Just intonation

Just intervals (and chords created by combining them) consist of tones from a single harmonic series of an implied fundamental.

In Western musical practice, bowed instruments such as violins, violas, cellos, and double basses are tuned using pure fifths or fourths.

In contrast, keyboard instruments are rarely tuned using only pure intervals—the desire for different keys to have identical intervals in Western music makes this impractical.

Some instruments of fixed pitch, such as electric pianos, are commonly tuned using equal temperament, in which all intervals other than octaves consist of irrational-number frequency ratios.

Ptolemy describes a variety of other just intonations derived from history (Pythagoras, Philolaus, Archytas, Aristoxenus, Eratosthenes, and Didymus) and several of his own discovery / invention, including many interval patterns in 3-limit, 5-limit, 7-limit, and even an 11-limit diatonic.

[citation needed] The prominent notes of a given scale may be tuned so that their frequencies form (relatively) small whole number ratios.

This twelve-tone scale is fairly close to equal temperament, but it does not offer much advantage for tonal harmony because only the perfect intervals (fourth, fifth, and octave) are simple enough to sound pure.

[9] The primary reason for its use is that it is extremely easy to tune, as its building block, the perfect fifth, is the simplest and consequently the most consonant interval after the octave and unison.

To build such a twelve-tone scale (using C as the base note), we may start by constructing a table containing fifteen pitches: The factors listed in the first row and column are powers of 3 and 5, respectively (e.g., ⁠ 1 / 9 ⁠ = 3−2 ).

For instance, in the first row of the table, there is an ascending fifth from D and A, and another one (followed by a descending octave) from A to E. This suggests an alternative but equivalent method for computing the same ratios.

For instance, one can obtain A, starting from C, by moving one cell to the left and one upward in the table, which means descending by a fifth and ascending by a major third: Since this is below C, one needs to move up by an octave to end up within the desired range of ratios (from 1:1 to 2:1): A 12 tone scale is obtained by removing one note for each couple of enharmonic notes.

Note that it is a diminished fifth, close to half an octave, above the tonic C, which is a discordant interval; also its ratio has the largest values in its numerator and denominator of all tones in the scale, which make it least harmonious: All are reasons to avoid it.

In Indian music, the just diatonic scale described above is used, though there are different possibilities, for instance for the sixth pitch (dha), and further modifications may be made to all pitches excepting sa and pa.[10] Some accounts of Indian intonation system cite a given 12 swaras being divided into 22 shrutis.

[11][12] According to some musicians, one has a scale of a given 12 pitches and ten in addition (the tonic, shadja (sa), and the pure fifth, pancham (pa), are inviolate (known as achala[13] in Indian music theory): Where we have two ratios for a given letter name or swara, we have a difference of 81:80 (22 cents), which is the syntonic comma[9] or the praman[13] in Indian music theory.

In "Revelation", Michael Harrison goes even further, and uses the tempo of beat patterns produced by some dissonant intervals as an integral part of several movements.

On her 1987 lecture album Secrets of Synthesis there are audible examples of the difference in sound between equal temperament and just intonation.

French horns can be tuned by shortening or lengthening the main tuning slide on the back of the instrument, with each individual rotary or piston slide for each rotary or piston valve, and by using the right hand inside the bell to adjust the pitch by pushing the hand in deeper to flatten the note, or pulling it out to sharpen the note while playing.

[15][page needed] For example, a composer who chooses to write in 7-limit just intonation will not employ ratios that use powers of prime numbers larger than 7.

[18] While these systems allow precise indication of intervals and pitches in print, more recently some composers have been developing notation methods for Just Intonation using the conventional five-line staff.

James Tenney, amongst others, preferred to combine JI ratios with cents deviations from the equal tempered pitches, indicated in a legend or directly in the score, allowing performers to readily use electronic tuning devices if desired.

[19][20] Beginning in the 1960s, Ben Johnston had proposed an alternative approach, redefining the understanding of conventional symbols (the seven "white" notes, the sharps and flats) and adding further accidentals, each designed to extend the notation into higher prime limits.

Johnston introduces new symbols for the septimal ( & ), undecimal (↑ & ↓), tridecimal ( & ), and further prime-number extensions to create an accidental based exact JI notation for what he has named "Extended Just Intonation" (Fig.

In 2000–2004, Marc Sabat and Wolfgang von Schweinitz worked in Berlin to develop a different accidental-based method, the Extended Helmholtz-Ellis JI Pitch Notation.

[23] Following the method of notation suggested by Helmholtz in his classic On the Sensations of Tone as a Physiological Basis for the Theory of Music, incorporating Ellis' invention of cents, and continuing Johnston's step into "Extended JI", Sabat and Schweinitz propose unique symbols (accidentals) for each prime dimension of harmonic space.

The Pythagorean pitches are then paired with new symbols that commatically alter them to represent various other partials of the harmonic series (Fig. 1).

A typically used convention is that cent deviations refer to the tempered pitch implied by the flat, natural, or sharp.

A complete legend and fonts for the notation (see samples) are open source and available from the Plainsound Music Edition website.

4 for "combined" symbol) Sagittal notation (from Latin sagitta, "arrow") is a system of arrow-like accidentals that indicate prime-number comma alterations to tones in a Pythagorean series.

At the same time, they provide some degree of practicality through their extension of staff notation, as traditionally trained performers may draw on their intuition for roughly estimating pitch height.

However, the use of unique symbols reduces harmonic ambiguity and the potential confusion arising from indicating only cent deviations.