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The harmonics produced by Erfurt bell (1497) [8] or any well-tuned bell: [9] strike note on E with hum note, minor third, fifth, octave or nominal, and major third and perfect fifth in the second octave. The tuning of a bell is completely dependent on its shape.
These partials are customarily given names such as hum, prime, minor third (or tierce), fifth (or quint), octave (or nominal), upper octave, etc. The strike note of the bell, which is determined by three partials (the octave, upper fifth, and the upper octave), is generally close to the pitch of the prime in a well-tuned bell."
The overtones on an English handbell are a 12th (an octave and a perfect fifth) above the fundamental, while Dutch handbells – such as Petit & Fritsen – focus on the overtone a minor 10th (an octave and a minor third) or a major 10th (an octave and a major third) above the fundamental.
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The extremes of the meantone systems encountered in historical practice are the Pythagorean tuning, where the whole tone corresponds to 9:8, i.e. (3:2) 2 / 2 , the mean of the major third (3:2) 4 / 4 , and the fifth (3:2) is not tempered; and the 1 ⁄ 3-comma meantone, where the fifth is tempered to the extent that three ...
The intervals of 5-limit just intonation (prime limit, not odd limit) are ratios involving only the powers of 2, 3, and 5.The fundamental intervals are the superparticular ratios 2/1 (the octave), 3/2 (the perfect fifth) and 5/4 (the major third).
In neither the twelve tone equitemperament nor the quarter-comma meantone is the fifth a rational fraction of the octave, but several tunings exist which approximate the fifth by such an interval; these are a subset of the equal temperaments ( "N TET" ), in which the octave is divided into some number (N) of equally wide intervals.
The diminished fifth is often called a tritone in modern tonal theory, but functionally and notationally it can only resolve inwards as a diminished fifth and is therefore not reckoned a tritone—that is, an interval composed of three adjacent whole tones—in mid-renaissance (early 16th-century) music theory. [19]