SafekipediaPythagorean tuning
Adapted from Wikipedia · Adventurer experience
Pythagorean tuning is a way to tune musical instruments using special ratios to create pleasant sounds. It is based on 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 tuning method 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 described by later writers like Ptolemy and Boethius. Musicians used Pythagorean tuning for many years 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.
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. 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 always the same.
For example, there are two kinds of semitones in Pythagorean tuning. In equal temperament, all semitones are the same size. The differences in interval sizes all relate to a small amount called ε. 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. The octave is a common interval in all tuning systems. Names like ditone and semiditone are special to Pythagorean tuning. These intervals use simple ratios of numbers, which make 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 | ||||||
| 0 | augmented seventh | A7 | ascending comma | Pythagorean comma (531441:524288) | diesis (128:125) | ||
| 0 | diminished second | d2 | descending comma | (524288:531441) | |||
| 1 | minor second | m2 | semitone, half tone, half step | diatonic semitone, minor semitone | limma (λείμμα) (256:243) | ||
| 1 | augmented unison | A1 | 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 started in 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 some musical challenges, it was once very common. It creates smooth 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.
Discography
Bragod is a duo that performs medieval Welsh music. They use special instruments like the crwth and a six-stringed lyre with Pythagorean tuning.
Other musicians like Gothic Voices and Lou Harrison have also used Pythagorean tuning in their performances and recordings.
Images
This article is a child-friendly adaptation of the Wikipedia article on Pythagorean tuning, available under CC BY-SA 4.0.
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