29 April 2013
Joe Monzo
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Hello everyone, Joe Monzo here, from San Diego. I am giving a special
presentation today on musical tuning. This is a highly technical
subject, and I have provided a glossary to everyone via email ahead of
time. I will try to keep the discussion non-technical as much as
possible, but the use of some of these terms is unavoidable, so
hopefully the glossary will help everyone to follow along.
Tuning quite obviously plays an important role in music. Unfortunately,
for over a century, one single tuning system has become an almost
universal standard in Western music, and by extension because of the
pervasive influence of Western culture, throughout the entire world.
That tuning system is known as "12-tone equal temperament",
commonly abbreviated to "12-et" or "12-tet" by tuning theorists. Some
tuning theorists prefer to recognize the mathematics of this tuning in
its abbreviation, calling it "12-edo" for "12 equal divisions of the
octave", and I will use this term myself in this discussion.
Of course, it is a good thing to have some kind of standard for tuning,
as it may apply to notation, terminology, and especially the building of
instruments. Indeed, it was the mass-production of instruments in Europe
and America during the 1800s which led to the desire for such a
standard. But I characterized this at the beginning as "unfortunate,"
and the reason I say that is because 12-edo has become _such_ a
universal standard that most musicians learn it as "the" tuning, as in
"the one and only tuning," without ever even thinking to question it.
The reality is that there is an infinity variety of frequencies
available for use in music, thus an infinite variety of musical pitches.
Using as an instrument the human voice, modern computerized instruments,
or certain instruments such as the unfretted strings (violin, cello,
etc.) or trombone, one may produce any pitches desired from this
infinite continuum. For most other instruments, however, a subset must
be selected, as for example the key systems on woodwind instruments or
the frets on a guitar can only produce a relatively small number of
pitches.
Before 1900, tuning was a subject of great interest among musicians. It
was around that time that 12-edo became firmly established as a
standard. Shortly afterwards, with Arnold Schoenberg using all 12
notes without regard to their former tonal functions and his influence
spreading that concept widely, most musicians ceased to learn anything
about tuning and simply accepted 12-edo as the one and only tuning.
This is a situation which has only recently begun to reverse. But
investigation of tuning has a very long background.
I will begin with a brief overview of the history of tuning.
The earliest documents we have which unequivocally specify the
mathematics of musical tunings are those of the ancient Greeks.
Pythagoreas is credited with most of the early advances in this area,
but certainly the knowledge is much older and probably was first written
down by the Sumerians about 2500 BC. The Babylonians conquered the
Sumerians and destroyed their civilization, but realized the value of
their culture and appropriate almost all of it as their own. Then during
the centuries around 500 BC the Persians and Greeks learned much of the
same knowledge from the Babylonians.
There was much discussion of tuning in the musical treatises of the
Greeks. When the Romans conquered Greece their culture was strongly
influenced by that of Greece, but all of the books about music continued
to be written in Greek, until very late in the Roman Empire. By far the
chief book about music in Latin from this time is that by Boethius,
written around 505 AD during the time of the German invasions which
ended the Western Roman Empire. Boethius's book remained the primary
reference in music-theory for about 1000 years.
It was after the Crusades, when Greek books were being brought to Italy
by returning soldiers , that scholars learned to read Greek and began
to realize the wealth of information that was contained in the Greek
books, thus slowly developing into the Renaissance. Many music-theorists
from 1300 to 1600 were very interested in tuning, and it became a very
active area of exploration and experimentation.
During the so-called "common-practice" period, roughly 1600 to 1900, the
meantone family of tunings, which I will discuss soon, became firmly
established as a standard in Western music. There were still many
varieties of tuning, but almost all of them were based in some way on
the concepts embodied in meantone.
The industrial revolution brought about the mass manufacture of
instruments, and 12-edo was finally established as the standard tuning.
12-edo belongs to the meantone family, as well as to many others,
and at first the notes were used according to the rules of tonal
music-theory, using 7-note diatonic subsets. As I stated above,
the use of all 12 pitches equally, stemming from the usage, theory and
teaching of
Schoenberg, eventually led to most musicians even becoming unaware that
there could be any other possibilities for tuning in music. With the
establishment of serial technique after World War II, use of any tunings
other than 12-edo nearly ceased, with the exception of a few microtonal
pioneers, perhaps most importantly Harry Partch.
The use of computers in music, especially since the era of personal
computers beginning in the early 1980s with the the IBM-PC and Apple
Macintosh, has greatly facilitated the exploration of alternate tuning
systems. I myself have created a Windows application called Tonescape,
which models the mathematics of tunings as geometrical diagrams and
allows the user to devise any tuning and compose with it. Widespread
adoption of the internet beginning in the late 1990s has also made it
easier for musicians interested in tuning to communicate with each
other, sharing ideas, compositions, and software.
For the rest of my discussion I will describe technical details
about tuning.
All sound is produced by vibration of the medium through which the sound
waves travel. Sound may travel through wood, water, the ground,
walls, indeed anything except a vacuum. For music we generally only
need to be concerned about the behavior of sound travelling through air.
The movement of air molecules excited by the sound's vibration is in the
form of a wave, radiating outward from the source of the vibration. In
each ear, our
eardrum (technically called the tympanum) vibrates in sympathy with the
air molecules, and picks up what our brain recognizes as sound. The
"amplitude" of a sound refers to the intensity of the vibration, and a
graph of the sound's amplitude over time describes its "waveform."
To produce what we recognize as a specific pitch, a waveform must be
"periodic" - that is, the wave shape must repeat a certain number of
times per second. An "aperiodic" shape is more random, and produces an
unpitched percussion sound. The number of repetitions in a periodic wave
is described as its "frequency," and this is measured as
cycles-per-second ("cps") which are normally abbreviated as "Hz", named
after the scientist Heinrich Hertz, who discovered electromagnetic
waves.
To measure frequencies, we must arbitrarily choose one particular
frequency as a reference. Today, this is usually "A-440", which refers
to the pitch whose frequency is 440 Hz and is named "A", the A above
middle-C which is written
on the second space of the treble staff . This is a standard
which was agreed upon by several countries in 1939.
In music, a pitch distance between two notes, and the sound made by those
two notes, is called an "interval." A "ratio" is a comparison of two
numerical quantities, and we use ratios to compare two frequencies.
Ratios are typical expressed as fractions in lowest terms, with the
number "1" representing the reference frequency. Thus, the simplest
interval is the "perfect prime" (also called "unison") which has the
ratio 1:1. Ratios may also be written as fractions, which helps to make
the mathematics more clear, but tuning theorists more commonly use the
colon to separate the numbers. The unison could thus also be written
1/1.
One of the most important characteristics of pitch is that we perceive
a phenomenon known as "octave-equivalence," which refers to the fact
that notes an octave apart, while clearly different pitches, have sonic
characteristics which are so similar that we give them the same
letter-name. The ratio of the octave is 2:1 -- so, for example, the
octave above A-440 is the next higher A (on a ledger-line above the
treble staff) at 880 Hz. Similarly, the octave below A-440, with the
ratio 1:2, is the A below middle-C at 220 Hz (on the top line of the
bass staff). Written as fractions, they would be 2/1 and 1/2,
respectively, making it clear that the frequencies are double and half
that of the reference pitch. It should also be easy to see that
reciprocal fractions indicate opposite directions of intervals: 2/1 goes
up, 1/2 goes down.
Ratios may be measured with either the highest or lowest note as the
reference, just as musical intervals may be calculated either up or down
without changing anything else about the interval. Tuning theorists will
typically invoke octave-equivalence, building a tuning system for one
reference octave and considering all ratios obtained by repeated
multiplication or division by 2 refer back to the same reference ratios.
Therefore, the ratio generally is an improper fraction whose value is
between 1 (the origin reference pitch) and 2 (the octave above).
Any number to the zero power equals 1 (n^0 = 1), and refers to the
origin reference pitch. Positive powers of 2 indicate octaves above, and
negative powers of 2 indicate octaves below. For example, 2^2 =4 and
is the same as the 4:1 ratio, which indicates a pitch 2 octaves above
the reference; 2^-3 = 1/8 and is the same as the 1:8 ratio, and
indicates a pitch 3 octaves below the reference.
I also want to mention here the concept of "cents," in which a semitone
of our standard 12-edo tuning is divided into 100 equal parts. Thus,
there are 1200 cents to the octave. This term was invented in the 1800s
and is in common use among tuning theorists, as a handy way to describe
the perceived size of intervals and relate it to our standard tuning.
This should be familiar to some of you who may use, for example,
electronic guitar tuners, and I will also be making use of the term
"cents" here . The cent-values of the semitones in 12-edo are thus:
0, 100, 200, 300, etc.
It should be evident that by using only powers of 2, we get nothing
but octaves -- there are no other notes with which to build a scale.
This was noticed in ancient times, and it was realized that an
additional number must be employed as a factor to create ratios which
will make a scale. The obvious next choice is to use 3. However, the 3:1
ratio is larger than 2:1 -- i.e., larger than an octave, and we will
want to construct a scale within one octave. 3:1 is actually the
interval which musicians call a 12th, that is, an octave plus a
perfect-5th. So to find the actual 5th, we need to make it one octave
smaller. If I do the math: 3/1 * 1/2 = 3/2. So 3:2 is the ratio of the
perfect-5th.
The earliest ancient writings, from Babylon and Greece, desribe tuning
lyre strings using the method of tuning-by-concords, in which
perfect-4ths and 5ths were tuned by ear until they produce no audible
beats. These intervals are mathematically the 4:3 and 3:2 ratio,
respectively. For example, using A-440 as a reference, the "E" a
perfect-5th above would have a ratio of 3:2, which is the same as 3/2
and thus equals 1.5, and so its frequency is 440 * 1.5 = 660 Hz. To find
the next note, the perfect-4th below that E would have a 3:4 ratio to
the E, and be a "B" with a frequency of 660 * 0.75 = 495 Hz; and so on.
This type of tuning was attributed to Pythagoras, although it was
in reality much older, but to this day it is still called "pythagorean tuning." It is
also known today as "3-limit" tuning, because the factors in all of the
ratios are 2 and 3, using no other numbers for factors.
The ancient Greeks realized that a stretched string could be measured
and then divided according to these ratios to produce the same pitches,
by placing bridges at the measured points along the single string .
Thus, the monochord (known as "canon" in Greek) was invented and used
for millenia as the primary tool for teaching music-theory.
It was also noticed that when using tuning-by-concords to create a
scale, the 13th note produced was very close in pitch to the origin
reference pitch, about 1/4 of a semitone (or 23.5 cents) higher than the
origin. This small interval was called a "comma," and is now known
specifically as the "pythagorean-comma." The ratio of the
pythagorean-comma is the very awkward number 531441:524288, which I find
much easier to write as 3^12:2^19 -- this describes transparently that
you go up 12 perfect-12ths (3^12, in the numerator of the ratio) and
down 19 octaves (2^19, in the denominator).
I will demonstrate the emergence of the pythagorean-comma by building
the whole tuning step-by-step, starting with A-440 =3^0 as the
reference and ignoring the powers of 2, since we are assuming
octave-equivalence (but they will show up in the denominators of the
ratios anyway), then multiplying the resulting ratio by 440 to obtain
the Hz value of each pitch. We have already obtained the perfect-5th E
at 660 Hz and, by going down a 4th from that, the major-2nd B at 495 Hz.
Continuing, and putting all the results in a table (the decimal places
for Hz are rounded off):
3^ 0 = 1:1 = 440 Hz = 0 cents = prime/unison = A 3^ 1 = 3:2 = 660 Hz = 702 cents = perfect-5th = E 3^ 2 = 9:8 = 495 Hz = 204 cents = major-2nd = B 3^ 3 = 27:16 = 742.5 Hz = 906 cents = major-6th = F# 3^ 4 = 81:64 = 556.9 Hz = 408 cents = major-3rd = C# 3^ 5 = 243:128 = 835.3 Hz = 1110 cents = major-7th = G# 3^ 6 = 729:512 = 626.5 Hz = 612 cents = augmented-4th = D# 3^ 7 = 2187:2048 = 469.9 Hz = 114 cents = augmented-prime = A# 3^ 8 = 6561:4096 = 704.8 Hz = 816 cents = augmented-5th = E# 3^ 9 = 19683:16384 = 528.6 Hz = 318 cents = augmented-2nd = B# 3^10 = 59049:32768 = 792.9 Hz = 1020 cents = augmented-6th = Fx 3^11 = 177147:131072 = 594.7 Hz = 522 cents = augmented-3rd = Cx 3^12 = 531441:524288 = 446.0 Hz = 23 cents = augmented-7th = Gx
For this last interval, "augmented-7th" is an anomalous name, because
since we are invokingoctave-equivalence, it is actually far smaller than any kind of 7th, andin fact is 446 Hz, only slightly higher in pitch than the reference
A-440 -- thus "pythagorean-comma" is a much better name for it.
In fact, the tuning of the reference "A" itself has varied much more widely
than that throughout its known history, being as low as 400 Hz during
the early Baroque period and as high as 456 in the 1880s.
The idea behind temperament is that pitches may be changed slightly,
so that the intervals formed are no longer mathematically the exact
ratios, but their size is close enough to the ratios that our ear
perceives them as being the same interval. The main reason for
temperament has been to formulate a tuning system which has a small
number of pitches that can fit comfortably under the hand on a keyboard,
but still represent a large enough variety of ratios so as to make the
harmony complex and interesting. Looking at the cents-values in the
table above, it can be seen that our standard 12-edo tuning is in fact
quite close to pythagorean tuning, but eliminates the problem of the
comma.
The 3:2 ratio (perfect-5th) is very close to 702 cents in size, and the
4:3 ratio (perfect-4th) is close to 498 cents. If each 5th in the
tuning-by-concords method is made 2 cents smaller, and each 4th is made
2 cents larger, then the pythagorean comma disappears and the 13th note
in the series really _is_ exactly the same as the origin pitch. This is
mathematically how we derive 12-edo tuning. The earliest evidence we
have of anyone actually deriving 12-edo mathematically comes from both
China and Europe at almost exactly the same time, in the 1500s; however,
I have some webpages which demonstrate that as long ago as 3000 BC the
Sumerian mathematical system was sophisticated enough that they could
have figured out a very close approximation to this tuning.
Aristoxenos was a very important ancient Greek theorist who refused to
quantify any intervals in his system with ratios, instead insisting that
musicians rely soley on their hearing to properly tune their music. But
he demonstrates tuning-by-concords in which the 13th note in the series
is supposed to be the same pitch as the origin note, so he manifestly
gets credit for being the first person to describe temperament.
Other ancient Greek writers described a variety of ratios for the
different genera (diatonic, chromatic, and enharmonic) they used.
When Boethius wrote his treatise around 505 AD, he used pythagorean
tuning for the measurement of the diatonic genus and his own unique
measurements for the other two genera. The so-called "Dark Ages" ensued
after the complete obliteration of the western Roman Empire by the
invading Germanic tribes, and when music-theory began to be written down
again after the reign of Charlemagne, in the 800s, Boethius's book was
widely recognized as the primary resource on music-theory. However, none
of the medieval writers said anything about the chromatic or enharmonic
genera, focusing entirely on the diatonic. Thus, pythagorean tuning
became firmly established as the only accepted tuning ... at least in
theory.
What was actually happening in practice during the Dark Ages is anyone's
guess. But it is
apparent that by the 1200s something very different was happening in
England. Many authors wrote about the "English countenance" (as we saw
on p. 98 of our Hanning text), which featured a predominance of parallel
3rds and 6ths. This indicates a shift in tuning practice, because in
pythagorean tuning the only consonant intervals are the octave, 4th, and
5th; 3rds and 6ths were considered to be dissonant because in that
tuning they _are_ rather dissonant (modern aficionados of pythagorean
tuning prefer to call them "active"). But extensive use of 3rds and 6ths
in singing would indicate that these intervals were being tuned to a
"sweeter" and more pleasing sound. In the 1300s English author Walter
Odington specified that 3rds were being tuned to the 80:64 ratio, which
reduces to 5:4, rather than the pythagorean 81:64. This is the first
explicit reference since ancient Greece to the 81:80 ratio, which is
today called the "syntonic-comma" and is about 21.5 cents. A major-3rd
with ratio 5:4 is about 386 cents, compared to the pythagorean
major-3rd of 408 cents. This is a quite audible difference.
The English style became very popular in continental Europe, and thus a
profound change in musical style was effected. Music-theorists became
more aware of these tuning subtleties, and also of the ancient Greek
books, and we find Marchetto of Padua
writing in his _Lucidarium_ of 1318 that the whole-tone was divided not
simply into two semitones, but rather into 5 "dieses," various
combinations of which made up _three_ different sizes of semitones. The
exact measurements of Marchetto are not entirely clear, and I have
devoted a whole webpage to a discussion of this, with some of my own
interpretations. But it is obvious that music by this time was becoming
more chromatic as well as more "sweet."
Finally, in 1482, Portuguese theorist Bartolomeo Ramos published a
method to divide a monochord string for a tuning with extensive use of
ratios which included factors of 2, 3, and 5. (I also have a webpage
devoted entirely to Ramos.) This is the first
description of a "5-limit" tuning since ancient Greece. This is also the
beginning of the more general idea that ratios with smaller numbers in
their terms are more pleasing to the ear than those with larger numbers.
This type of tuning eventually came to be known as "just-intonation,"
nowadays usually abbreviated as "JI."
However, very soon it became apparent that trying to use a complete
5-limit system in practice would engender some difficult circumstances.
For one thing, there was no longer simply "a" whole-tone, but rather two
different sizes of whole-tone: the pythagorean 9:8 ratio of 204 cents,
and a smaller 10:9 ratio of 182 cents. This is not really a problem when
singing or playing melody, but by the Renaissance chordal harmony had
become an important aspect of music, and if both of these ratios
represented a "B," choosing between them for a particular chord meant
the difference between chords that could be either sweetly consonant or
grindingly dissonant. Indeed, when performing this repertoire with close
attention to such details of tuning, one can sometimes hear composers
making imaginative and effective use of consonance and dissonance,
effects which are completely lost when rendering these pieces in our
modern 12-edo tuning.
Another, more drastic, problem was that of "commatic drift," where a
choir singing a capella would keep each chord perfectly in tune
harmonically, using 5-limit ratios, but the overall pitch would drift
downward (flat) because of the use of common-tones and the various
changes in ratios used for notes
which have the same letter-name. Thus, beginning on the reference "A"
chord, at the next cadence the same notated chord would be a
syntonic-comma lower.
By the early 1500s theorists were already describing a form of
"temperament" in which each 5th in the pythagorean chain would be
narrowed slightly, so that going up the chain A:E:B:F#:C# would result
in a C# which relates to A by the 5:4 (i.e., 80:64) ratio rather than
the pythagorean 81:64. According to this method of tuning, each 5th is
made 1/4 of a syntonic-comma (about 5 cents) smaller than the "perfect"
3:2 ratio. The 9:8 ratio, "B" in our example, being the second note in
the chain-of-5ths, would be narrowed by 1/2 comma, putting it exactly
between the two "Bs" of ratios 10:9 and 9:8. This is the origin of
the name "meantone" -- the mean whole-tone midway between the two
5-limit JI versions of the whole-tone.
The precise name of this tuning is "1/4-comma meantone," and it was
used in practice starting around 1511, and lasted several centuries.
The "perfect" 5th is a little less than 697 cents, enough to produce
audible beating and thus not quite "perfect."
Various other "shavings" of the 5th were tried,
and the first mathematically exact description
of a temperament is that of 2/7-comma meantone, by Zarlino in 1558.
This interesting tuning shows the preference of Renaissance composers
for 5-limit ratios: while the error from 5-limit JI is slightly greater
for the 5th and 4th, just over 6 cents, it is the meantone which has
the least possible amount of error for the 3rds and 6ths, and the
error for all four of those intervals is exactly the same: 1/7-comma,
about 3 cents.
At the same time, Renaissance composers and theorists were becoming
more and more intrigued by the ancient Greek genera. Indeed, the
work of Marchetto which I already mentioned may have been a result
of this, but we have no way of knowing for sure. But in 1555 Nicola
Vicentino published a very ingenious tuning which theorists today
refer to as "adaptive-JI," in which two chains-of-5ths are tuned
in 1/4-comma meantone, but the chains are 1/4-comma apart. The result
is that, by choosing the root and 3rd of a triad from one chain and
the 5th from the other chain, every triad can be played with exact
5-limit JI ratios as measured from the root of the chord, but the
melodic drift is never more than 1/4-comma (about 5 cents).
By the time of Mozart, the preferred tuning was 1/6-comma meantone,
in which the narrowing of the 5th is less than 4 cents, and finally
with the adoption of 12-edo, which is remarkably close to 1/11-comma
meantone, the amount of tempering is reduced further still to 2 cents.
The result is that 3rds and 6ths are no longer as sweet as they were
during the Renaissance. But if we learn about tuning and give
careful consideration to the tunings intended by Renaissance composers,
their music takes on a whole new sound and feeling.
Thank you.
========
REFERENCES
Barker, A. (1989). Greek musical writings. (Vol. 2). New York: Cambridge University Press.
Bower, C. (1989). Boethius: Fundamentals of music. New Haven: Yale University Press.
Chesnut, J. (1977). Mozart's teaching of intonation. Jouranl of the American musicological society, 30(2), 254-271.
Heartz, D., Mann, A., Oldman, C., & Hertzmann, E. (1965). Thomas Attwoods theorie- und kompositionsstudien bei Mozart. (Wolfgang Amadeus Mozart: Neue Ausgabe saemtlicher Werke ed., Vol. Ser. X, Werkgruppe 30, Bd. 1). Kassel.
Helmholtz, H. (1954). A. Ellis (Ed.), On the sensations of tone (2nd ed.). New York: Dover.
Herlinger, J. (1985). The Lucidarium of Marchetto of Padua (1317-18). Chicago: University of Chicago Press.
Meibom, M. (1652). Antiquae musicae auctores septem, graece et latine. (Vol. 1). Amsterdam: Apud Ludovicum Elzevirium.
Partch, H. (1979). Genesis of a music: An account of a creative work, its roots, and its fulfillments. (2nd, enlarged paperback ed.). New York: Da Capo Press.
Strunk, O. (1998). Source readings in music history. (rev. ed.). New York: W. W. Norton & Co.
Vicentino, N. (1996). M. Maniates & C. Palisca (Eds.), Ancient music adapted to modern practice. New Haven: Yale University Press. (Original published 1555).
WEBPAGES BY JOE MONZO
Aron, P. (1523). Toscanello in musica. English translation by Perretti, L. (2003).
http://tonalsoft.com/monzo/aron/toscanello/aron_toscanello.htm
Monzo, J. (2002). Speculations on Sumerian tuning.
http://tonalsoft.com/monzo/sumerian/sumerian-tuning.aspx
Monzo, J. (2002). Vicentino's adaptive-JI of 1555.
http://tonalsoft.com/monzo/vicentino/vicentino.aspx
Monzo, J. (2003). The measurement of Aristoxenus's divisions of the tetrachord.
http://tonalsoft.com/monzo/aristoxenus/aristoxenus.aspx
Monzo, J. (2003). Ramos's division of the monochord.
http://tonalsoft.com/monzo/ramos/ramos.aspx
Monzo, J. (2007). Mozart's tuning: 55-edo and its close relative, 1/6-comma meantone.
http://tonalsoft.com/monzo/55edo/55edo.aspx
Monzo, J. (2007). Speculations on Marchetto of Padua's "Fifth-Tones".
http://tonalsoft.com/monzo/marchetto/marchetto.aspx
Zarlino, . (1558). Le institutione harmoniche. English translation by Perretti, L. (2003).
http://tonalsoft.com/monzo/zarlino/1558/zarlino1558-2.aspx
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