
in setting an internationally agreed pitch. These points are discussed
in more detail in A Brief History of Musical Tuning by Jonathan Tennenbaum
Sound is a wave associated with the transmission of mechanical energy
through a supporting medium. It can be shown experimentally that sound
cannot travel through a vacuum. The energy available in a sound wave
disturbs the medium in a periodic manner. Periodicity is important if a
sound wave is to carry information. In air, the disturbance propagates
as the successive compression and decompression (the latter sometimes
called rarefaction) of small regions in the medium. If we generate a
pure note and place a detector (our ear, for example) at a point in the
surrounding medium, a distance from the source, the number of
compression-decompression sequences arriving at the detector during a
chosen time interval is called the frequency. The time interval between
successive maximal compressions is called the period. The product of
the frequency and the wavelength is the velocity. You
are probably aware that the speed of sound is far lower than the speed
of light (the speed of light is 299,792,458 metres per second). When,
in the middle of a thunder storm, the flash of lightning is followed,
noticeably later, by a clap of thunder, we take ever greater comfort
the longer the delay. At ground level and at 0° C. the speed of sound
is approximately 331.5 metres per second (c. 1,194 km or 760 miles per
hour). This is approximately equivalent to 1 mile every 5 seconds (or
very roughly 1 km every 3 seconds). The wavelength of the note we call
a'=440Hz. proves to be about 753 mm (about 30 inches). It has long been
established, and was described thus by Rayleigh, "that within certain
wide limits the velocity of sound is independent, or at least very
nearly independent, of its intensity, and also of its pitch (that is,
its rate of vibration)". In general terms this must be the case
otherwise how could music remain coherent even when it has travelled
some considerable distance from performer to listener. The credit
for the first correct published account of the vibration of strings is
usually given to Marin Mersenne (1588-1648) although Galileo Galilei
(1564-1642) published a remarkable discussion of the vibration of
bodies in 1638, derived from his study of the pendulum and of the
relationship between pendulum length and frequency of vibration.
Although this appeared two years after Mersenne published his Harmonicorum Liber,
Galilei's discoveries pre-date those of Mersenne. Wallis (1616-1703)
and Joseph Sauveur (1653-1716) noticed that along a vibrating string
there are points where there is no motion and others where the movement
is particularly violent. Sauveur coined the term 'node' for the former
and 'loop' for the latter, although, today, we use the term 'antinode'
instead of 'loop' and also suggested the terms 'fundamental' and
'harmonic', applied to frequencies that are integer multiples of a
particular frequency. In the discussions that follow, we have adopted
the convention that the fundamental is the first harmonic although, in
some books, the first harmonic is the name given to the second, not the
first, note in the harmonic series. By the 16th-century, it was clear
that the interval relationships between notes, applied to the
frequencies of those notes, was identical to the ratios discovered by
the Greek from their study of the sounding length of vibrating strings. We have prepared an article entitled the Physics of Musical Instruments - A Brief History to which you may wish to refer for further details on this topic. Our
appreciation of pitch stability has changed as some instrument
notorious for their pitch and tuning instability have been replaced
with instruments that are much more stable. For example, modern
electronic instruments are almost entirely insensitive to changes in
ambient temperature, while even the humble modern piano, with its full
metal frame, is a much more stable platform than the half metal half
wood framed pianos made three quarters of a century ago, or than the
harpsichords, clavichords and spinets made three centuries earlier.
Similarly, the relative uniformity of pitch standards around the world,
makes it much easier for the modern musician to travel and perform
abroad.
'History of Musical Pitch' - a table prepared by Mr. A. J. Ellis and published in 1880 (with additions from later publications) ::  units, hertz or Hz, are equivalent to vibrations per second;c''
(one octave above middle C, C5 in scientific notation) is calculated
from a' (A4 in scientific notation) using equal temperament;if another note was originally measured this has been converted to a' using equal temperament;all pitches assume an ambient temperature of 59° Fahrenheit (15° centigrade).The speed of sound in air increases as the square-root of temperature.
The speed of sound in air at 0° centigrade is 331.5 m/s, and it increases by 0.6 m/s for each increase of 1° centigrade | a' (or A4) (in hertz) | c'' (or C5) (in hertz) | Place | Date | Description | 376.3 | 447.5 | Lille, France | 1700 (anté) | Pitch taken by Delezenne from an old dilapidated organ of l'Hospice Comtesse | 378.8 | 450.5 | Paris, France | 1766 | Pitch calculated from data given by Dom Bédos in L'Art du Facteur d'Orgues | 380.0 | 451.9 | Heidelberg, Germany | 1511 | Pitch calculated from data given by Arnold Schlick | 392.2 | 469.1 | St. Petersburg, Russia | 1739 | Euler's clavichord | 395.8 | 470.7 | Versailles, France | 1789 | Organ of the palace chapel | 398.0 | 473.3 | Berlin, Germany | 1775 | Pitch estimated from a flute described by Jean Henri Lambert in Observations sur les Flûtes, pub. Académie Royal des Sciences, Berlin | 400.0 | 475.7 | Paris, France | c. 1756 | Pitch estimated from a flute made by T Lot, one of the five 'maîtres constructeurs' of wind-instruments in Paris, France | 401.3 | 477.8 | Paris, France | 1648 | Mersenne's Spinet | 404.0 | 480.4 | Paris, France | 1699 | Paris Opera A | 405.8 | 482.6 | Paris, France | 1713 | Sauveur's calculation | 407.9 | 485.0 | Hamburg, Germany | 1762 | Organ of St. Michael's Church, Hamburg | 409.0 | 486.4 | Paris, France | 1783 | Tuning fork of Pascal Taskin, court tuner | 415.5 | 494.1 | Dresden, Germany | 1722 | Organ of St. Sophia | 419.6 | 499.0 | Seville, Spain | 1785 & 1790 | Organ of Seville cathedral | 421.6 | 501.3 | Vienna, Austria | 1780 | supposed to be Mozart's pitch | 422.5 | 502.4 | London, England | 1751 | Handel's tuning fork | 423.5 | 503.6 | London, England | 1711 | an existing tuning fork of John Shore | 425.5 | 506.0 | Paris, France | 1829 | Pianoforte at the Paris Opera | 427.6 | 508.5 | Paris, France | 1823 | Opèra Comique | 430.8 | 536.4 | Paris, France | 1830 | Opera pitch as related by Drouet, the celebrated French flautist | 432.0 | 513.7 | Brussels, Belgium | 1876 | Proposed pitch standard | 435.0 | 517.3 | Paris, France | 1859 | The French 'Diapason Normal',
set in law by the French government acting with the advice of Halvy,
Meyerbeer, Auber, Ambroise Thomas and Rossini, although the mean of
several forks set to this pitch lies slightly higher at a'=435.4 which is equivalent to c''=517.8 | 437.0 | 519.7 | Paris and Toulouse, France | 1836 & 1859 | The earlier was the pitch of the Italian Opera in Paris, the later that of the Conservatoire in Toulouse | 440.0 | 523.25 | Paris, France | 1829 | Orchestral pitch of the Paris Opera | 440.0 | 523.25 | Stuttgart, Germany | 1838 | Proposed pitch standard, Stuttgart congress (actually a'=440.2 when corrected to table temperature); also Scheibler's standard. | 441.0 | 524.4 | Rome, Italy | 1725 (anté) | Pitch calculated from a flute made by Biglioni and possibly brought from Rome by J. J. Quantz when he left Rome in 1725 | 444.0 | 528.0 | London, England | 1860 | Standard intended for the Society of Arts - (however a fork set to this standard by J.H. Griesbach has a measured pitch of a'=445.7, equivalent to a c''=530.1) | 444.5 | 528.6 | Madrid, Spain | 1858 | Theatre Royale, Madrid | 444.5 | 528.6 | London, England | c. 1810 | Pitch of a flute made by Henry Potter | 444.6 | 528.7 | London, England | 1877 | Organ in St. Paul's Cathedral | 444.8 | 528.9 | Turin, Weimer, Würtemberg | 1859 | Measurements made for the French Commission | 445.7 | 530.1 | London, England | 1860 | see 440.0 above | 446.0 | 530.4 | Paris, France; Dresden and Pesth, Germany, | 1859 | Pleyel's Piano taken by Delezenne and the pitches at the Opera houses of Dresden and Pesth | 447.11 | 531.7 | London, England | 1845 | Pitch calculated from a fork said to be at the pitch of the Royal Philharmonic Society | 448.0 | 532.8 | Hamburg, Germany | 1839 & 1840 | Opera | 448.0 | 532.8 | Paris, France | 1854 | Opéra Comique | 448.0 | 532.8 | Paris, France | 1858 | Grand Opèra | 448.0 | 532.8 | Liège, Belgium | 1859 | Conservatoire | 450.0 | 535.1 | London, England | 1850 to 1885 | An average of the pitches of London orchestras during this period | 450.5 | 536.7 | Lille, France | 1848 & 1854 | Lille Opera, measured during performance | 451.0 | 536.3 | Brussels, Belgium | 1879 | Pitch standard proposed for the Begian Army | 451.5 | 536.9 | St. Petersburg, Russia | 1858 | Opera | 451.7 | 537.2 | Milan, Italy | 1867 | La Scala Opera | 451.8 | 537.3 | Berlin, Germany | 1859 | Opera | 451.9 | 537.4 | London, England | 1878 | British Army Regulations |
452.0 | 537.5 | Lille, France | 1859 | Conservatoire |
452.0 | 537.5 | London, England | 1889 | Official Pitch at the 'Inventions' Exhibition in 1885 - the highest pitch used intentionally by English orchestras up to 1890 |
452.5 | 538.2 | London, England | 1846 to 1854 | Mean
pitch of the Philharmonic Band under Sir Michael Costa. His Majesty's
Rules and Regulations required Army Bands to play at the Philharmonic
pitch, and a fork tuned to a'=452.5 in 1890 is preserved as the standard for the Military Training School at Kneller Hall |
453.3 | 539.0 | London, England | 1837 (anté) | Pitch calculated from a flute made by Rudall and Rose possibly as early as 1827 |
454.08 | 540.0 | London, England | 1874 | Old Philharmonic Pitch, instigated by Sir Charles Hall |
454.7 | 540.8 | London, England | 1874 | Fork representing the highest pitch adopted for Philharmonic concerts | 454.7 | 540.8 | London, England | 1879 | Steinway's English pitch; also Messrs. Bryceson's pitch |
455.3 | 541.5 | London, England | 1879 | Messrs. Erard's pitch | 455.5 | 541.7 | Brussels, Belgium | 1859 | Band of the Guides | 456.1 | 542.4 | London, England | 1857 | Fork set to the French Society of Pianoforte Makers | 457.2 | 543.7 | New York, USA | 1879 | Pitch used by Messrs. Steinway in America |
456.0 | 542.30 | Vienna, Austria | 1859 | Viennese 'high pitch' |
457.6 | 544.2 | Vienna, Austria | c. 1640 | Great Franciscan organ |
460.0 | 547.05 | Vienna, Austria | 1880 | Old Austrian Military Pitch |
461.0 | 548.3 | London, England | 1838 (anté) | Actual pitch of a flute said to be tuned to a'=453.3 | 474.1 | 563.8 | Durham, England | 1683 | Cathedral Organ by Bernhardt Smith | 474.1 | 563.8 | London, England | 1708 | Organ of the Chapel Royal by Bernhardt Smith | 480.8 | 571.8 | Hamburg, Germany | 1543 & 1879 | Organ at the church of St. Catherine | 484.1 | 575.7 | Lübeck, Germany | 1878 | Cathedral, small organ | 489.2 | 581.8 | Hamburg, Germany | 1688 & 1693 | Organ at the church of St. Jacob | 505.6 | 601.4 | Paris, France | 1636 | Mersenne's church pitch |
506.9 | 602.9 | Halberstadt, Germany | 1361 | Cathedral Organ |
567.6 | 675.2 | Paris, France | 1636 | Mersenne's chamber pitch |
570.7 | 678.7 | Germany | 1619 | Pitch called Kammerton (chamber pitch) by Praetorius; also called North German church pitch |
pitches in use in England in the 1920s : taken from Notes on Concertina Pitch |
note | Normal (-20 cents to ISO) | New Philharmonic (-4 cents to ISO) | Stuttgart/ISO | Society of Arts (+22 cents to ISO) | Old Philharmonic (+54 cents to ISO) |
A | 434.91 | 438.95 | 440 | 445.68 | 454.08 |
A# | 460.77 | 465.05 | 466.16 | 472.18 | 481.09 |
B | 488.17 | 492.70 | 493.88 | 500.25 | 509.69 |
C | 517.20 | 522.00 | 523.25 | 530.00 | 540.00 |
C# | 547.95 | 553.04 | 554.37 | 561.52 | 572.11 |
D | 580.54 | 585.93 | 587.33 | 594.90 | 606.13 |
D# | 615.06 | 620.77 | 622.25 | 630.28 | 642.17 |
E | 651.63 | 657.68 | 659.26 | 667.76 | 680.36 |
F | 690.38 | 696.79 | 698.46 | 707.47 | 720.81 |
F# | 731.43 | 738.22 | 739.99 | 749.53 | 763.68 |
G | 774.92 | 782.12 | 783.99 | 794.10 | 809.09 |
G# | 821.00 | 828.62 | 830.61 | 841.32 | 857.20 |
A | 869.82 | 877.90 | 880.00 | 891.35 | 908.17 |
A# | 921.55 | 930.10 | 932.33 | 944.35 | 962.17 |
B | 976.34 | 985.40 | 987.77 | 1000.51 | 1019.38 |
C | 1034.40 | 1044.00 | 1046.50 | 1060.00 | 1080.00 |
Harmonic Series ::  Sauveur,
following on from work, published in 1673, by two Oxford men, William
Noble and Thomas Pigot, noted that a vibrating string produces sounds
corresponding to several of its harmonics at the same time. The
dynamical explanation for this was first published in 1755 by Daniel
Bernouilli (1700-1782). He described how a vibrating string can sustain
a multitude of simple harmonic oscillations. We call this the
'superposition principle'. The harmonics are multiples of the
'fundamental frequency', also called the 'first harmonic' or
'generator'. So for a string with a fundamental frequency of 440Hz.,
that is fixed at both ends, the harmonics are integral multiples of
440Hz.; i.e. 440Hz. (1 times 440Hz.), 880Hz. (2 times 440Hz.), 1320Hz.
(3 times 440Hz.), 1760Hz. (4 times 440Hz.) and so on. The first 15
harmonics are given below, their frequencies set out in the second
column. The third column, headed 'normalized', is the result of
dividing the frequency of the harmonic by powers of 2 (transposing the
sound down one octave for each power of 2) so that it lies within a
single octave (between 440Hz. and 800Hz.). The nearest note in the
chromatic scale on A is given in column 4 while the column headed %
shows how close the normalized frequency is to the frequency of the
nearest equal-tempered note diatonic to A. Harmonic | Frequency | Normalized | Note name | Closeness in % | 1 | 440Hz. | 440Hz | A | 100% | 2 | 880Hz. | 440Hz | A | 100% | 3 | 1320Hz. | 660Hz | E | 100% | 4 | 1760Hz. | 440Hz | A | 100% | 5 | 2200Hz. | 550Hz | C# | 99% | 6 | 2640Hz. | 660Hz | E | 100% | 7 | 3080Hz. | 770Hz | G | 98% | 8 | 3520Hz. | 440Hz | A | 100% | 9 | 3960Hz. | 495Hz | B | 100% | 10 | 4400Hz. | 550Hz | C# | 99% | 11 | 4840Hz. | 605Hz | D | 103% | 12 | 5280Hz. | 660Hz | E | 100% | 13 | 5720Hz. | 715Hz | F# | 97% | 14 | 6160Hz. | 770Hz | G | 98% | 15 | 6600Hz. | 825Hz | G# | 99% |
We
can extract a complete diatonic scale on A from the first 15 harmonics.
The D is somewhat sharp while the F#, in particular, is very flat. It
would not be impractical to tune a stringed instrument to play diatonic
melodies in the key of A using this scale. You will see that
the perfect fifth appears in this harmonic series as the third
harmonic. The ratio of the frequencies of the third and second harmonic
is (1320:880) which is (3:2). However the fourth, the note D, which
should have a frequency in ratio to A of (4:3) (1.33333), actually
comes out as 1.375. A more serious problem is the absence of an
interval one could call a tone or a semitone. The Greeks defined their
tone as the difference between a perfect fifth and a perfect fourth,
but the fourth is not perfect in this scale. There is no way of
deriving chromatic scales either by starting from A or by starting from
another note, say, the perfect fifth, E.
Pythagorean Series ::  Can
the perfect fifth, one of the three intervals (octave, fifth, and
fourth) which have been considered to be consonant throughout history
by essentially all cultures, form a logical base for building a
chromatic scale; for example, one starting from the note C? Such a
sequence would progress as follows: C G D A E B F# C# G# D# A# F C If
one applies the ratio (3:2) twelve times to 440 and normalizes the
result by dividing by powers of 2 the result is sharp by a ratio called
the Pythagorean or ditonic comma (524288:531441). This scheme
unfortunately substitutes one problem for another. Here, the third and
the octave are both too large.
To clarify the distinction between tuning and temperament we quote from Pierre Lewis's article Understanding Temperaments.
A tuning is laid out with nothing but pure intervals, leaving the Pythagorean or ditonic comma to fall as it must. A temperament
involves deliberately mistuning some intervals to obtain a distribution
of the comma that will lead to a more useful result in a given context.
Solutions can be grouped into three main classes:
- Tunings (Pythagorean, just intonation)
- Regular temperaments where all fifths but the wolf fifth are tempered the same way
- Irregular temperaments where the quality of the fifths around
the circle changes, generally so as to make the more common keys more
consonant
Temperaments are further classified as circulating or closed
if they allow unlimited modulation, i.e. enharmonics are usable (equal
temperament, most irregular temperaments), non-circulating or open
otherwise (tunings, most regular temperaments).
The choice of a particular solution depends on many factors such as
- the needs of the music (harmonic vs melodic, modulations)
- the tastes of the musicians and listeners
- the instrument to be tuned (organ vs harpsichord - tuning the
former is much more work so one needs a more convenient solution),
- aesthetic (Gothic's tense thirds and pure fifths vs the
stable, pure thirds of the Renaissance and Baroque) and theoretical
considerations, and ease of tuning (equal temperament is one of the
more difficult)
|
Pythagorean
intervals and their derivations (also called by modern theorists, the
3-limit system because all ratios are powers only of 2 and/or 3)
| Interval | Ratio | Derivation | Cents* | Unison | (1:1) | Unison 1:1 | 0.00 | Minor Second | (256:243) | Octave - Major Seventh | 90.22 | Major Second | (9:8) | (3:2)^2 | 203.91 | Minor Third | (32:27) | Octave - Major Third | 294.13 | Major Third | (81:64) | (3:2)^4 | 407.82 | Fourth | (4:3) | Octave - Fifth | 498.04 | Augmented Fourth | (729:512) | (3:2)^6 | 611.73 | Fifth | (3:2) | (3:2)^1 | 701.96 | Minor Sixth | (128:81) | Octave - Major Third | 792.18 | Major Sixth | (27:16) | (3:2)^3 | 905.87 | Minor Seventh | (16:9) | Octave - Major Second | 996.09 | Major Seventh | (243:128) | (3:2)^5 | 1109.78 | Octave | (2:1) | Octave (2:1) | 1200.00 | *
The cent is a logarithmic measure of a musical interval invented by
Alexander Ellis. It first appears in the appendix he added to his
translation of Helmhotz's 'On the Sensations of Tone' [1875]. A cent is
the logarithmic division of the equitempered semitone into 100 equal
parts. It is therefore the 1200th root of 2, a ratio approximately
equal to (1:1.0005777895) The formula for calculating the 'cents-value' of any interval ratio is: cents = log10(ratio) * [1200 / log10(2)] or
cents = 1200 × log2 (ratio)
Intervals expressed in cents are added while those expressed in ratio
form must be multiplied: for example, a perfect fourth plus a perfect
fifth equals an octave. In ratio form, (4:3) times (3:2) = (12:6) =
(2:1), in cents, 498.04 + 701.96 = 1200 |
A number of proposals were adopted to 'improve' the Pythagorean scale.
For instance, the Greek major tone, represented by the ratio (9:8)
could be married to the semitone, represented by the ratio (256:243)
and a scale of five whole tones plus two semitones could be formed. Now
the octave is exact but the thirds are still sharp and, because the
sharps and flats are not enharmonic, there are problems when changing
key. Another
solution employed a pure fourth (4:3) and set the octave as a pure
fourth above a perfect fifth, before using the ratio (9:8) to fill in
the remaining tones. The remaining semitones were chosen on the basis
of taste. Unfortunately, the third is still sharp! A further
solution was to slightly narrow the fifth in every or in only some of
the notes arising from the circle of fifths, so absorbing the comma of
Pythagoras. This kind of solution made it possible to move from one key
to any other and formed the basis of the well-tempered system promoted
in 1722 and again in 1724 when Bach published his "Well-Tempered
Clavier". The series of keyboard preludes and fugues was written as
much to show the characteristic colour of different keys as to
demonstrate that, using this tuning system, a composer was no longer
prevented from exploring every minor and major key. Historical Temperaments are considered in more detail at the end of this lesson.
Meantone Scale ::  Sometimes
called the mesotonic scale, the meantone (also written mean-tone) scale
was particularly favoured by organists and explains why organ music
from the period 1500 to the 19th century was written in a relatively
small number of keys, those that this scale favoured. Arnolt Schlick's Spiegel der Orgelmacher und Organisten
(1511) described both the practice of and formulae for mean-tone tuning
which makes it clear that it was already in use. Pietro Aron produced a
more thorough analysis in Toscanello in Musica (1523), which sufficed for all practical purposes. The earliest complete description was published by Francisco de Salinas in De Musica libri septem (1577). How was it set? Based
on C, the method relied on using the first five notes from the circle
of fifths from C, namely C, G, D, A, E and setting a pure third between
C-E by narrowing the fifths by a small amount - from a ratio of (3:2)
to a ratio of (2.99:2). D, the note between C and E was set so that the
ratio between D and C was identical to that between E and D, so placing
D in the mean position between C and E, hence the scale's name. What
happened after this to complete the chromatic scale introduced a number
of variants which only the more studious of our readers are likely to
pursue. Suffice it to point out that the results generally work well in
the keys C, G, D, F and B flat but outside these serious problems arise
and composers writing for this system avoided keys more distant from C. Pietro
Aron's description of meantone tuning is the best known. All but one of
the fifths are flattened from the pure (3:2) ratio by 1/4 of the
syntonic comma. The remaining fifth ends up being sharp by 1 3/4 of the
syntonic comma (the wolf). The syntonic comma is the ratio (9:8)
divided by (10:9), which is the ratio between a pure C-D interval and a
pure D-E interval. In a pure harmonic series starting at CCC (bottom C
on a 16' voice), middle C is 8 times the fundamental, middle D is 9
times the fundamental, and middle E is 10 times the fundamental. The
result of this procedure is a scale with 8 pure major 3rds and 4
diminished 4ths. But there were other meantone procedures known in the
16th and 17th centuries, especially by 2/7th comma, in which the minor
3rds are pure and the major 3rds beat, and 1/3rd comma. In the the
mid-18th century, several instrument-makers and theoreticians used a
1/6th comma meantone temperament, particularly Gottfried Silbermann and
Vallotti. A bizarre fact is that equal temperament is really meantone
by 1/12th comma, that is every fifth is narrowed by 1/12 of the
syntonic comma and the interval between C - D and between D - E are
equal. So, all the modern pianos you have ever heard are in meantone
temperament!
Equal Temperament ::  It
must have been a brave man who first pointed out to a world wedded to
centuries of mean, natural and Pythagorean tuning, that a scale could
be formed using a universal ratio for a semitone such that successive
application of this ratio generated the notes of a chromatic scale
before completing the octave with its harmonic ratio of (2:1), and that
using such a system one might play in tune in any key. This universal
ratio is the twelfth root of 2. This tuning system, called 12EDO
(Equal Divisions of the Octave) by modern tuning theorists, found
favour amongst lutenists who, having tuned the instrument's strings to
different notes, could fret each at an identical point from the nut to
produce parallel equal-tempered scales something that would be
impossible using any other temperament. Unfortunately, as Nicola
Vicentino, the inventor of the archicembalo with six rows of keys that
enabled six different versions of any scale to be performed complete
with temperamental adjustment, observed, this produced horrible clashes
between the lute tuned to an equal-tempered scale performed with a
keyboard tuned using mean-tone temperament. At the time, keyboard
players found the equal-tempered scale more 'sour' than the other
systems in the five keys commonly used, and because most composers
worked only in a limited number of keys the benefits to be had from the
equal-tempered system in more distant keys were not at all obvious.
This probably helped delay its acceptance until such time as enough
'new' ears had become used to it, or enough composers had explored more
distant keys with it in mind. It is still surprising that the
system may have been known in Europe as early as the fifteenth century
(some have suggested that equal temperament was first explained by Chu
Tsai-yü in a paper entitled A New Account of the Science of the Pitch Pipes
published in 1584). However, Henricus Grammateus had already drawn up a
fairly close approximation in 1518, and Zarlino corrected Vincenzo
Galilei's plan for a twelve-stringed equal-tempered lute (Galilei had
invoked Aristoxenus as his inspiration in this project). Even though
the mathematician and music theorist Mersenne produced a correct and
systematic description in 1635, equal temperament was not adopted until
150 years later in Germany and Austria, while Britain and France
delayed for over two centuries. Today we take it and its convenience
for granted. The equal-tempered system cannot be derived from
rational relationships because the twelfth root of 2, like the square
root of 2, is irrational. The theoretical equal temperament frequencies for the A=440Hz. tuning pitch are: Tuning Pitch: A=440Hz. | A | 27.50Hz. | 55.00Hz. | 110.00Hz. | 220.00Hz. | 440.00Hz. | 880.00Hz. | 1760.00Hz. | 3520.00Hz. | A# | 29.13Hz. | 58.27Hz. | 116.54Hz. | 233.08Hz. | 466.16Hz. | 932.32Hz. | 1864.65Hz. | 3729.31Hz. | B | 30.86Hz. | 61.73Hz. | 123.47Hz. | 246.94Hz. | 493.88Hz. | 987.76Hz. | 1975.53Hz. | 3951.06Hz. | C | 32.70Hz. | 65.40Hz. | 130.81Hz. | 261.62Hz. | 523.25Hz. | 1046.50Hz. | 2093.00Hz. | 4186.00Hz. | C# | 34.64Hz. | 69.29Hz. | 138.59Hz. | 277.18Hz. | 554.36Hz. | 1108.73Hz. | 2217.46Hz. | | D | 36.70Hz. | 73.41Hz. | 146.83Hz. | 293.66Hz. | 587.33Hz. | 1174.65Hz. | 2349.31Hz. | | D# | 38.89Hz. | 77.78Hz. | 155.56Hz. | 311.12Hz. | 622.25Hz. | 1244.50Hz. | 2489.01Hz. | | E | 41.20Hz. | 82.40Hz. | 164.81Hz. | 329.62Hz. | 659.25Hz. | 1318.51Hz. | 2637.02Hz. | | F | 43.65Hz. | 87.30Hz. | 174.61Hz. | 349.22Hz. | 698.45Hz. | 1396.91Hz. | 2793.82Hz. | | F# | 46.24Hz. | 92.49Hz. | 184.99Hz. | 369.99Hz. | 739.98Hz. | 1479.97Hz. | 2959.95Hz. | | G | 48.99Hz. | 97.99Hz. | 195.99Hz. | 391.99Hz. | 783.99Hz. | 1567.98Hz. | 3135.96Hz. | | G# | 51.91Hz. | 103.82Hz. | 207.65Hz. | 415.30Hz. | 830.60Hz. | 1661.21Hz. | 3322.43Hz. | |
Just Intonation ::
Barbour writes, in Tuning and Temperament, "it is
significant that the great music theorists ... presented just
intonation as the theoretical basis of the scale, but temperament as a
necessity".
However, the natural or harmonic scale is being explored again
in the twentieth century through the work of Harry Partch, Lou Harrison
and others who, with the advantages of modern technology, have sought
to explore musical systems that were abandoned more for their practical
limitations than for any lack of aesthetic interest. One only has to
consider the complexity of a piano built to perform music based on a
microtonal system, or remind ourselves of Nicola Vicentino's
archicembalo, instruments that have been made and played, to appreciate
that the equal-tempered scale brings with it certain advantages.
An interesting 31-note equal temperament, 31EDO (Equal
Divisions of the Octave), produces a scale that is much closer to just
intonation than the 12-note equal temperament (12EDO) discussed in the
previous section. The thirds, (2(8/31) = 1.1958733 and 2(10/31) = 1.2505655),
are much nearer just intonation than those of 12-note equal
temperament, although the perfect fourth and fifth are less good than
12EDO but still acceptable (2(18/31) = 1.4955179). The 31 notes can be mapped onto the 35 note names of the Western notational system.
Steps = 31 * log2 (f/f0) where f is the frequency in 31EDO
Note | Interval above C
| Steps | Cents |
C | Perfect unison | 0 | 0 |
C# | Augmented unison | 2 | 77 |
C## | Doubly augmented unison | 4 | 155 |
Dbb & B## | Diminished second | 1 | 39 |
Db | Minor second | 3 | 116 |
D | Major second | 5 | 194 |
D# | Augmented second | 7 | 271 |
D## & Fbb | Doubly augmented second | 9 | 348 |
Ebb | Diminished third | 6 | 232 |
Eb | Minor third | 8 | 310 |
E | Major third | 10 | 387 |
E# | Augmented third | 12 | 465 |
Fb | Diminished fourth | 11 | 426 |
F | Perfect fourth | 13 | 503 |
F# | Augmented fourth | 15 | 581 |
F## | Doubly augmented fourth | 17 | 658 |
Gbb & E## | Doubly diminished fifth | 14 | 542 |
Gb | Diminished fifth | 16 | 619 |
G | Perfect fifth | 18 | 697 |
G# | Augmented fifth | 20 | 774 |
G## | Doubly augmented fifth | 22 | 852 |
Abb | Diminished sixth | 19 | 735 |
Ab | Minor sixth | 21 | 813 |
A | Major sixth | 23 | 890 |
A# | Augmented sixth | 25 | 968 |
A## & Cbb | Doubly augmented sixth | 27 | 1045 |
Bbb | Diminished seventh | 24 | 929 |
Bb | Minor seventh | 26 | 1006 |
B | Major seventh | 28 | 1084 |
B# | Augmented seventh | 30 | 1161 |
Cb | Diminished octave | 29 | 1123 |
C | Perfect octave | 31 | 1200 |
It is undeniable, though, that just intonation should be explored in
greater detail and I recommend readers wishing to do this go to The Just Intonation Network (check out the references listed below) Below, for the reference of tuning enthusiasts, is
Kyle Gann's Anatomy of An Octave, which contains all pitches that meet any one of the following six criteria: All ratios between whole numbers 32 and lower All ratios between 31-limit numbers up to 64 (31-limit meaning that the numbers contain no prime-number factors larger than 31) Harmonics up to 128 (each whole number divided by the closest inferior power of 2) All ratios between 11-limit numbers up to 128 All ratios between 5-limit numbers up to 1024 Certain historically important ratios such as the schisma and Pythagorean comma The table is similar to, but much briefer than, that found in Alain Danielou's encyclopedic but long out-of-print Comparative Table of Musical Intervals. Interval Ratio | Cents equivalent | Interval Name (if any) |
(1:1) | 0.000 | tonic |
(32805:32768) | 1.954 | schisma ((3 to the 8th/2 to the 12th) x 5/8) |
(126:125) | 13.795 | |
(121:120) | 14.367 | |
(100:99) | 17.399 | |
(99:98) | 17.576 | |
(81:80) | 21.506 | syntonic comma |
(531441:524288) | 23.460 | Pythagorean comma (3 to the 12th/2 to the 19th) |
(65:64) | 26.841 | 65th harmonic |
(64:63) | 27.264 | |
(63:62) | 27.700 | |
(58:57) | 30.109 | |
(57:56) | 30.642 | |
(56:55) | 31.194 | Ptolemy's enharmonic |
(55:54) | 31.768 | |
(52:51) | 33.618 | |
(51:50) | 34.284 | |
(50:49) | 34.977 | |
(49:48) | 35.698 | |
(46:45) | 38.052 | inferior quarter-tone (Ptolemy) |
(45:44) | 38.907 | |
(128:125) | 41.059 | diminished second (16/15 x 24/25) |
(525:512) | 43.408 | enharmonic diesis (Avicenna) |
(40:39) | 43.831 | |
(39:38) | 44.970 | superior quarter-tone (Eratosthenes) |
(77:75) | 45.561 | |
(36:35) | 48.770 | superior quarter-tone (Archytas) |
(250:243) | 49.166 | |
(35:34) | 50.184 | equal temperament (ET) 1/4-tone approximation |
(34:33) | 51.682 | |
(33:32) | 53.273 | 33rd harmonic |
(32:31) | 54.964 | inferior quarter-tone (Didymus) |
(125:121) | 56.305 | |
(31:30) | 56.767 | superior quarter-tone (Didymus) |
(30:29) | 58.692 | |
(29:28) | 60.751 | |
(57:55) | 61.836 | |
(28:27) | 62.91 | inferior quarter-tone (Archytas) |
(80:77) | 66.170 | |
(27:26) | 65.337 | |
(26:25) | 67.900 | 1/3-tone (Avicenna) |
(51:49) | 69.261 | |
(126:121) | 70.100 | |
(25:24) | 70.672 | minor 5-limit semitone (half-step) |
(24:23) | 73.681 | |
(117:112) | 75.612 | |
(23:22) | 76.956 | |
(67:64) | 79.307 | 67th harmonic |
(22:21) | 80.537 | hard semitone (1/2-step) (Ptolemy, Avicenna, Safiud) |
(21:20) | 84.467 | |
(81:77) | 87.676 | |
(20:19) | 88.801 | |
(256:243) | 90.225 | Pythagorean semitone (half-step) |
(58:55) | 91.946 | |
(135:128) | 92.179 | limma ascendant |
(96:91) | 92.601 | |
(19:18) | 93.603 | |
(55:52) | 97.107 | |
(128:121) | 97.364 | |
(18:17) | 98.955 | equal temperament (ET) semitone (half-step), approximation |
2 to the 1/12th | 100.000 | equal temperament (ET) semitone (half-step), exact |
(35:33) | 101.867 | |
(52:49) | 102.880 | |
(86:81) | 103.698 | |
(17:16) | 104.955 | overtone semitone (half-step) |
(33:31) | 108.237 | |
(49:46) | 109.381 | |
(16:15) | 111.731 | major 5-limit semitone (half-step) |
(31:29) | 115.458 | |
(77:72) | 116.234 | |
(15:14) | 119.443 | Cowell just semitone (half-step) |
(29:27) | 123.712 | |
(14:13) | 128.298 | |
(69:64) | 130.229 | 69th harmonic |
(55:51) | 130.726 | |
(27:25) | 133.238 | alternate Renaissance semitone (half-step) |
(121:112) | 133.810 | |
(13:12) | 138.573 | 3/4-tone (Avicenna) |
(64:59) | 140.828 | |
(38:35) | 142.373 | |
(63:58) | 143.159 | |
(88:81) | 143.498 | |
(25:23) | 144.353 | |
(62:57) | 145.568 | |
(135:124) | 147.145 | |
(49:45) | 147.433 | |
(12:11) | 150.637 | undecimal "median" semitone (1/2-step) |
(59:54) | 153.307 | |
(35:32) | 155.140 | 35th harmonic |
(23:21) | 157.493 | |
(57:52) | 158.940 | |
(34:31) | 159.920 | |
(800:729) | 160.897 | |
(56:51) | 161.916 | |
(11:10) | 165.004 | |
(54:49) | 168.219 | |
(32:29) | 170.423 | |
(21:19) | 173.268 | |
(31:28) | 176.210 | |
(567:512) | 176.646 | |
(51:46) | 178.642 | |
(71:64) | 179.697 | 71st harmonic |
(10:9) | 182.404 | minor whole-tone |
(49:40) | 186.340 | |
(39:35) | 187.343 | |
(29:26) | 189.050 | |
(125:112) | 190.115 | |
(48:43) | 190.437 | |
(19:17) | 192.558 | |
(160:143) | 194.468 | |
(28:25) | 196.198 | |
(121:108) | 196.771 | |
(55:49) | 199.987 | |
2 to the 1/6th | 200.000 | equal-tempered whole-tone, exact |
(64:57) | 200.532 | |
(9:8) | 203.910 | major whole-tone |
(62:55) | 207.404 | |
(44:39) | 208.843 | |
(35:31) | 210.104 | |
(26:23) | 212.253 | |
(112:99) | 213.598 | |
(17:15) | 216.687 | |
(25:22) | 221.309 | |
(58:51) | 222.667 | |
(256:225) | 222.463 | |
(33:29) | 223.696 | |
(729:640) | 225.416 | |
(57:50) | 226.840 | |
(73:64) | 227.789 | 73rd harmonic |
(8:7) | 231.174 | septimal whole-tone |
(63:55) | 235.104 | |
(55:48) | 235.685 | |
(39:34) | 237.527 | |
(225:196) | 238.886 | |
(31:27) | 239.171 | |
(147:128) | 239.607 | |
(169:147) | 241.449 | |
(23:20) | 241.961 | |
(2187:1900) | 243.547 | |
(38:33) | 244.240 | |
(144:125) | 244.969 | diminished third (6/5 x 24/25) |
(121:105) | 245.541 | |
(15:13) | 247.741 | |
(52:45) | 250.313 | |
(37:32) | 251.344 | 37th harmonic |
(81:70) | 252.680 | |
(125:108) | 253.076 | |
(22:19) | 253.805 | |
(51:44) | 255.602 | |
(196:169) | 256.596 | consonant interval (Avicenna) |
(29:25) | 256.950 | |
(36:31) | 258.874 | |
(93:80) | 260.679 | |
(57:49) | 261.816 | |
(64:55) | 262.368 | |
(7:6) | 266.871 | septimal minor third |
(90:77) | 270.080 | |
(75:64) | 274.582 | augmented second (9/8 x 25/24) |
(34:29) | 275.378 | |
(88:75) | 276.736 | |
(27:23) | 277.591 | |
(20:17) | 281.358 | |
(33:28) | 284.447 | |
(46:39) | 285.802 | |
(13:11) | 289.210 | |
(58:49) | 291.925 | |
(45:38) | 292.721 | |
(32:27) | 294.135 | Pythagorean minor third |
(19:16) | 297.513 | overtone minor third |
2 to the 1/4th | 300.000 | equal-tempered minor third, exact |
(25:21) | 301.847 | |
(31:26) | 304.508 | |
(105:88) | 305.777 | |
(55:46) | 309.368 | |
(6:5) | 315.641 | 5-limit minor third |
(77:64) | 320.144 | 77th harmonic |
(35:29) | 325.562 | |
(29:24) | 327.622 | |
(75:62) | 329.550 | |
(98:81) | 329.832 | |
(121:100) | 330.008 | |
(23:19) | 330.761 | |
(63:52) | 332.208 | |
(40:33) | 333.041 | |
(17:14) | 336.130 | |
(243:200) | 337.148 | |
(62:51) | 338.125 | |
(28:23) | 340.552 | |
(39:32) | 342.483 | 39th harmonic |
(128:105) | 342.905 | |
(8000:6561) | 343.304 | |
(11:9) | 347.408 | undecimal "median" third |
(60:49) | 350.617 | |
(49:40) | 351.351 | |
(38:31) | 352.477 | |
(27:22) | 354.547 | |
(16:13) | 359.472 | |
(79:64) | 364.537 | 79th harmonic |
(100:81) | 364.807 | |
(121:98) | 364.984 | |
(21:17) | 365.826 | |
(99:80) | 368.914 | |
(26:21) | 369.747 | |
(57:46) | 371.194 | |
(31:25) | 372.408 | |
(36:29) | 374.333 | |
(56:45) | 378.602 | |
(96:77) | 381.811 | |
(8192:6561) | 384.360 | Pythagorean "schismatic" third |
(5:4) | 386.314 | 5-limit major third |
(64:51) | 393.090 | |
(49:39) | 395.183 | |
(44:35) | 396.192 | |
(39:31) | 397.447 | |
(34:27) | 399.090 | |
2 to the 1/3rd | 400.000 | equal-tempered major third, exact |
(63:50) | 400.108 | |
(121:96) | 400.681 | |
(29:23) | 401.303 | |
(125:99) | 403.713 | |
(24:19) | 404.442 | |
(512:405) | 405.866 | |
(62:49) | 407.384 | |
(81:64) | 407.820 | Pythagorean major third |
(19:15) | 409.244 | |
(33:26) | 412.745 | |
(80:63) | 413.578 | |
(14:11) | 417.508 | |
(51:40) | 420.612 | |
(125:98) | 421.289 | |
(23:18) | 424.364 | |
(32:25) | 427.373 | diminished fourth |
(41:32) | 429.062 | 41st harmonic |
(50:39) | 430.160 | |
(77:60) | 431.875 | |
(9:7) | 435.084 | septimal major third |
(58:45) | 439.353 | |
(49:38) | 440.154 | |
(40:31) | 441.278 | |
(31:24) | 443.081 | |
(1323:1024) | 443.517 | |
(128:99) | 444.772 | |
(22:17) | 446.363 | |
(57:44) | 448.150 | |
(162:125) | 448.879 | |
(35:27) | 449.275 | |
(83:64) | 450.047 | 83rd harmonic |
(100:77) | 452.484 | |
(13:10) | 454.214 | |
(125:96) | 456.986 | augmented third (5/4 x 25/24) |
(30:23) | 459.994 | |
(64:49) | 462.348 | |
(98:75) | 463.069 | |
(17:13) | 464.428 | |
(72:55) | 466.278 | |
(55:42) | 466.867 | |
(38:29) | 467.936 | |
(21:16) | 470.781 | septimal fourth |
(46:35) | 473.152 | |
(25:19) | 475.114 | |
(320:243) | 476.539 | |
(29:22) | 478.259 | |
(675:512) | 478.492 | |
(33:25) | 480.646 | |
(45:34) | 485.286 | |
(85:64) | 491.269 | 85th harmonic |
(4:3) | 498.045 | perfect fourth |
2 to the 5/12ths | 500.000 | equal-tempered perfect fourth, exact |
(75:56) | 505.757 | |
(51:38) | 509.415 | |
(43:32) | 511.518 | 43rd harmonic |
(121:90) | 512.412 | |
(39:29) | 512.905 | |
(35:26) | 514.612 | |
(66:49) | 515.621 | |
(31:23) | 516.761 | |
(27:20) | 519.551 | |
(23:17) | 523.319 | |
(42:31) | 525.745 | |
(19:14) | 528.687 | |
(110:81) | 529.812 | |
(87:64) | 531.532 | 87th harmonic |
(34:25) | 532.328 | |
(49:36) | 533.761 | |
(15:11) | 536.951 | |
(512:375) | 539.104 | |
(26:19) | 543.015 | |
(63:46) | 544.462 | |
(48:35) | 546.835 | |
(1000:729) | 547.211 | |
(11:8) | 551.318 | undecimal tritone (11th harmonic) |
(62:45) | 554.812 | |
(40:29) | 556.737 | |
(29:21) | 558.796 | |
(112:81) | 561.006 | |
(18:13) | 563.382 | |
(25:18) | 568.717 | augmented fourth (4/3 x 25/24) |
(89:64) | 570.880 | 89th harmonic |
(32:23) | 571.726 | |
(39:28) | 573.657 | |
(46:33) | 575.022 | |
(88:63) | 578.582 | |
(7:5) | 582.512 | septimal tritone |
(108:77) | 585.721 | |
(1024:729) | 588.270 | low Pythagorean tritone |
(45:32) | 590.224 | high 5-limit tritone |
(38:27) | 591.648 | |
(31:22) | 593.718 | |
(55:39) | 595.170 | |
(24:17) | 597.000 | |
Square root of 2 | 600.000 | equal-tempered tritone, exact |
(99:70) | 600.088 | |
(17:12) | 603.000 | |
(44:31) | 606.304 | |
(125:88) | 607.623 | |
(27:19) | 608.352 | |
(91:64) | 609.354 | 91st harmonic |
(64:45) | 609.776 | low 5-limit tritone |
(729:512) | 611.730 | high Pythagorean tritone |
(57:40) | 613.154 | |
(77:54) | 614.279 | |
(10:7) | 617.488 | septimal tritone |
(63:44) | 621.418 | |
(33:23) | 624.999 | |
(56:39) | 626.343 | |
(23:16) | 628.274 | 23rd harmonic |
(36:25) | 631.283 | diminished fifth (3/2 x 24/25) |
(121:84) | 631.855 | |
(49:34) | 632.719 | |
(13:9) | 636.618 | |
(81:56) | 638.994 | |
(55:38) | 640.141 | |
(42:29) | 641.204 | |
(29:20) | 643.263 | |
(45:31) | 645.211 | |
(93:64) | 646.991 | 93rd harmonic |
(16:11) | 648.682 | |
(51:35) | 651.794 | |
(729:500) | 652.789 | |
(35:24) | 653.185 | |
(19:13) | 656.985 | |
(375:256) | 660.896 | |
(22:15) | 663.049 | |
(47:32) | 665.507 | 47th harmonic |
(72:49) | 666.258 | |
(25:17) | 667.672 | |
(81:55) | 670.188 | |
(28:19) | 671.313 | |
(31:21) | 674.255 | |
(189:128) | 674.691 | |
(34:23) | 676.681 | |
(40:27) | 680.449 | dissonant "wolf" 5-limit fifth |
(46:31) | 683.263 | |
(95:64) | 683.827 | 95th harmonic |
(49:33) | 684.403 | |
(52:35) | 685.412 | |
(58:39) | 687.095 | |
(125:84) | 688.160 | |
(112:75) | 694.243 | |
(121:81) | 694.816 | |
2 to the 7/12ths | 700.000 | equal-tempered perfect fifth, exact |
(3:2) | 701.955 | perfect fifth |
(121:80) | 716.322 | |
(50:33) | 719.380 | |
(97:64) | 719.895 | 97th harmonic |
(1024:675) | 721.508 | |
(44:29) | 721.766 | |
(243:160) | 723.461 | |
(38:25) | 724.886 | |
(35:23) | 726.865 | |
(32:21) | 729.219 | |
(29:19) | 732.064 | |
(84:55) | 733.149 | |
(55:36) | 733.748 | |
(26:17) | 735.572 | |
(75:49) | 736.931 | |
(49:32) | 737.652 | 49th harmonic |
(23:15) | 740.006 | |
(192:125) | 743.014 | diminished sixth (8/5 x 24/25) |
(20:13) | 745.786 | |
(77:50) | 747.516 | |
(54:35) | 750.752 | |
(125:81) | 751.121 | |
(17:11) | 753.637 | |
(99:64) | 755.228 | 99th harmonic |
(48:31) | 756.946 | |
(31:20) | 758.722 | |
(45:29) | 760.674 | |
(14:9) | 764.916 | septimal minor sixth |
(120:77) | 768.125 | |
(39:25) | 769.855 | |
(25:16) | 772.627 | augmented fifth |
(36:23) | 775.636 | |
(11:7) | 782.492 | undecimal minor sixth |
(63:40) | 786.422 | |
(52:33) | 787.283 | |
(101:64) | 789.854 | 101st harmonic |
(30:19) | 790.756 | |
(128:81) | 792.180 | Pythagorean minor sixth |
(49:31) | 792.644 | |
(405:256) | 794.134 | |
(19:12) | 795.558 | |
(46:29) | 798.726 | |
(100:63) | 799.892 | |
2 to the 2/3rds | 800.000 | equal-tempered minor sixth, exact |
(27:17) | 800.910 | |
(62:39) | 802.553 | |
(35:22) | 803.822 | |
(51:32) | 806.910 | 51st harmonic |
(8:5) | 813.686 | 5-limit minor sixth |
(6561:4096) | 815.640 | Pythagorean "schismatic" sixth |
(77:48) | 818.189 | |
(45:28) | 821.427 | |
(103:64) | 823.801 | 103rd harmonic |
(29:18) | 825.667 | |
(50:31) | 827.600 | |
(121:75) | 828.053 | |
(21:13) | 830.253 | |
(55:34) | 832.706 | |
(34:21) | 834.175 | |
(81:50) | 835.193 | |
(125:77) | 838.797 | |
(13:8) | 840.528 | overtone sixth |
(57:35) | 844.328 | |
(44:27) | 845.483 | |
(31:19) | 847.523 | |
(80:49) | 848.662 | |
(49:30) | 849.413 | |
(18:11) | 852.592 | undecimal "median" sixth |
(105:64) | 857.095 | 105th harmonic |
(64:39) | 857.517 | |
(23:14) | 859.448 | |
(51:31) | 861.905 | |
(400:243) | 862.852 | |
(28:17) | 863.870 | |
(33:20) | 866.959 | |
(38:23) | 869.239 | |
(81:49) | 870.168 | |
(48:29) | 872.409 | |
(53:32) | 873.505 | 53rd harmonic |
(58:35) | 874.438 | |
(63:38) | 875.223 | |
(128:77) | 879.856 | |
(107:64) | 889.760 | 107th harmonic |
(5:3) | 884.359 | 5-limit major sixth |
(57:34) | 894.513 | |
(52:31) | 895.524 | |
(42:25) | 898.153 | |
(121:72) | 898.726 | |
2 to the 3/4ths | 900.000 | equal-tempered major sixth, exact |
(32:19) | 902.487 | |
(27:16) | 905.865 | Pythagorean major sixth |
(49:29) | 908.107 | |
(22:13) | 910.790 | |
(39:23) | 914.208 | |
(56:33) | 915.553 | |
(17:10) | 918.641 | |
(109:64) | 921.821 | 109th harmonic |
(46:27) | 922.442 | |
(75:44) | 923.264 | |
(29:17) | 924.622 | |
(128:75) | 925.418 | diminished seventh (16/9 x 24/25) |
(77:45) | 929.920 | |
(12:7) | 933.129 | septimal major sixth |
(55:32) | 937.632 | 55th harmonic |
(31:18) | 941.126 | |
(441:256) | 941.562 | |
(50:29) | 943.084 | |
(19:11) | 946.195 | |
(216:125) | 946.924 | |
(121:70) | 947.496 | |
(45:26) | 949.730 | |
(26:15) | 952.259 | |
(111:64) | 953.299 | 111th harmonic |
(125:72) | 955.031 | augmented sixth (5/3 x 25/24) |
(33:19) | 955.760 | |
(40:23) | 958.039 | |
(54:31) | 960.864 | |
(96:55) | 964.323 | |
(110:63) | 964.896 | |
(7:4) | 968.826 | septimal minor seventh |
(58:33) | 976.304 | |
(225:128) | 976.537 | |
(51:29) | 977.368 | |
(44:25) | 978.725 | |
(30:17) | 983.313 | |
(113:64) | 984.215 | 113th harmonic |
(99:56) | 986.402 | |
(23:13) | 987.747 | |
(62:35) | 989.896 | |
(39:22) | 991.165 | |
(55:31) | 992.631 | |
(16:9) | 996.090 | Pythagorean small minor seventh |
(57:32) | 999.468 | 57th harmonic |
2 to the 5/6ths | 1000.000 | equal-tempered minor seventh |
(98:55) | 1000.020 | |
(25:14) | 1003.802 | |
(34:19) | 1007.442 | |
(52:29) | 1010.986 | |
(88:49) | 1013.666 | |
(115:64) | 1014.588 | 115th harmonic |
(9:5) | 1017.596 | 5-limit large minor seventh |
(56:31) | 1023.790 | |
(38:21) | 1026.732 | |
(29:16) | 1029.577 | 29th harmonic |
(49:27) | 1031.823 | |
(20:11) | 1034.996 | |
(51:28) | 1038.121 | |
(729:400) | 1039.103 | |
(31:17) | 1040.080 | |
(42:23) | 1042.507 | |
(117:64) | 1044.438 | 117th harmonic |
(64:35) | 1044.860 | |
(4000:2187) | 1045.266 | |
(11:6) | 1049.363 | undecimal "median" seventh |
(90:49) | 1052.572 | |
(57:31) | 1054.432 | |
(46:25) | 1055.684 | |
(81:44) | 1056.502 | |
(35:19) | 1057.627 | |
(59:32) | 1059.172 | 59th harmonic |
(24:13) | 1061.427 | |
(50:27) | 1066.772 | |
(63:34) | 1067.780 | |
(13:7) | 1071.702 | |
(119:64) | 1073.781 | 119th harmonic |
(54:29) | 1076.326 | |
(28:15) | 1080.557 | |
(58:31) | 1084.542 | |
(15:8) | 1088.269 | 5-limit major seventh |
(62:33) | 1091.763 | |
(32:17) | 1095.045 | |
(49:26) | 1097.163 | |
(66:35) | 1098.133 | |
2 to the 11/12ths | 1100.000 | equal-tempered major seventh, exact |
(17:9) | 1101.045 | |
(121:64) | 1102.636 | 121st harmonic |
(125:66) | 1105.668 | |
(36:19) | 1106.397 | |
(256:135) | 1107.821 | |
(55:29) | 1108.094 | |
(243:128) | 1109.775 | Pythagorean major seventh |
(19:10) | 1111.199 | |
(40:21) | 1115.533 | |
(61:32) | 1116.885 | 61st harmonic |
(21:11) | 1119.463 | |
(44:23) | 1123.084 | |
(23:12) | 1126.319 | |
(48:25) | 1129.338 | |
(121:63) | 1129.900 | |
(123:64) | 1131.017 | 123rd harmonic |
(25:13) | 1132.100 | |
(77:40) | 1133.830 | |
(52:27) | 1134.703 | |
(27:14) | 1137.039 | septimal major seventh |
(56:29) | 1139.249 | |
(29:15) | 1141.308 | |
(60:31) | 1143.233 | |
(31:16) | 1145.036 | 31st harmonic |
(64:33) | 1146.727 | |
(33:17) | 1148.318 | | (243:125) | 1150.834 | |
(35:19) | 1151.230 | |
(39:20) | 1156.169 | |
(125:64) | 1158.941 | augmented seventh (15/8 x 25/24) |
(88:45) | 1161.094 | |
(45:23) | 1161.991 | |
(96:49) | 1164.303 | |
(49:25) | 1165.066 | |
(51:26) | 1166.424 | |
(108:55) | 1168.233 | |
(55:28) | 1168.847 | |
(57:29) | 1169.891 | |
(63:32) | 1172.736 | 63rd harmonic |
(160:81) | 1178.494 | |
(99:50) | 1182.601 | |
(125:63) | 1186.205 | |
(127:64) | 1186.422 | 127th harmonic |
(2:1) | 1200.000 | octave |
Naming Intervals ::  The following table lists the names of the most common intervals using a number of modern conventions.
The standard system for comparing
intervals of different sizes is with cents, based on a logarithmic
scale where the octave is divided into 1200 equal parts. In equal
temperament, each semitone is exactly 100 cents. To remind our readers
of the formula given earlier, the value in cents for the interval f1 to f2 is 1200×log2(f2/f1).
In diatonic set theory, specific and generic
intervals are distinguished. Specific intervals are the interval class
or number of semitones between scale degrees or collection members, and
generic intervals are the number of scale steps between notes of a
collection or scale.
# semitones
|
Interval
class |
Generic
interval |
Common
diatonic name |
Comparable
just interval |
Comparison of interval width in cents (to nearest integer) |
equal
temperament |
just
intonation |
quarter-comma
meantone |
Pythagorean
tuning |
0 |
0 |
0 |
perfect unison |
(1:1) |
0 |
0 |
0 |
0 |
1 |
1 |
1 |
minor second |
(16:15) |
100 |
112 |
117 |
90 |
2 |
2 |
1 |
major second |
(9:8) |
200 |
204 |
193 |
204 |
3 |
3 |
2 |
minor third |
(6:5) |
300 |
316 |
310 |
294 |
4 |
4 |
2 |
major third |
(5:4) |
400 |
386 |
386 |
408 |
5 |
5 |
3 |
perfect fourth |
(4:3) |
500 |
498 |
503 |
498 |
6 |
6 |
3
4 |
augmented fourth
diminished fifth |
(45:32)
(64:45) |
600 600 |
590
610 |
579
621 |
612 |
7 |
5 |
4 |
perfect fifth |
(3:2) |
700 |
702 |
697
wolf fifth 737 |
702 |
8 |
4 |
5 |
minor sixth |
(8:5) |
800 |
814 |
814 |
792 |
9 |
3 |
5 |
major sixth |
(5:3) |
900 |
884 |
889 |
906 |
10 |
2 |
6 |
minor seventh |
(16:9) |
1000 |
996 |
1007 |
996 |
11 |
1 |
6 |
major seventh |
(15:8) |
1100 |
1088 |
1083 |
1110 |
12 |
0 |
0 |
perfect octave |
(2:1) |
1200 |
1200 |
1200 |
1200 |
Reference:
Interval (music)
Historical Temperaments ::  Prior
to the almost universal adoption of the equal temperament system of
tuning where the interval between successive semitones is a constant
and the ratio for the octave is set at 2:1, musicians and theorists
produced numerous solutions for bending the natural Pythagorean scale
to practical use. That this was an impossible task, particularly if one
wished to modulate to all the possible major or minor scales, was
demonstrated time and again by composers such as Willaert who used
their works to demonstrate the shortcomings of any of the temperaments
then in use. Written in four parts, the vocal work Quid non ebrietas,
starts on the key note before taking the singers through a sequence of
perfect fifths that, if they use Pythagorean tuning based on perfect
fifths, leaves them sharp by the Pythagorean comma when they return to
the key note at the end of the piece. If the singers choose instead to
use just intonation, they reach the end flat to the desired key note.
Of course, most of these problems could be ignored so long as composers
chose to remain reasonably close to the key in which the work started.
Composers like Willaerts, Nicola Vicentino and Carlo Gesualdo pushed
the boundaries of temperament so hard that special instruments had to
be invented to handle the complexities of tempered scales as key notes
changed. Vicentino invented the archicembalo with its six rows of keys.
He also inspired Fabio Colonna's sambuca in which the octave was divided into thirty-one parts. The
temperaments we set out below were commonly used before the widespread
introduction of equal temperament. Each was an attempt to rid the
'so-called' natural scale of its problems under modulation. We give
some information below about the common 'historical temperaments' used
when setting keyboard instruments for historically informed performance. PYTHAGOREAN - see table below Strictly,
not a temperament but a tuning because natural intervals are not
adjusted but allowed to fall where they may, it dates back to 500 BC.
This simple scale creates eleven pure fifths around the circle, leaving
the entire Pythagorean comma between G# and Eb There are four
pure major thirds at B-D#, F#-A#, Db-F, and Ab-C, but these are not
particularly useful. The remainder are quite harsh. van ZWOLLE - see table below Arnout van Zwolle (1400-1466) modified the Pythagorean scale by placing the comma
between B and F#. This moved the thirds to D-F#, A-C#, E-G#, and B-Eb,
which were more useful. This gives pure major triads on D, A and E. MEANTONE - see table below The
best known of the old scales (see more below), this scale emphasizes
pure thirds by making the fifths narrow. It was certainly in use by the
end of the 15th century, if not earlier. It has the greatest number of
pure triads of any of the scales on this disk. All whole steps are
equally spaced, one half of a major third apart. It also has a very
prominent "wolf" between G# and Eb. If the circle is extended down to
Ab, the pitch is very different from the G#, Some baroque keyboards had
a split pair of black keys that allowed the musician to choose G# or Ab. SILBERMANN I, II - see table below Organ
builder Gottfried Silbermann (1678-1734) tried several variants to
narrow the "wolf" and make his instruments useable in more keys. None
of the intervals of these two scales are pure. RAMEAU - see table below Jean
Phillipe Rameau. (1683-1764) modified the meantone scale to provide
three pure fifths. This very pleasant scale almost completely
eliminates the harsh "wolf" of the meantone while preserving most of
its pure harmony. WERKMEISTER III, IV, V, VI - see table below Organ
builder and mathematician Andreas Werkmeister (1645-1706) devoted much
of his life to the study of temperament and suggested many. different
scales. The best known of these are included on this disk. His goal was
to place the best thirds in those keys with the fewest incidentals. It
is very likely that Bach (l685-1750) wrote his famous "Well-tempered
Klavier" pieces for one of these temperaments. KIRNBERGER II, lII - see table below Composer
and music theorist Johann Philipp Kirnberger (1721-1783) suggested
several temperaments. The two scales here offer a large number of pure
fifths. The first has pure thirds at C-E, G-B, and D-F# but the fifths
at D-A and A-E are somewhat harsh. Kirnberger later proposed an
alternate scale with smoother fifths, but only one pure third at C-E. ITALIAN 18th Century One
of the many variations commonly in use in the 18th century that
emphasized a pure third at C-E and distributed the "wolf" around the
circle of fifths. There is only one pure interval in this scale. EQUAL TEMPERED - see table above This
scale is so common in the 20th century that many musicians and
instrument makers tend not to know that there are alternatives.
Dividing the Pythagorean comma equally around the circle of
fifths is not a recent idea. Equal temperament was probably known in
the 1700s or earlier, but was not considered a satisfactory scale due
to the impurity of all intervals. In the late 18th and 19th centuries,
composers increasingly explored modulation to many different keys. They
found that most temperaments were unsatisfactory because of the
significant tonal changes involved in changing keys. The equal-tempered
scale was begrudgingly recognized as an acceptable compromise that
worked equally well in every key. It is only through over a century of
dominance that this scale has become the one that we are accustomed to
- the scale that sounds "in tune" to us today. It is actually one type
of 'meantone' tuning because the third lies exactly midway between the
root and the fifth. Table from Alternate Temperaments: Theory and Philosophy by Terry Blackburn and calculated from a = 440 Hz. | Kirnberger II | c | 262.37 | c# | 276.40 | d | 295.16 | d# | 310.95 | e | 327.96 | f | 349.82 | f# | 368.95 | g | 393.55 | g# | 414.60 | a | 440.00 | a# | 466.43 | b | 491.93 | c | 524.73 | | Kirnberger III | 263.18 | 277.26 | 294.25 | 311.92 | 328.98 | 350.91 | 370.10 | 393.55 | 415.89 | 440.00 | 467.88 | 493.47 | 526.36 | | Werckmeister III | 263.40 | 277.50 | 294.33 | 312.18 | 330.00 | 351.21 | 369.99 | 393.77 | 416.24 | 440.00 | 468.27 | 495.00 | 526.81 | | Werckmeister IV | 263.11 | 275.93 | 294.66 | 311.83 | 330.00 | 350.81 | 369.58 | 392.88 | 413.90 | 440.00 | 469.86 | 492.77 | 526.21 | | Werckmeister V | 261.63 | 276.56 | 294.33 | 311.13 | 328.88 | 350.02 | 369.99 | 392.44 | 413.43 | 440.00 | 466.69 | 493.33 | 523.25 | | Werckmeister VI | 262.77 | 276.83 | 292.77 | 312.03 | 330.00 | 350.36 | 370.53 | 393.39 | 415.24 | 440.00 | 468.05 | 495.00 | 525.54 | | Van Biezen | 262.51 | 277.18 | 294.00 | 311.83 | 329.26 | 350.81 | 369.58 | 392.88 | 415.77 | 440.00 | 467.75 | 492.76 | 525.03 | | Bach (Klais) | 262.76 | 276.87 | 294.30 | 311.46 | 328.70 | 350.37 | 369.18 | 393.70 | 415.30 | 440.00 | 467.18 | 492.26 | 525.53 | | Just (Barbour) | 264.00 | 275.00 | 297.00 | 316.80 | 330.00 | 352.00 | 371.25 | 396.00 | 412.50 | 440.00 | 475.20 | 495.00 | 528.00 | | | Pythagorean | c | 260.74 | c# | 278.44 | d | 293.33 | d# | 309.03 | e | 330.00 | f | 347.65 | f# | 371.25 | g | 391.11 | g# | 417.66 | a | 440.00 | a# | 463.54 | b | 495.00 | c | 521.48 | | van Zwolle | 260.74 | 274.69 | 293.33 | 309.03 | 330.00 | 347.65 | 366.25 | 391.11 | 417.66 | 440.00 | 463.54 | 495.00 | 521.48 | | Meantone (-1/4) | 263.18 | 275.00 | 294.25 | 314.84 | 328.98 | 352.00 | 367.81 | 393.55 | 411.22 | 440.00 | 470.79 | 491.93 | 526.36 | | Silbermann(-1/6) | 262.37 | 276.14 | 293.94 | 312.89 | 329.32 | 350.55 | 368.95 | 392.73 | 413.35 | 440.00 | 486.36 | 492.25 | 524.73 | | Salinas (-1/3) | 264.00 | 273.86 | 294.55 | 316.80 | 328.64 | 353.46 | 366.67 | 394.36 | 409.10 | 440.00 | 473.24 | 490.92 | 528.00 | | Zarlino (-1/7) | 263.53 | 274.51 | 294.38 | 315.68 | 328.83 | 358.63 | 367.32 | 393.90 | 410.31 | 440.00 | 471.84 | 491.50 | 527.06 | | Rossi (-1/5) | 262.69 | 275.68 | 294.06 | 313.67 | 329.18 | 351.13 | 368.49 | 393.06 | 412.50 | 440.00 | 469.33 | 492.55 | 525.38 | | Rossi (-1/9) | 262.91 | 275.38 | 294.14 | 314.19 | 329.09 | 351.51 | 368.19 | 393.28 | 411.93 | 440.00 | 469.98 | 492.27 | 525.82 | | Rameau(syntonic) | 263.18 | 276.71 | 294.25 | 310.31 | 328.98 | 352.00 | 368.95 | 393.55 | 415.07 | 440.00 | 467.39 | 491.93 | 526.36 | |
Information on Temperaments ::
Books:
Temperament by Stuart Isacoff published Faber and Faber (originally by Alfred Knopf)
Tuning and Temperament by J. Murray Barbour published Michigan State College Press
Fundamentals of Musical Acoustics by Arthur H. Benade published Dover Publications
The Structure of Recognizable Diatonic Tunings by Easley Blackwood published Princeton University Press
Treatise on Harpsichord Tuning by Jean Denis published Cambridge University Press
Tuning: Containing the Perfection of Eighteenth-Century
Temperament, the Lost Art of Nineteenth-Century Temperament and the
Science of Equal Temperament by Owen H. Jorgensen published Michigan State University Press
Lutes, Viols and Temperaments by Mark Lindley published Cambridge University Press
Intervals, Scales and Temperaments by Lt. S. Lloyd and Hugh Boyle published MacDonald, London
Musical Temperaments by Erich Neuwirth published Springer Verlag
Musicalische Temperatur by Andreas Werckmeister published Diapason Press
Online
Understanding Temperaments by Pierre Lewis from which the references and comments below have been taken
A very complete bibliography by Manuel Op de Coul
Other tuning-related weblocations by Stichting Huygens-Fokker
Also: WannaLearn, OpenHere
A very thorough, well-researched and clear discussion of Pythagoras's tuning
by Margo Schulter, also with very complete discussion of later
temperaments: a must-read for anyone seriously interested in tunings
and temperaments (esp. for the medieval period)
Bach's musical temperament - by Dr. Kellner
The Just Intonation System of Nicola Vicentino
Temperament: A Beginner's Guide by Stephen Bicknell: less technical, with a lot more on historical perspective, and with some suggestions for CDs
Temperament, A Beginner's Guide by Stephen Bicknell (also here) - update of above?
Comparison of temperaments by Andrew Purdam (also here)
Alternate Temperaments: Theory and Philosophy by Terry Blackburn (also here)
History of Tuning and Temperament by Howard Stoess
Historical Tunings on the Modern Historical Concert Grand by Edward Foote
The Meantone temperament home page
Meantone and Temperament in Bach's Time by Daniel Pyle, adapted by Ben Chi
Meantone Temperaments by Graham Breed, also other pages, e.g. Graham's microtonal software
Lucy tuning by Charles Lucy (see some notes below)
Just intonation network
Definitions of tuning terms by Joseph L. Monzo
Just intonation by Kyle Gann, also An Introduction to Historical Tunings by Kyle Gann
Applet on temperaments by Keith Griffin
Tuning/temperament S/W compiled by Nicholas S. Lander
Fred Nachbaur's MIDI tempering utilities
Keyboard temperament analyzer/calculator by Bradley Lehman
12-tone equal temperament
Christiaan Huygens and 31-tone equal temperament
72-tone equal temperament
Justesse a cappella … la renaissance par Yves Ouvrard, Jean-Pierre Vidal, Olivier Bettens
The Mathematics of Tuning and Temperament by David Bartlett
Algorithms for Mapping Diatonic Keyboard Tunings and Temperaments by Kenneth P. Scholtz
Ear Training
Tuning in Medieval Welsh String Music by Robert Evans
Pitches, Scales and Modes in Han (Chinese) traditional music
Tuning Indian Instruments
Timbre/Tone Colour ::  All
musical instrument have acoustical properties determined by their form
and material of construction. Musical instruments require intervention
from an actuator (or performer) to provide the energy that will
initiate the production of sound. Sound is a form of mechanical energy
that requires a medium through which to propagate or travel. A sound
travels from a source, through a medium to a detector. For us the
detector is the human ear. If the sound is to be considered musical
with a specific pitch or tone quality, rather than just 'noise', the
mechanical energy has to radiate from the instrument as regular
disturbances, what we call 'periodic' vibrations. The vibrator
producing fluctuations, oscillations, pulsations or undulations (these
terms are all equivalent) will be different on different instruments
and the initiation and resonance may arise from two separate processes.
We say that the sound producing system has two parts - the initiator and the resonator. Examples of initiators: - String - violin, guitar, piano, psaltry, harp
- Reed - clarinet, oboe, bassoon, English horn.
- Lips - trumpet, trombone, French horn, tuba.
- Membrane - drum, tambourine
- Wood - wood block, xylophone.
- Metal - bells, cymbals.
- Electronic instruments - speakers that can produce vibrations
Examples of resonators: - Wooden box which may be hollow or solid - violin, guitar, piano (sounding board)
- Tubing - (brass, silver, wood, pipe-like) - trumpet, trombone, French, horn, flugel horn, tuba, trombone.
- Chest, oral, nasal and throat cavities - human voice.
- Electronic instruments - amplifier, tuned circuits.
The
character of the sound each instrument produces is, therefore, partly
due to vibrations associated with the process of initiation and partly
due to the characteristic vibrations that are generated by the
resonator, initially sustained but usually decaying once energy is no
longer supplied to the system. If, on a stringed instrument, the bow is
continuously drawn across a string, the instrument is described as
being in continuous-control mode; i.e. onset - sustain. If, however, on
the same instrument, a string is plucked with a finger, the instrument
is then said to be in envelope-based mode; i.e. onset - sustain -
decay. In general and when the process of initiation is mechanical and
occurs over a relatively short time, short relative to the persistance
of the resonance response that follows, a note has a clear starting or
'onset' sound (arising from the initiator) which is distinguishable
from the sound that follows (that arising from the resonator). For
example, the 'tonguing' sound that begins notes produced on
wind-instruments is distinguishable from the sustained resonance
associated with the remainder of the note. The percussive initiation of
a note produced on a piano, the sound of the hammer striking the
string, is distinguishable from the sound that rings on should you keep
the piano key depressed for any length of time. The mechanical
processes involved in sound production on musical instruments include
plucking or bowing (on violin, viola, cello string bass, harpsichord),
blowing (on clarinet, oboe, trumpet, trombone, recorder, voice) or
striking (on drums, piano, clavichord, xylophone). It has been found
that if the onset is removed from recordings of sounding musical
instruments it become much more difficult to distinguish one from
another. External factors, too, can influence 'timbre' - for example,
if an instrument moves in a room relative to the listener. To
summarise, timbre is the spectrotemporal pattern of a generated sound
indicating the way the energy in the system is distributed between
different harmonics or frequency components and the way that
distribution is changing over time. The instruments of the orchestra, viewed as mechanical systems, can be classified in the following manner:
- Strings
- Bowed: Violin, viola, cello, double bass, bowed psaltry
- Plucked: Violin, viola, cello, double bass, lute, harp, citern, sitar, shamisen, mandolin, harpsichord
- Hammered: Zither, dulcimer, plucked psaltry
- Struck: Piano, clavichord
- Woodwinds
- Blown Flute: Transverse flute, recorder
- Blown Single reeds: Clarinet, bass clarinet, saxophone
- Blown Double reeds: Oboe, bassoon, contra bassoon, crumhorn
- Brass
- Blown: Cornet, trumpet, French horn, trombone, Flugel horn, tuba
-
Percussion
- Struck Tuned
- Bells, chimes
- Glockenspiel
- Xylophone, vibraphone, marimba
- Timpani
- Struck Untuned
- Bass and snare drums
- Cymbals
- Tam-tam
- Gong
- Claves, maracas, bongos, tambourine, whip, triangle, woodblock, bells
Reference:
Examine Timbre
Classification of Common Musical Instruments ::
Aerophones (Wind Instruments - Vibrating Air) | Chordophones (Stringed Instruments - Vibrating Strings) | Idiophones (Vibrating Instruments) | Membranophones (Vibrating Membrane Instruments) | Electrophones (Electronically Created Sounds) | Free Aerophones | Free Aerophones (Moving object vibrates air to create sound) | | | Bull-roarer Buzzer | Free Reed Aerophones (Vibrating reeds without resonators) | | | Accordion Harmonica Harmonium Mouth Organ Sheng |
| Flutes (Flue Voiced) (Air column split by lip of of the instrument) | Open Tube | | End Blown Single Flutes | | | Kaval | | End Blown Multiple Flutes | | | Panpipes Antara | | Whistle Blown | | | Boatswains Whistle Flageolet Recorder
| | Side Blown | | | Flute (transverse) | Closed Tube | | | Ocarina | Keyboard | | | Organ |
| Reedpipes (Vibration of reeds) | Double Pipes Triple Pipes Hornpipes Bladder pipe | | | Crumhorn | Bagpipes | | | Musette | Shawm | | | Oboe Cor Anglais or English Horn | Rackett | | | Sordone Sordun | Bassanelli Bassoon | | | Curtal Bassoon Contrabassoon High-pitched Bassoon Single-reed Bassoon | Sarrusophone Single-reed Reedpipes | | | Clarinet Saxophone | Free-reed Aerophones |
| Lip Vibrated Aerophones (Vibration of Lips) | Horn Fingerhole Horns | | | Cornett Serpent | Trumpet | | Conical Bore | | | Alphorn Bugle Cornet Euphonium Flugelhorn Tuba | | Cylindrical Bore | | | English Baritone Horn Sousaphone Trombone |
| Zithers | Ground Zithers Musical Bows | | Aeolian Bows Pluriare | Stick Zithers | | Vinã Bladder and String | Raft Zither Trough Zither Frame Zither Tube Zither Board Zither Long Zither | | Individually Bridged Long Zither Fretted Long Zither | Box Zither | | Monochord String Drum Trumpet Marine Psaltery Aeolian Harp Fretted Zither Bell Harp | | | Autoharp Dulcimer Zither |
| Keyboard Chordophones | Keyboard | | Experimental Keyboard Transposing Keyboard | Chekker Dulce Melos Clavichord | | Cimbal d'amour | Harpsichord | | Clavicytherium Gut-strung Harpsichords & Enharmonic Harpsichords Spinet Virginal Claviorganum Piano-organ | Bowed Keyboard Instruments | Pianoforte | | Harpsichord Piano Tangent Piano Sustaining Piano Pedalboard Piano Enharmonic Piano Player Piano Fortepiano Piano |
| Lyres | | Lyre Crwth |
| Harps | Ground Bows Harps | | Harp | Harp Zither Harp Lute |
| Lutes | Long-necked Lute Short-necked Lute Archlute | | Theorbo Theorbo Lute Chitarrone Angelica | Mandora Mandolin Sitar Cittern | | Archcittern | Bandora | | Orpharion Penorcon Polyphant Balalaika Charango Colascione | Guitar | | Gittern Vihuela Spanish Guitar Bass Guitar Bandurria Ukulele | Organistrum (hurdy-gurdy) |
| Bowed Chordophones | Bow Fiddle Rebec | | Kit Folk Rebec Polnische Geige | Lira da Braccio | | Lira da Gamba | Viola da Gamba Violone Baryton Viola | | Violino d'amore | Violin | | Viola Tenor Violin Violoncello or 'Cello Double Bass | | Concussion Idiophones | Clappers | | Claves Slapstick | Castanets Cymbals |
| Percussion Idiophones | Stamped Idiophones Percussion Beams | | Marimba Bell Lyre Celesta Gender Glockenspiel Orchestra Bells Saron Vibraphone | Percussion Disks | | Gong | Percussion Sticks | | Triangles | | Xylophone | | | Crystallophones Lithophones Metallophones | Percussion Tubes | | Stamping Tubes Slit Drums Tubular Bells & Chimes Angklung | Percussion Vessels | | Percussion Gourds & Pots The Echeion Steel Drums Bells | | | Temple Blocks Wood Block |
| Shaken Idiophones (Rattles) | Vessel Rattles | | Pellet Bells | | Gourd Rattles | | | Maracas | | Basketry Rattles Hollow Ring Rattles | Suspension Rattles | | Stick Rattles Sistrum Strung Rattles | Frame, Pendant, and Sliding Rattles | | Sistro |
| Scraped Idiophones | Scrapers | | Güiro | Cog Rattles | | Cog Rattle Ratchet Washboard
|
| Split Idiophone |
| Plucked Idiophones | | Jew's Harp | Thumb Piano | | Music Box Sansa Mbira |
| Friction Idiophones | Friction Sticks Friction-bar Pianos Friction Vessels | | Musical Glasses Glass Armonica Musical Saw |
| Predrum Membranophones | Ground drums Pot drums |
| Tubular Drums | Frame Drums Shallow Drums Cylinder Drums Hourglass Drums Conical Drums Goblet Drums Barrel Drums Rattle Drums Water Drums Drum Kit / Drum Set Talking Drums Drum Chimes | | Bass Drum Bongos Conga Snare Drum Tenor Drum Timbales Tom-tom |
| Kettledrums | | Timpani |
| Friction Drums |
| Mirliton | | Kazoo |
| Synthesizers | | Moog Theremin Ondes Martenot Trautonium | |
References:
List of musical instruments by Hornbostel-Sachs number
Taxonomy of Musical Instruments by Henry Doktorski

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