ogata@leviathan.cs.umd.edu (Jefferson Ogata) (03/15/91)
Well, I've been getting so many requests for this, I decided to just post
it. It's long, and not comprehensive, but I hope it's useful to some
people. I was surprised at the number of requests; it seems a lot of
people are interested in different tunings.
Feel free to email or post with comments or corrections. I'll be happy
to include further information in this compilation, and I intend to
add some more stuff myself later. Someone else has volunteered to send
me notes on the Hindu scale.
I didn't provide a bibliography, but the references are given in the text.
- jeff
As there has been some interest in non-(twelve-tone equal-tempered) scales
lately, I decided to dig out some numbers. I have used a variety of scales
myself, but some of the pitches I got out of books I don't own, and others
were already programmed into synths. Not having a frequency counter, it
would be difficult for me to get the actual pitches or deviations for these
scales, so I dug up some books in the music library here at U of MD.
The topics are in the following order: equal-tempered and Just twelve-
tone scales, Pythagorean twelve-tone and Hindu 22-tone scales, Chinese
and Japanese scales, Balinese and Javanese scales, scales of the Shona
people of Zimbabwe. I will try to follow up this post with a bit more
info in a few days.
I have used several of the scales in discussion; I recorded some
music in Just scales last year. I wish I could provide a table for
this interesting 19-tone tuning on my Proteus/1, but I don't know
what the values are (I'll see if I can find out).
Some basic scale theory:
The octave is the standard for measurement of intervals by Western
scholars. An octave is the interval attained by multiplying the
frequency of the lower pitch by exactly 2. Octaves are divided into
cents. A cent is 1/1200 of an octave, or 1/100 of an equal- tempered
semitone. To raise a pitch by one cent, multiply its frequency by
2 ** (1/1200), where ** denotes "to the power of". To raise a pitch
by one semitone (== 100 cents) multiply its frequency by
2 ** (100/1200) (== 2 ** (1/12)). In general, to raise a pitch by
n cents, multiply its frequency by 2 ** (n/1200).
To determine how many cents frequency Y is above frequency X, take
1200 * log2 (Y/X). If you don't have a calculator with log2, take
1200 * (logx (Y/X)) / (logx 2) where logx is the log you do have on
your calculator.
The tables I give are all measured in cents. Where I give a table for
a scale, I give a version where every pitch is in cents above the
tonic.
When tuning a synthesizer to one of these scales, pick a tonic for
reference. For each note that you decide to map to one of the pitches
in question, note how many semitones above the tonic it is. Now
subtract 100 cents from the target interval for each semitone. For
example: suppose the tonic is A, I need to map a pitch 340 cents
above the tonic to the C. I subtract 300 cents and tune C 40 cents
sharp. To map a pitch 685 cents above the tonic to the E, I subtract
700 cents and tune E 15 cents flat.
The "standard" of the octave, while common, is not universal; check out
the stuff on the Shona mbira tunings at the end.
When two sustained notes are played together, a third implied tone
arises, called a beat or interference beat. The frequency of the beat
tone is equal to the difference in frequency between the two sounding
tones. Frequently this beat lands in subsonic frequencies (i.e. < ~20
Hz), and people have traditionally avoided such beats because they
are often disturbing to human physiology (perhaps it is a sign of an
earthquake that triggers an emotional response). For an example of
the problem of beats, play two notes together a semitone apart on a
piano in a low octave. You may be able to perceive a low-frequency
tone, and the overall sound will probably be unpleasant. Now play the
same two notes in a high octave. The beat will no longer sound
offensive, since it is no longer a subsonic; higher notes on an
equal-tempered instrument are farther apart frequency-wise than
lower notes. Intervals that are separated by a factor of low integer
ratios (e.g. the Just perfect fifth 3/2) have beats that are easy
to keep out of subsonic frequencies. For example, the 3/2 ratio
always has a beat that is exactly one octave lower than the low
tone. The 2/1 or octave ratio has a beat that is exactly equal to
the low tone. This is why octaves sound so good; the beat reinforces
the chord.
Note that subsonics are not always unpleasant; some frequencies are
very soothing; other frequencies sound good by themselves. Subsonics
between ~1 and ~20 Hz often sound annoying when added to music.
Here is a table from _The Gamelan Music of Java and Bali_ by Donald A.
Lentz (1965, University of Nebraska Press LCCCN 65-10545) pp. 24-25. I have
chopped this table into bits, as it is in a wide format in the book. The
table is based on a tonic of C and gives a lot of cent values. The cent
values are given with no fractional parts. Comments are mine.
The equal-tempered twelve-tone (Tempered) scale is the standard for modern
Western music. The Just (perfect) scale is the ancient beat-canceling
integral-ratio scale commonly used until the eighteenth century in Western
music. It is still used by some performers of ancient music for the sake of
authenticity and in certain contexts by other performers. The actual pitches
of the Just scale depend on which note is taken as the tonic of the scale.
The ratio and cent values, however, remain constant regardless of the tonic.
Lentz apparently made this table in C so he could provide note names.
-I- -II- -III-
Cents Equal Just
above Tem- (Using
Funda- pered C as a
mental Tonic)
Name Interval Ratio Interval Name
0 C Unison 1/1 Unison C
100 Db Half step
200 D Whole step
204 9/8 Whole step D
300 Db Minor 3rd
386 5/4 Major 3rd E
400 E Major 3rd
498 4/3 Perfect 4th F
500 F Perfect 4th
600 F# Aug. 4th
700 G Perf. 5th
702 3/2 Perf. 5th G
800 G# Aug. 5th
884 5/3 Maj. 6th A
900 A Maj. 6th
1000 A# Aug. 6th
1088 15/8 Maj. 7th B
1100 B Maj. 7th
1200 C Perf. 8ve 2/1 Perf. 8ve C
The Pythagorean scale is derived from perfect fifths alone. The intonation
of the Renaissance period used eight ascending fifths and three descending
fifths. When 12 perfect 3/2 fifths have been ascended and the octave has
been corrected back to the original register, the resulting ratio is
531441/524288 (== (3/2) ** 12 / 2 ** 7). This interval is known as a comma
and is equal to 24 cents.
The Hindu scale is based both on ascending fifths and ascending fourths.
The eleven pitches from ascending fifths and the eleven pitches from
ascending fourths are combined to produce a 22-tone scale. There are
quite a few details about this scale that I won't try to summarize here.
The Pramana is the distance between Srutis 4 and 5, which are the two
whole-tone intervals of the Just scale: 9/8 (between I and II) and
10/9 (between II and III).
-I- -IV- -V-
Cents Pythagorean Hindu
above
Funda-
mental Cycle Sruti
No. Interval Name Ratio No. Name
0 0 Unison C 1/1 1 Sa
22 81/80 Pramana
24 12 Comma
90 Minor 2nd (Limma) 256/243 2 Ri
112 16/15 3 Ri
114 7 Aug. Prime C#
182 10/9 4 Ri
204 2 Maj. 2nd D 9/8 5 Ri
294 -3 Min. 3rd Eb 32/27 6 Ga
316 6/5 7 Ga
318 9 Aug. 2nd D#
386 5/4 8 Ga
408 4 Maj. 3rd E 81/64 9 Ga
498 -1 Perf. 4th F 4/3 10 Ma
520 27/20 11 Ma
590 45/32 12 Ma
600 6 Aug. 4th F# 64/45 13 Pa
702 1 Perf. 5th G 3/2 14 Fa
792 128/81 15 Dha
814 8/5 16 Dha
816 8 Aug. 5th G#
884 5/3 17 Dha
906 3 Maj. 6th A 27/16 18 Dha
996 -2 Min. 7th Bb 16/9 19 Ni
1018 9/5 20 Ni
1020 10 Aug. 6th A#
1088 15/8 21 Ni
1110 5 Maj. 7th B 243/128 22 Ni
I am skipping Columns VI and VII of the table, which give blown fifths
and string-length division values. The blown fifths are Chinese intervals
arrived at by cutting bamboo tubes of particular lengths. These tubes
produce small fifths that range between 670 and 680 cents. A study of these
tubes by Dr. E. M. von Hornbostel claimed that the average interval is 678
cents. This produces a cycle of 23 fifths, and the comma after 23 fifths is
6 cents. Here are some quotes from pp. 27-28 of Lentz.
"Two theoretical systems evolved in China, one derived from the
Cyclic Pentatonic and the other from the division of string lengths.
They are found combined in the highest form of Ch'in music. The Cyclic
Pentatonic, arrived at mathematically, is important in Chinese musical
thought. The conception of the twelve liis (tones) dates back to the Han
dynasty. They are formed by building empirical fifths in a manner
similar to that used in Pythagorean tuning. Methods of arriving at these
fifths included the use of twelve tubes. Levis indicates that a stopped
bamboo tube 230 millimeters long, 8.12 millimeters in diameter, and
vibrating at 366 vibrations per second was the Yellow Bell (Huang Chong),
the standard established by the Bureau of Weights and Measures in 239
B.C. Tube number two was made two-thirds the length of the reference
tube; number three was made equal to two-thirds of number two and then
doubled to bring the tone within the ambit of an octave. This process was
repeated for each of the twelve tubes. The fifths produced by these tubes
were small compared to Western fifths. Various musicologists place them
between 670 and 680 cents as compared to the Just fifth of 702 cents."
"In Java musicians and gamelan makers, when queried about the small
fifth, explained it as coming from nature but gave no specific examples.
At dawn one morning in a small village in Java, a bird was singing with
a call of an octave and a small fifth. This I recorded. The same song
was frequently heard later and again recorded for confirmation of
interval. It is a fascinating bit of fancy that the fifth in the bird
call when measured on the stroboscope varied only two or three cents
from the one of 678 cents. There are also indications that the small
fifth might have been a standard in Sumeria and Egypt."
"In music for the ch'in, a zither-type instrument, the seven
strings are tuned to the Cyclic Pentatonic. Each string has frets or
nodes dividing it into the following lengths: 1/2, 1/3, 2/3, 1/4,
3/4, 1/5, 4/5, 1/6, 5/6, 1/8, and 7/8. The resultant cent values are
listed in column VII of Chart IV. Each string employs the principle of
Just intonation, but many microtonic intervals result when this theory of
string length division is combined with the above Cyclic Pentatonic
procedure, which is used to tune the seven open strings. There is no
counterpart in Western Tempered music. Many Oriental stringed instruments
use frets which are movable. The accuracy of placing the fret, which is
done by ear, can greatly affect the pitch and thus produce noticeable
deviances in a system using natural intervals of microtonic size. The
tones of the present-day scales of Japanese koto and gekkin music evolved
with variance from the Ch'in principle.
"Western musicians think of the fifth as being an interval of 700 to
702 cents. The deviation of only two cents betwen Just, Pythagorean, and
Tempered fifths is so small that these sizes are accepted as being the
true fifth. This is not the case in Oriental music, even though some
Western musicologists try to explain their fifths as anomalies from the
Western norm. The conception of the fifths is in many cases very
different. For the most part they are smaller than the Western fifth. In
Chinese music, another common theoretical fifth of 693 cents, as
contrasted with the Cyclic fifth of 678 cents, results from combining
three of the characteristic large seconds derived from string-length
division (see column VII in Chart IV). Each of the large seconds has a
value of 231 cents and a ratio of 8/7."
"Fifths of varying sizes are produced on different pipes when the
end-correction factor is not considered, thus not fitting a theoretical
system. These convert to a standard when duplicated. This procedure of
duplication is found in China along with the theoretical fifth, and
although one cannot find positive documentation of it for the gamelans of
Java and Bali, it is highly possible that it became a factor in the
varying sizes of the fifth there too."
On p. 33 the gamelan scale is described:
"Three basic tones and two or four secondary tones are the
background of the gamelan tonal system. The main tone, called dong in
Bali, is supported by two tones, one a fifth above (called dang) and a
the other a fifth below (called dung). The secondary tones are a fifth
above (d`eng) and a fifth below (ding) the supporting tones. By bringing
the five tones within an octave, the following scale results: dong,
d`eng, dung, dang, ding.
"...For convenience the tone names of Western notation will be
used, with C arbitrarily chosen as a starting tone. But it should be
recalled that the Oriental fifths are variable in size and in all
probability will not correspond to a Western fifth.
This results in a scale named C D F G Bb.
"In the Balinese-Javanese five-tone scale, a large interval,
approximately a minor third, which will vary in size from one gamelan to
the next, occurs between the second and third and the fourth and fifth
degrees."
Using a fifth of 678 cents, we can generate one example of this scale:
Degree Cents above tonic
I 0 == unison
II 156 == two fifths minus one octave
III 522 == octave minus one fifth
IV 678 == one fifth
V 1044 == two octaves minus two fifths
In _Musics of Vietnam_ by Pham Duy, Edited by Dale R. Whiteside (1975,
Southern Illinois University Press) Duy explains that theoretically the
Khmer scale of South Vietnam is divided into seven equally-spaced tones.
This would make cent values as follows:
Degree Cents above tonic
I 0
II 171
III 342
IV 514
V 685
VI 857
VII 1028
However, Duy notes that in practice the intervals in the Khmer scale are not
exactly equal.
In _The Soul of Mbira_ by Paul F. Berliner (1978, 1981, University of
California Press), Berliner describes a number of tunings used for the
mbira, which is a kalimba-like African instrument common in Zimbabwe.
Apparently each region has its own tuning, and different instrument makers
tune their instruments differently. The prevailing theory of Shona mbira
tuning is that "mbira makers and players use a distinctive, well defined
scale, with only slight variation in different parts of the country....It
can be described as a seven-note scale, with all the intervals equal."
(p. 66) However, Berliner found in a sample of tunings that the variation
was very large, varying between 37 to 286 cents between adjacent scale
degrees, and not equal at all. The various mbira players select their
instruments based on a variety of factors, including tuning, which they
refer to collectively as the "chuning" of the instrument.
"...I asked several musicians who owned these mbira...to select from
a set of fifty-four forks tuned 4 c.p.s. apart (from 212 c.p.s. to 424
c.p.s.) the individual forks which each thought matched the tuning of the
keys on his respective instrument. The fact that they sometimes said that
the pitch of an mbira key fell between two tuning forks demonstrated that
the musicians could discern fine variations in tuning."
Here is one of the tunings Berliner gives:
Between Mbira interval in cents
C and D 185
D and E 204
E and F 204
F and G 163
G and A 158
A and B 137
B and C 251
This gives the following table:
Degree Cents above tonic
I 0
II 185
III 389
IV 593
V 756
VI 914
VII 1051
oct. 1302
Note that the octave is not a factor of two in this tuning. Apparently the
"octaves" are highly variable in mbira tunings. Berliner gives some tables
of the variations from a "true" octave he found in various mbiras.
The octave also is not exact in Scottish and Irish bagpipe music, since the
high overtone used for the octave on the canter pipe is somewhat off. The
high notes of a bagpipe melody tend to come out weaker and a little bit
flat.
This concludes the summary of information I collected; I hope to follow
this up with some other examples in a few days. Happy tuning.
--
Jefferson Ogata ogata@cs.umd.edu
University Of Maryland Department of Computer Science