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In music, just intonation or pure intonation is the tuning of musical intervals as whole number ratios (such as 3:2 or 4:3) ... Just Intonation Explained by Kyle Gann;
Most just intonation tunings have the problem that they cannot modulate to a different key (a very common means of expression throughout the common practice period of music) without discarding many of the tones used in the previous key, thus for every key to which the musician wishes to modulate, the instrument must provide a few more strings ...
Played in 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 seven notes in a C major scale ...
The frequencies of the harmonic series, being integer multiples of the fundamental frequency, are naturally related to each other by whole-numbered ratios and small whole-numbered ratios are likely the basis of the consonance of musical intervals (see just intonation). This objective structure is augmented by psychoacoustic phenomena.
At the same time, syntonic-diatonic just intonation was posited first by Ramos and then by Zarlino as the normal tuning for singers. However, meantone presented its own harmonic challenges. Its wolf intervals proved to be even worse than those of the Pythagorean tuning (so much so that it often required 19 keys to the octave as opposed to the ...
For example, in 19 ET, E ♯ and F ♭ are the same pitch; and in just intonation for C major, C ♯ D are within 8.1 ¢, and so can be tempered to be identical. Many musical instruments are capable of very fine distinctions of pitch, such as the human voice, the trombone, unfretted strings such as the violin, and lutes with tied frets.
The harmonic seventh interval is a minor seventh tuned in the 7:4 pitch ratio, one of the possible "just ratios" defined for this interval in just intonation (slightly below the width of a minor seventh as tuned in equal temperament). With just intonation on all notes of the harmonic seventh chord, the ratio between the frequencies of the ...
In this system the perfect fifth is flattened by one quarter of a syntonic comma ( 81 : 80 ), with respect to its just intonation used in Pythagorean tuning (frequency ratio 3 : 2 ); the result is 3 / 2 × [ 80 / 81 ] 1 / 4 = 4 √ 5 ≈ 1.49535, or a fifth of 696.578 cents. (The 12th power of that value is 125, whereas 7 octaves ...