The Phoenix Project
The Frequencies of Music 

This is a virtual piano showing the frequencies in cycles per second (Hz), of each of the 88 keys on a piano (that is, note frequencies of each note found on a standard piano), with the 49th note, the fifth A (called A4), tuned to 440 cycles per second (referred to as A440). This distribution of frequencies is known as equal temperament, that is, each successive pitch is derived by multiplying the previous by the twelfth root of two. For example, A4 is normally tuned to 440 Hz. To get the next semitone (A♯4), multiply 440 Hz by the twelfth root of two. To go from A4 to B4 (up two semitones), multiply 440 by the twelfth root of two squared. For other tuning schemes refer to musical tuning.

The following equation will give the frequency F of the n^th piano key, as shown in the table:

Systems for the twelve-note chromatic scale

It is impossible to tune the twelve-note chromatic scale so that all intervals are "perfect"; many different methods with their own various compromises have thus been put forward. The main ones are:

• Just intonation

In Just Intonation the frequencies of the scale notes are related to one another by simple numeric ratios, a common example of this being 1:1, 9:8, 5:4, 4:3, 3:2, 5:3, 15:8, 2:1 to define the ratios for the 7 notes in a C major scale. In theory a variety of approaches are possible, such as basing the tuning of pitches on the harmonic series (music), which are all whole number multiples of a single tone. In practice however this quickly leads to potential for confusion depending on context, especially in the larger system of 12 chromatic notes used in the West. For instance, a major second may end up either in the ratio 9:8 or 10:9. For this reason, just intonation may be less suitable system for use on keyboard instruments or other instruments where the pitch of individual notes is not flexible. (On fretted instruments like guitars and lutes, multiple frets for one interval can be practical.)

• Pythagorean tuning

A Pythagorean tuning is technically a type of just intonation, in which the frequency ratios of the notes are all derived from the number ratio 3:2, a ratio of central importance to the School of Pythagoras in Ancient Greece. Using this approach for example, the 12 notes of the Western chromatic scale would be tuned to the following ratios: 1:1, 256:243, 9:8, 32:27, 81:64, 4:3, 729:512, 3:2, 128:81, 27:16, 16:9, 243:128, 2:1. Also called "3-limit" because there are no prime factors other than 2 and 3, this Pythagorean system was of primary importance in Western musical development in the Medieval and Renaissance periods. As a concept it was further developed by Safi ad-Din al-Urmawi, who divided the octave into seventeen parts (limmas and commas) and used in the Turkish and Persian tone systems.^{[citation needed]}

• Meantone temperament

A system of tuning which averages out pairs of ratios used for the same interval (such as 9:8 and 10:9), thus making it possible to tune keyboard instruments. Next to the twelve-equal temperament, which some would not regard as a form of meantone, the best known form of this temperament is quarter-comma meantone, which tunes major thirds justly in the ratio of 5:4 and divides them into two whole tones of equal size. To do this, eleven perfect fifths in each octave are flattened by a quarter of a syntonic comma, with the remaining fifth being left very sharp (such an unacceptably out-of-tune fifth is known as a wolf interval). However, the fifth may be flattened to a greater or lesser degree than this and the tuning system will retain the essential qualities of meantone temperament; examples include the 31-equal fifth and Lucy tuning.

• Both just intonation and meantone temperament can be regarded as forms of regular temperament.
• Well temperament

Any one of a number of systems where the ratios between intervals are unequal, but approximate to ratios used in just intonation. Unlike meantone temperament, the amount of divergence from just ratios varies according to the exact notes being tuned, so that C-E will probably be tuned closer to a 5:4 ratio than, say, D♭-F. Because of this, well temperaments have no wolf intervals. A well temperament system is usually named after whoever first came up with it.

• Equal temperament

(a special case of mean-tone temperament), in which adjacent notes of the scale are all separated by logarithmically equal distances (100 cents): A harmonized C major scale in equal temperament (.ogg format, 96.9KB). This is the most common tuning system used in Western music, and is the standard system for tuning a piano. Since this scale divides an octave into twelve equal-ratio steps and an octave has a frequency ratio of two, the frequency ratio between adjacent notes is then the twelfth root of two, 2^1/12, or ~1.05946309...

• Syntonic temperament

A tuning system which subsumes nearly all of the above tuning systems.^[2] For example, of the regular temperaments, "equal temperament" is the syntonic tuning in which the tempered perfect fifth (P5) is 700 cents wide; 1/4-comma meantone is the syntonic tuning in which the P5 is 696.6 cents wide; Pythagorean tuning is the syntonic tuning in which the P5 is 702 cents wide; 5-equal is the syntonic tuning in which the P5 is 720 cents wide; and 7-equal is the tuning in which the P5 is 686 cents wide. All of these syntonic tunings have identical fingering on an isomorphic keyboard.^[3] So do many irregular tunings such as well temperaments and Just Intonation tunings.^[4]

• Tempered timbres

A timbre's partials (also known as harmonics or overtones) can be tempered such that each of the timbre's partials aligns with a note of a given tempered tuning. This alignment of tuning and timbre is the ultimate source of consonance,^[5] of which one notable example is the alignment between the partials of a harmonic timbre and a Just Intonation tuning. Hence, using tempered timbres, one can achieve a degree of consonance, in any tempered tuning, that is comparable to the consonance acheived by the combination of Just Intonation tuning and harmonic timbres. Tempering timbres in real time, to match a tuning that can change smoothly in real time, using the tuning-invariant fingering of an isomorphic keyboard, is a central component of Dynamic Tonality (ibid., Milne et al., 2009).

Tuning systems that are not produced with exclusively just intervals are usually referred to as temperaments.

• Natural overtone scale, a scale derived from the harmonic series. This scale is present on the Ancient Chinese Guqin and is regarded the first musical scale.
• Slendro, a pentatonic scale used in Indonesian music.
• Pelog, the other main gamelan scale, with three hemitonic pentatonic modes superimposed to form a scale with seven notes to the octave
• 43-tone scale, created by Harry Partch, an American composer who wrote musical and dramatic works in just intonation
• Bohlen-Pierce scale
• LucyTuning, a meantone system advocated by Charles Lucy, related to the number Pi and writings of John Harrison.
• Alpha and beta scales of Wendy Carlos
• Quarter tone scale, first presented by Mikha'il Mishaqah, used in the theory of Arabic music tone systems. From this the heptatonic scales consisting of minor, neutral, and major seconds of maqamat are chosen, this system was first promoted by al-Farabi using a 25 tone scale.
• Thirteenth Sound
• 19 equal temperament
• 22 equal temperament
• 31 equal temperament
• 53 equal temperament
• 88 equal temperament
• Schismatic temperament
• Miracle temperament
• Stretched tuning makes an octave represent slightly more than a doubling in frequency. It is usually applied to keyboard instruments with tines or thick strings, where the ratio of harmonic to fundamental can be slightly greater than a true integer ratio (most notably the piano, and some electric pianos).
• Hexany

This table lists open strings on some common string instruments and their standard tunings.

Last updated Saturday, May 09, 2009 11:46 AM
You are here:

This website was designed by PHOENIX TECHNOLOGIES - 615-847-8222 or 931-583-2509
Copyright © 2009- Phoenix Technologies Inc. All rights reserved.