In
music
Music is the arrangement of sound to create some combination of Musical form, form, harmony, melody, rhythm, or otherwise Musical expression, expressive content. Music is generally agreed to be a cultural universal that is present in all hum ...
, a ditone (, from , "of two tones") is the
interval of a
major third
In music theory, a third is a Interval (music), musical interval encompassing three staff positions (see Interval (music)#Number, Interval number for more details), and the major third () is a third spanning four Semitone, half steps or two ...
. The size of a ditone varies according to the sizes of the two tones of which it is compounded. The largest is the Pythagorean ditone, with a ratio of 81:64, also called a comma-redundant major third; the smallest is the interval with a ratio of 100:81, also called a comma-deficient major third.
Pythagorean tuning
The Pythagorean ditone is the major third in
Pythagorean tuning
Pythagorean tuning is a system of musical tuning in which the frequency ratios of all intervals are determined by choosing a sequence of fifthsBruce Benward and Marilyn Nadine Saker (2003). ''Music: In Theory and Practice'', seventh editi ...
, which has an interval ratio of 81:64, which is 407.82
cents. The Pythagorean ditone is evenly divisible by two
major tone
Major most commonly refers to:
* Major (rank), a military rank
* Academic major, an academic discipline to which an undergraduate student formally commits
* People named Major, including given names, surnames, nicknames
* Major and minor in musi ...
s (9/8 or 203.91 cents) and is wider than a just major third (5/4, 386.31 cents) by a
syntonic comma
In music theory
Music theory is the study of theoretical frameworks for understanding the practices and possibilities of music. ''The Oxford Companion to Music'' describes three interrelated uses of the term "music theory": The first i ...
(81/80, 21.51 cents). Because it is a comma wider than a "perfect" major third of 5:4, it is called a "comma-redundant" interval.
"The major third that appears commonly in the
ythagoreansystem (C–E, D–F, etc.) is more properly known as the Pythagorean ditone and consists of two major and two minor semitones (2M+2m). This is the interval that is extremely sharp, at 408c (the ''pure'' major third is only 386c)."
It may also be thought of as four justly tuned
fifths minus two
octave
In music, an octave (: eighth) or perfect octave (sometimes called the diapason) is an interval between two notes, one having twice the frequency of vibration of the other. The octave relationship is a natural phenomenon that has been referr ...
s.
The
prime factor
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
ization of the 81:64 ditone is 3^4/2^6 (or 3/1 * 3/1 * 3/1 * 3/1 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2).
Just intonation
In
Didymus's diatonic and
Ptolemy
Claudius Ptolemy (; , ; ; – 160s/170s AD) was a Greco-Roman mathematician, astronomer, astrologer, geographer, and music theorist who wrote about a dozen scientific treatises, three of which were important to later Byzantine science, Byzant ...
's
syntonic tunings, the ditone is a just major third with a ratio of 5:4, made up of two unequal tones—a
major and a minor tone of 9:8 and 10:9, respectively. The difference between the two systems is that Didymus places the minor tone below the major, whereas Ptolemy does the opposite.
Meantone temperament
In
meantone temperament
Meantone temperaments are musical temperaments; that is, a variety of Musical tuning#Tuning systems, tuning systems constructed, similarly to Pythagorean tuning, as a sequence of equal fifths, both rising and descending, scaled to remain within th ...
s, the major tone and minor tone are replaced by a "mean tone" which is somewhere in between the two. Two of these tones make a ditone or major third. This major third is exactly the just (5:4) major third in quarter-comma meantone. This is the source of the name: the note exactly halfway between the bounding tones of the major third is called the "
mean
A mean is a quantity representing the "center" of a collection of numbers and is intermediate to the extreme values of the set of numbers. There are several kinds of means (or "measures of central tendency") in mathematics, especially in statist ...
tone".
Equal temperament
Modern writers occasionally use the word "ditone" to describe the interval of a major third in
equal temperament
An equal temperament is a musical temperament or Musical tuning#Tuning systems, tuning system that approximates Just intonation, just intervals by dividing an octave (or other interval) into steps such that the ratio of the frequency, frequencie ...
. For example, "In modern acoustics, the equal-tempered semitone has 100 cents, the tone 200 cents, the ditone or major third 400 cents, the perfect fourth 500 cents, and so on. …”
[Manuel Pedro Ferreira, "Proportions in Ancient and Medieval Music", in ''Mathematics and Music: A Diderot Mathematical Forum'', edited by Gerard Assayag, Hans Georg Feichtinger, and José Francesco Rodrigues, 1–17 (Berlin, Heidelberg, and New York: Springer, 2002): 5. .]
See also
*
Tritone
In music theory, the tritone is defined as a interval (music), musical interval spanning three adjacent Major second, whole tones (six semitones). For instance, the interval from F up to the B above it (in short, F–B) is a tritone as it can be ...
*
Tone
Tone may refer to:
Visual arts and color-related
* Tone (color theory), a mix of tint and shade, in painting and color theory
* Tone (color), the lightness or brightness (as well as darkness) of a color
* Toning (coin), color change in coins
* ...
*
Pythagorean interval
In musical tuning theory, a Pythagorean interval is a musical interval with a Interval ratio, frequency ratio equal to a Exponentiation, power of two divided by a power of three, or vice versa.Benson, Donald C. (2003). ''A Smoother Pebble: Mathe ...
References
{{Intervals, state=expanded
Major intervals
3-limit tuning and intervals
Just tuning and intervals
Thirds (music)