Pythagorean tuning

Pythagorean tuning is a way of tuning musical instruments that uses special ratios to make pleasing sounds. It is based on the idea of using "pure" or perfect fifths, which have a frequency ratio of 3:2. This ratio comes from the next harmonic of a vibrating string after the octave, making it a natural and easy interval to tune by ear.

This system of tuning dates back to Ancient Mesopotamia and is often linked to the Ancient Greeks, especially Pythagoras from the sixth century BC. Even though it is named after him, some of these ideas were actually described by later writers like Ptolemy and Boethius. Musicians used Pythagorean tuning for many years, up until the beginning of the 16th century.

The Pythagorean scale is made by using only pure perfect fifths and octaves. In Greek music, this tuning helped create scales called tetrachords, which span an octave. There are two main types: extended Pythagorean tuning, which has no limit to the number of fifths, and 12-tone Pythagorean temperament, which is limited to twelve tones per octave. However, this limitation can cause some intervals, like the diminished sixth, to sound very out of tune, creating what is called a "wolf fifth".

Method

Pythagorean tuning uses a special way to tune musical notes by using perfect fifths. A perfect fifth is a musical interval with a frequency ratio of 3:2, which is the next simplest ratio after the octave (2:1). For example, starting from the note D, you can create other notes by moving up or down in perfect fifths. This creates a sequence of notes that covers a wide range of frequencies.

Because notes that are twice as high or low sound the same (called octaves), we adjust some notes to fit them into a smaller range, usually within one octave. This adjustment has been used since ancient times. In Pythagorean tuning, some notes that seem the same in other tuning systems, like A♭ and G♯, actually have slightly different frequencies. This small difference is called the Pythagorean comma. Because of this, one interval in the tuning, called the wolf interval, sounds very out of tune. This means that Pythagorean tuning works well for most music, but there will always be one interval that does not sound right.

Note	Interval from D	Formula	=	=	Frequency ratio	Size (cents)	12-TET-dif (cents)
D	unison	1 1 {\displaystyle {\frac {1}{1}}}	3 0 × 2 0 {\displaystyle 3^{0}\times 2^{0}}	3 0 2 0 {\displaystyle {\frac {3^{0}}{2^{0}}}}	1 1 {\displaystyle {\frac {1}{1}}}	0.00	0.00
E♭	minor second	( 2 3 ) 5 × 2 3 {\displaystyle \left({\frac {2}{3}}\right)^{5}\times 2^{3}}	3 − 5 × 2 8 {\displaystyle 3^{-5}\times 2^{8}}	2 8 3 5 {\displaystyle {\frac {2^{8}}{3^{5}}}}	256 243 {\displaystyle {\frac {256}{243}}}	90.22	−9.78
E	major second	( 3 2 ) 2 × 1 2 {\displaystyle \left({\frac {3}{2}}\right)^{2}\times {\frac {1}{2}}}	3 2 × 2 − 3 {\displaystyle 3^{2}\times 2^{-3}}	3 2 2 3 {\displaystyle {\frac {3^{2}}{2^{3}}}}	9 8 {\displaystyle {\frac {9}{8}}}	203.91	3.91
F	minor third	( 2 3 ) 3 × 2 2 {\displaystyle \left({\frac {2}{3}}\right)^{3}\times 2^{2}}	3 − 3 × 2 5 {\displaystyle 3^{-3}\times 2^{5}}	2 5 3 3 {\displaystyle {\frac {2^{5}}{3^{3}}}}	32 27 {\displaystyle {\frac {32}{27}}}	294.13	−5.87
F♯	major third	( 3 2 ) 4 × ( 1 2 ) 2 {\displaystyle \left({\frac {3}{2}}\right)^{4}\times \left({\frac {1}{2}}\right)^{2}}	3 4 × 2 − 6 {\displaystyle 3^{4}\times 2^{-6}}	3 4 2 6 {\displaystyle {\frac {3^{4}}{2^{6}}}}	81 64 {\displaystyle {\frac {81}{64}}}	407.82	7.82
G	perfect fourth	2 3 × 2 {\displaystyle {\frac {2}{3}}\times 2}	3 − 1 × 2 2 {\displaystyle 3^{-1}\times 2^{2}}	2 2 3 1 {\displaystyle {\frac {2^{2}}{3^{1}}}}	4 3 {\displaystyle {\frac {4}{3}}}	498.04	−1.96
A♭	diminished fifth	( 2 3 ) 6 × 2 4 {\displaystyle \left({\frac {2}{3}}\right)^{6}\times 2^{4}}	3 − 6 × 2 10 {\displaystyle 3^{-6}\times 2^{10}}	2 10 3 6 {\displaystyle {\frac {2^{10}}{3^{6}}}}	1024 729 {\displaystyle {\frac {1024}{729}}}	588.27	−11.73
G♯	augmented fourth	( 3 2 ) 6 × ( 1 2 ) 3 {\displaystyle \left({\frac {3}{2}}\right)^{6}\times \left({\frac {1}{2}}\right)^{3}}	3 6 × 2 − 9 {\displaystyle 3^{6}\times 2^{-9}}	3 6 2 9 {\displaystyle {\frac {3^{6}}{2^{9}}}}	729 512 {\displaystyle {\frac {729}{512}}}	611.73	11.73
A	perfect fifth	3 2 {\displaystyle {\frac {3}{2}}}	3 1 × 2 − 1 {\displaystyle 3^{1}\times 2^{-1}}	3 1 2 1 {\displaystyle {\frac {3^{1}}{2^{1}}}}	3 2 {\displaystyle {\frac {3}{2}}}	701.96	1.96
B♭	minor sixth	( 2 3 ) 4 × 2 3 {\displaystyle \left({\frac {2}{3}}\right)^{4}\times 2^{3}}	3 − 4 × 2 7 {\displaystyle 3^{-4}\times 2^{7}}	2 7 3 4 {\displaystyle {\frac {2^{7}}{3^{4}}}}	128 81 {\displaystyle {\frac {128}{81}}}	792.18	−7.82
B	major sixth	( 3 2 ) 3 × 1 2 {\displaystyle \left({\frac {3}{2}}\right)^{3}\times {\frac {1}{2}}}	3 3 × 2 − 4 {\displaystyle 3^{3}\times 2^{-4}}	3 3 2 4 {\displaystyle {\frac {3^{3}}{2^{4}}}}	27 16 {\displaystyle {\frac {27}{16}}}	905.87	5.87
C	minor seventh	( 2 3 ) 2 × 2 2 {\displaystyle \left({\frac {2}{3}}\right)^{2}\times 2^{2}}	3 − 2 × 2 4 {\displaystyle 3^{-2}\times 2^{4}}	2 4 3 2 {\displaystyle {\frac {2^{4}}{3^{2}}}}	16 9 {\displaystyle {\frac {16}{9}}}	996.09	−3.91
C♯	major seventh	( 3 2 ) 5 × ( 1 2 ) 2 {\displaystyle \left({\frac {3}{2}}\right)^{5}\times \left({\frac {1}{2}}\right)^{2}}	3 5 × 2 − 7 {\displaystyle 3^{5}\times 2^{-7}}	3 5 2 7 {\displaystyle {\frac {3^{5}}{2^{7}}}}	243 128 {\displaystyle {\frac {243}{128}}}	1109.78	9.78
D	octave	2 1 {\displaystyle {\frac {2}{1}}}	3 0 × 2 1 {\displaystyle 3^{0}\times 2^{1}}	2 1 3 0 {\displaystyle {\frac {2^{1}}{3^{0}}}}	2 1 {\displaystyle {\frac {2}{1}}}	1200.00	0.00

Note	C		D		E		F		G		A		B		C
Ratio	1⁄1		9⁄8		81⁄64		4⁄3		3⁄2		27⁄16		243⁄128		2⁄1
Step	—	9⁄8		9⁄8		256⁄243		9⁄8		9⁄8		9⁄8		256⁄243		—

Sizes of intervals

In Pythagorean tuning, intervals can start from any note, so there are twelve versions of each type of interval. Most interval types have two different sizes, except for unisons and octaves, which are the same no matter where they start.

For example, there are two kinds of semitones: a smaller one of about 90.225 cents and a larger one of about 113.685 cents. In equal temperament, all semitones are the same size, about 100 cents. In Pythagorean tuning, eleven fifths are about 701.955 cents each, but the twelfth, called the wolf fifth, is much smaller, about 678.495 cents. This difference creates variations in the sizes of other intervals as well.

The differences in interval sizes all relate to a small amount called ε, which is the difference between the Pythagorean fifth and the average fifth. These differences make some intervals slightly wider or narrower than their usual equivalents. This small difference is known as the Pythagorean comma.

Pythagorean intervals

Main articles: Pythagorean interval and Interval (music)

In Pythagorean tuning, four special intervals have unique names. These names help us understand how musical notes relate to each other in this tuning system. For example, the octave is a common interval in all tuning systems, but some names like ditone and semiditone are special to Pythagorean tuning. These intervals are tuned using simple ratios of numbers, making the music sound clear and harmonious.

Number of semitones	Generic names				Specific names
	Quality and number		Other naming conventions		Pythagorean tuning (pitch ratio names)	5-limit tuning	1/4-comma meantone
	Full	Short	Other naming conventions		Pythagorean tuning (pitch ratio names)	5-limit tuning	1/4-comma meantone
0	augmented seventh	A7	ascending comma		Pythagorean comma (531441:524288)		diesis (128:125)
0	diminished second	d2	descending comma		(524288:531441)		diesis (128:125)
1	minor second	m2	semitone, half tone, half step	diatonic semitone, minor semitone	limma (λείμμα) (256:243)
1	augmented unison	A1	semitone, half tone, half step	chromatic semitone, major semitone	apotome (ἀποτομή) (2187:2048)
2	major second	M2	tone, whole tone, whole step		epogdoön (ἐπόγδοον), sesquioctavum (9:8)
3	minor third	m3			semiditone (32:27)	sesquiquintum (6:5)
4	major third	M3			ditone (δίτονον) (81:64)	sesquiquartum (5:4)
5	perfect fourth	P4	diatessaron (διατεσσάρων)		epitrite (ἐπίτριτος), sesquitertium (4:3)
6	diminished fifth	d5
6	augmented fourth	A4			tritone (τρίτονον) (729:512)
7	perfect fifth	P5	diapente (διαπέντε)		hemiolion (ἡμιόλιον), sesquialterum (3:2)
12	(perfect) octave	P8	diapason (διαπασών)		duplex (2:1)

History and usage

The Pythagorean tuning system dates back to Ancient Mesopotamia and was later linked to the Greek philosopher Pythagoras. Ancient Greeks used this tuning, which they borrowed from Mesopotamian music theory. A similar tuning system, called the Shí-èr-lǜ scale, was also used in ancient China.

While Pythagorean tuning is rarely used today because of certain musical challenges, it was once widespread. It creates very smooth and pleasing sounds for fifths, but less smooth sounds for thirds. Over time, other tuning systems like meantone temperament and equal temperament became more popular, especially as music grew more complex. However, Pythagorean tuning is still sometimes used by singers and flexible instruments like the violin family for certain musical passages.