Pythagorean interval - AbsoluteAstronomy.com

Musical tuning

In music, there are two common meanings for tuning:* Tuning practice, the act of tuning an instrument or voice.* Tuning systems, the various systems of pitches used to tune an instrument, and their theoretical bases.-Tuning practice:...

theory, a Pythagorean interval is a musical interval with frequency ratio

Interval ratio

In music, an interval ratio is a ratio of the frequencies of the pitches in a musical interval. For example, a just perfect fifth is 3:2 , 1.5, and may be approximated by an equal tempered perfect fifth which is 27/12, 1.498...

equal to a power

Exponentiation

Exponentiation is a mathematical operation, written as an, involving two numbers, the base a and the exponent n...

of two divided by a power of three, or vice versa

Vice Versa

Vice Versa: A Lesson to Fathers is a novel by F. Anstey, first published in 1882.The title originates from the Latin phrase, "vice versa", meaning "the other way around".-Plot summary:...

. For instance, the perfect fifth

Perfect fifth

In classical music from Western culture, a fifth is a musical interval encompassing five staff positions , and the perfect fifth is a fifth spanning seven semitones, or in meantone, four diatonic semitones and three chromatic semitones...

with ratio 3/2 (equivalent to 3¹/2¹) and the perfect fourth

Perfect fourth

In classical music from Western culture, a fourth is a musical interval encompassing four staff positions , and the perfect fourth is a fourth spanning five semitones. For example, the ascending interval from C to the next F is a perfect fourth, as the note F lies five semitones above C, and there...

with ratio 4/3 (equivalent to 2²/3¹) are Pythagorean intervals.

All the intervals between the notes of a scale are Pythagorean if they are tuned using the Pythagorean tuning

Pythagorean tuning

Pythagorean tuning is a system of musical tuning in which the frequency relationships of all intervals are based on the ratio 3:2. This interval is chosen because it is one of the most consonant...

system. However, some Pythagorean intervals are also used in other tuning systems. For instance, the above mentioned Pythagorean perfect fifth and fourth are also used in just intonation

Just intonation

In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a just interval. The two notes in any just interval are members of the same harmonic series...

Interval table

Name	Short	Other name(s)	Ratio Interval ratio In music, an interval ratio is a ratio of the frequencies of the pitches in a musical interval. For example, a just perfect fifth is 3:2 , 1.5, and may be approximated by an equal tempered perfect fifth which is 27/12, 1.498...	Factors	Derivation	Cents Cent (music) The cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each...	ET Cents	Fifths
diminished second Diminished second In modern Western tonal music theory a diminished second is the interval between notes on two adjacent staff positions, or having adjacent note letters, whose alterations cause them, in ordinary equal temperament, to have no pitch difference, such as B and C or B and C...	d2		524288/531441	2¹⁹/3¹²		-23.460	0	-12
(perfect) unison Unison In music, the word unison can be applied in more than one way. In general terms, it may refer to two notes sounding the same pitch, often but not always at the same time; or to the same musical voice being sounded by several voices or instruments together, either at the same pitch or at a distance...	P1		1/1		1/1	0.000	0	0
Pythagorean comma Pythagorean comma In musical tuning, the Pythagorean comma , named after the ancient mathematician and philosopher Pythagoras, is the small interval existing in Pythagorean tuning between two enharmonically equivalent notes such as C and B , or D and C...			531441/524288			23.460	0	12
minor second Minor second In modern Western tonal music theory a minor second is the interval between two notes on adjacent staff positions, or having adjacent note letters, whose alterations cause them to be one semitone or half-step apart, such as B and C or C and D....	m2	limma, diatonic semitone, minor semitone	256/243	2⁸/3⁵		90.225	100	-5
augmented unison Augmented unison In modern Western tonal music theory an augmented unison is the interval between two notes on the same staff position, or having the same note letter, whose alterations cause them, in ordinary equal temperament, to be one semitone apart. In other words, it is a unison where one note has been raised...	A1	apotome, chromatic semitone, major semitone	2187/2048	3⁷/2¹¹		113.685	100	7
diminished third Diminished third In classical music from Western culture, a diminished third is the musical interval produced by narrowing a minor third by a chromatic semitone. For instance, the interval from A to C is a minor third, three semitones wide, and both the intervals from A to C, and from A to C are diminished thirds,...	d3	tone, whole tone, whole step	65536/59049	2¹⁶/3¹⁰		180.450	200	-10
major second Major second In Western music theory, a major second is a musical interval spanning two semitones, and encompassing two adjacent staff positions . For example, the interval from C to D is a major second, as the note D lies two semitones above C, and the two notes are notated on adjacent staff postions...	M2	tone, whole tone, whole step	9/8	3²/2³	3·3/2·2	203.910	200	2
semiditone	m3	(Pythagorean minor third Minor third In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the minor third is one of two commonly occurring thirds. The minor quality specification identifies it as being the smallest of the two: the minor third spans three semitones, the major... )	32/27	2⁵/3³		294.135	300	-3
augmented second Augmented second In classical music from Western culture, an augmented second is an interval produced by widening a major second by a chromatic semitone. For instance, the interval from C to D is a major second, two semitones wide, and both the intervals from C to D, and from C to D are augmented seconds, spanning...	A2		19683/16384	3⁹/2¹⁴		317.595	300	9
diminished fourth Diminished fourth In classical music from Western culture, a diminished fourth is an interval produced by narrowing a perfect fourth by a chromatic semitone. For example, the interval from C to F is a perfect fourth, five semitones wide, and both the intervals from C to F, and from C to F are diminished fourths,...	d4		8192/6561	2¹³/3⁸		384.360	400	-8
ditone Ditone In music, a ditone is the interval 81:64, which is 407.82 cents. It is the major third in Pythagorean tuning. The ditone is evenly divisible by two major tones and is wider than a just major third by a syntonic comma...	M3	(Pythagorean major third Major third In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the major third is one of two commonly occurring thirds. It is qualified as major because it is the largest of the two: the major third spans four semitones, the minor third three... )	81/64	3⁴/2⁶	27·3/16·2	407.820	400	4
perfect fourth Perfect fourth In classical music from Western culture, a fourth is a musical interval encompassing four staff positions , and the perfect fourth is a fourth spanning five semitones. For example, the ascending interval from C to the next F is a perfect fourth, as the note F lies five semitones above C, and there...	P4	diatessaron, sesquitertium	4/3	2²/3	2/3	498.045	500	-1
augmented third Augmented third In classical music from Western culture, an augmented third is an interval produced by widening a major third by a chromatic semitone. For instance, the interval from C to E is a major third, four semitones wide, and both the intervals from C to E, and from C to E are augmented thirds, spanning...	A3		177147/131072	3¹¹/2¹⁷		521.505	500	11
diminished fifth	d5	tritone	1024/729	2¹⁰/3⁶		588.270	600	-6
augmented fourth	A4	tritone	729/512	3⁶/2⁹		611.730	600	6
diminished sixth Diminished sixth In classical music from Western culture, a diminished sixth is an interval produced by narrowing a minor sixth by a chromatic semitone. For example, the interval from A to F is a minor sixth, eight semitones wide, and both the intervals from A to F, and from A to F are diminished sixths, spanning...	d6		262144/177147	2¹⁸/3¹¹		678.495	700	-11
perfect fifth Perfect fifth In classical music from Western culture, a fifth is a musical interval encompassing five staff positions , and the perfect fifth is a fifth spanning seven semitones, or in meantone, four diatonic semitones and three chromatic semitones...	P5	diapente, sesquialterum	3/2		3/2	701.955	700	1
minor sixth Minor sixth -Subminor sixth:In music, a subminor sixth or septimal sixth is an interval that is noticeably narrower than a minor sixth but noticeably wider than a diminished sixth.The sub-minor sixth is an interval of a 14:9 ratio or alternately 11:7....	m6		128/81	2⁷/3⁴		792.180	800	-4
augmented fifth Augmented fifth In classical music from Western culture, an augmented fifth is an interval produced by widening a perfect fifth by a chromatic semitone. For instance, the interval from C to G is a perfect fifth, seven semitones wide, and both the intervals from C to G, and from C to G are augmented fifths,...	A5		6561/4096	3⁸/2¹²		815.640	800	8
diminished seventh Diminished seventh In classical music from Western culture, a diminished seventh is an interval produced by narrowing a minor seventh by a chromatic semitone. For instance, the interval from A to G is a minor seventh, ten semitones wide, and both the intervals from A to G, and from A to G are diminished sevenths,...	d7		32768/19683	2¹⁵/3⁹		882.405	900	-9
major sixth Major sixth In classical music from Western culture, a sixth is a musical interval encompassing six staff positions , and the major sixth is one of two commonly occurring sixths. It is qualified as major because it is the largest of the two...	M6		27/16	3³/2⁴	9·3/8·2	905.865	900	3
minor seventh Minor seventh In classical music from Western culture, a seventh is a musical interval encompassing seven staff positions , and the minor seventh is one of two commonly occurring sevenths. The minor quality specification identifies it as being the smallest of the two: the minor seventh spans ten semitones, the...	m7		16/9	2⁴/3²		996.090	1000	-2
augmented sixth Augmented sixth In classical music from Western culture, an augmented sixth is an interval produced by widening a major sixth by a chromatic semitone. For instance, the interval from C to A is a major sixth, nine semitones wide, and both the intervals from C to A, and from C to A are augmented sixths, spanning...	A6		59049/32768	3¹⁰/2¹⁵		1019.550	1000	10
diminished octave Diminished octave In classical music from Western culture, a diminished octave is an interval produced by narrowing a perfect octave by a chromatic semitone. As such, the two notes are denoted by the same letter but have different accidentals...	d8		4096/2187	2¹²/3⁷		1086.315	1100	-7
major seventh Major seventh In classical music from Western culture, a seventh is a musical interval encompassing seven staff positions , and the major seventh is one of two commonly occurring sevenths. It is qualified as major because it is the larger of the two...	M7		243/128	3⁵/2⁷	81·3/64·2	1109.775	1100	5
augmented seventh Augmented seventh In classical music from Western culture, an augmented seventh is an interval produced by widening a major seventh by a chromatic semitone. For instance, the interval from C to B is a major seventh, eleven semitones wide, and both the intervals from C to B, and from C to B are augmented sevenths,...	A7	(octave − comma)	1048576/531441			1176.540	1200	-12
(perfect) octave Octave In music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"...	P8	diapason	2/1		2/1	1200.000	1200	0
diminished ninth	d9	(octave + comma)	531441/262144	3¹²/2¹⁸		1223.460	1200	12

Notice that the terms ditone and semiditone are specific for Pythagorean tuning, while tone and tritone are used generically for all tuning systems. Interestingly, despite its name, a semiditone (3 semitones, or about 300 cents) can hardly be viewed as half of a ditone (4 semitones, or about 400 cents).

12-tone Pythagorean scale

The table shows from which notes some of the above listed intervals can be played on a instrument using a repeated-octave 12-tone scale (such as a piano) tuned with D-based symmetric Pythagorean tuning. Further details about this table can be found in Size of Pythagorean intervals.

Fundamental intervals

The fundamental intervals are the superparticular ratios 2/1, 3/2, and 4/3. 2/1 is the octave

Octave

In music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"...

or diapason (Greek

Ancient Greek

Ancient Greek is the stage of the Greek language in the periods spanning the times c. 9th–6th centuries BC, , c. 5th–4th centuries BC , and the c. 3rd century BC – 6th century AD of ancient Greece and the ancient world; being predated in the 2nd millennium BC by Mycenaean Greek...

for "across all"). 3/2 is the perfect fifth

Perfect fifth

, diapente ("across five"), or sesquialterum. 4/3 is the perfect fourth

Perfect fourth

, 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 , which is considered to be unstable...

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

In Western music theory, a major second is a musical interval spanning two semitones, and encompassing two adjacent staff positions . For example, the interval from C to D is a major second, as the note D lies two semitones above C, and the two notes are notated on adjacent staff postions...

. 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, Størmer's theorem, named after Carl Størmer, 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

In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the major third is one of two commonly occurring thirds. It is qualified as major because it is the largest of the two: the major third spans 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 chromatic diesis, the comma of Didymus, the Ptolemaic comma, or the diatonic comma is a small comma type interval between two musical notes, equal to the frequency ratio 81:80, or around 21.51 cents...

(81/80). Likewise, the difference between the tone and the perfect fourth is the semiditone, a narrow minor third

Minor third

In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the minor third is one of two commonly occurring thirds. The minor quality specification identifies it as being the smallest of the two: the minor third spans three semitones, the major...

, 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 stack of perfect fifths, but in meantone, each fifth is narrow compared to the ratio 27/12:1 in 12 equal temperament, the opposite of...

.

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

An equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. As pitch is perceived roughly as the logarithm of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for...

, they are not equal in Pythagorean tuning, and their difference, 531441/524288, is known as the Pythagorean comma

Pythagorean comma

In musical tuning, the Pythagorean comma , named after the ancient mathematician and philosopher Pythagoras, is the small interval existing in Pythagorean tuning between two enharmonically equivalent notes such as C and B , or D and C...

Contrast with modern nomenclature

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).

External links

Neo-Gothic usage by Margo Schulter

The source of this article is wikipedia, the free encyclopedia. The text of this article is licensed under the GFDL.

Interval table

12-tone Pythagorean scale

Fundamental intervals

Contrast with modern nomenclature

See also

External links