Pythagorean interval

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Pythagorean interval

The intervals of Pythagorean tuning

Pythagorean tuning

Pythagorean tuning is a system of musical tuning in which the frequency relationships of all interval are based on the ratio sesquialterum. Its name comes from medieval texts which attribute its discovery to Pythagoras, but its use has been documented as long ago as 3500 B.C....
are just intervals

Just intonation

In music, just intonation is any musical tuning in which the frequency of notes are related by ratios of whole numbers. Any interval tuned in this way is called a just interval; in other words, the two notes are members of the same harmonic series ....
involving only powers of two and three.

The fundamental intervals are the superparticular ratios

Superparticular number

Superparticular numbers, also called epimoric ratios, are improper vulgar fractions of the formSuperparticular numbers were written about by Nicomachus in his treatise "Introduction to Arithmetic"....
2/1, 3/2, and 4/3.

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Encyclopedia

Names	Ratio	Cents Cent (music) The cent is a logarithmic scale unit of measure used for musical interval . Typically cents are used to measure extremely small intervals, or to compare the sizes of comparable intervals in different tuning systems, and in fact the interval of one cent is much too small to be heard between successive notes....	ET Cents
(perfect) unison	1/1	0.00	0
comma	531441/524288	23.46	0
minor second limma minor semitone	256/243	90.22	100
augmented unison apotome major semitone	2187/2048	113.69	100
diminished third	65536/59049	180.45	200
major second tone	9/8	203.91	200
minor third semiditone	32/27	294.13	300
augmented second	19683/16384	317.60	300
diminished fourth	8192/6561	384.36	400
major third ditone	81/64	407.82	400
perfect fourth diatessaron sesquitertium	4/3	498.04	500
augmented third	177147/131072	521.51	500
diminished fifth	1024/729	588.27	600
augmented fourth tritone	729/512	611.73	600
diminished sixth	262144/177147	678.49	700
perfect fifth diapente sesquialterum	3/2	701.96	700
minor sixth	128/81	792.18	800
augmented fifth	6561/4096	815.64	800
diminished seventh	32768/19683	882.40	900
major sixth	27/16	905.87	900
minor seventh	16/9	996.09	1000
augmented sixth	59049/32768	1019.55	1000
diminished octave	4096/2187	1086.31	1100
major seventh	243/128	1109.78	1100
(perfect) octave diapason	2/1	1200.00	1200
augmented seventh octave + comma	531441/262144	1223.46	1200

The intervals of Pythagorean tuning

Pythagorean tuning

Just intonation

Superparticular number

Octave

In music, an octave The octave is occasionally referred to as a diapason.The octave above an indicated note is sometimes abbreviated 8va, and the octave below 8vb....
or diapason (Greek

Ancient Greek

Ancient Greek is the historical stage in the development of the Greek language spanning across the Archaic Greece , Classical Greece , and Hellenistic civilization periods of ancient Greece and the classical antiquity....
for "across all"). 3/2 is the perfect fifth

Perfect fifth

The perfect fifth is the musical interval between a note and the note seven semitones above it on the musical scale. For example, the note G lies a perfect fifth above C; D is a perfect fifth above G, C is a perfect fifth above F, and so on....
, diapente ("across five"), or sesquialterum. 4/3 is the perfect fourth

Perfect fourth

The perfect fourth is a musical interval which spans four diatonic scale scale degree. It consists of the note and the note five semitones above it on the musical scale....
, diatessaron ("across four"), or sesquitertium. These three intervals and their octave equivalents, such as the perfect eleventh and twelfth, are the only absolute consonances

Consonance and dissonance

In music, a consonance is a harmony, Chord , or interval considered stable, as opposed to a dissonance ? considered unstable . The strictest definition of consonance may be only those sounds which are pleasant, while the most general definition includes any sounds which are used freely....
of the Pythagorean system. All other intervals have varying degrees of dissonance, ranging from smooth to rough.

The difference between the perfect fourth and the perfect fifth is the tone or major second

Major second

A major second , also called a whole step or a whole tone,One source says step is "chiefly US."The preferred usage has been argued since the 19th century:...
. This has the ratio 9/8, and it is the only other superparticular ratio of Pythagorean tuning, as shown by Størmer's theorem

Størmer's theorem

In number theory, St?rmer's theorem, named after Carl St?rmer, gives a finite bound on the number of consecutive pairs of smooth numbers that exist, for a given degree of smoothness, and provides a method for finding all such pairs using Pell equations....
.

Two tones make a ditone, a dissonantly wide major third

Major third

A major third is one of two commonly occurring musical intervals that span three diatonic scale degrees, the other being the minor third. It is denoted 'major' because it is the larger of the two: the major third is a leap of four semitones, the minor third three....
, ratio 81/64. The ditone differs from the just major third (5/4) by the syntonic comma

Syntonic comma

In music theory, the syntonic comma , also known as the comma of Didymus the Musician or Ptolemy comma, is a small interval between two musical notes, equal to the frequency ratio 81:80, or around 21.51 Cent s....
(81/80). Likewise, the difference between the tone and the perfect fourth is the semiditone, a narrow minor third

Minor third

A minor third is a Interval of three semitones. It is the smaller of two commonly occurring musical intervals compounded of two steps of the diatonic scale....
, 32/27, which differs from 6/5 by the syntonic comma. These differences are "tempered out" or eliminated by using compromises in meantone temperament

Meantone temperament

Meantone temperament is a musical temperament, which is a system of musical tuning. In general, a meantone is constructed the same way as Pythagorean tuning, as a chain of perfect fifths, but in a meantone, each fifth is narrowed by the same amount in order to make the other intervals, like the major third, closer to their ideal just intonat...
.

The difference between the minor third and the tone is the minor semitone or limma of 256/243. The difference between the tone and the limma is the major semitone or apotome ("part cut off") of 2187/2048. Although the limma and the apotome are both represented by one step of 12-pitch equal temperament

Equal temperament

Equal temperament is a musical temperament, or a system of Musical tuning in which every pair of adjacent notes has an identical frequency ratios....
, they are not equal in Pythagorean tuning, and their difference, 531441/524288, is known as the Pythagorean comma

Pythagorean comma

The Pythagorean comma , named after the ancient mathematician and philosopher Pythagoras, is the Microtonal music Pythagorean interval defined as the difference between a Pythagorean apotome and a Limma, e.g....
.

There is a one-to-one correspondence between interval names (number of scale steps + quality) and frequency ratios. This contrasts with equal temperament, in which intervals with the same frequency ratio can have different names (e.g., the diminished fifth and the augmented fourth); and with other forms of just intonation, in which intervals with the same name can have different frequency ratios (e.g., 9/8 for the major second from C to D, but 10/9 for the major second from D to E).

Pythagorean interval

See also

External links