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Although E-sharp minor is usually notated as F minor, it could be used on a local level, such as bars 17 to 22 in Johann Sebastian Bach's The Well-Tempered Clavier, Book 1, Prelude and Fugue No. 3 in C-sharp major. (E-sharp minor is the mediant minor key of C-sharp major.) The scale-degree chords of E-sharp minor are: Tonic – E-sharp minor
The above scale allows a minor tone to occur next to a semitone which produces the awkward ratio 32:27 for D→F, and still worse, a minor tone next to a fourth giving 40:27 for D→A. Flattening D by a comma to 10:9 alleviates these difficulties but creates new ones: D→G becomes 27:20, and D→B becomes 27:16.
Harmonic fragment scales form a rare exception to this issue. In tunings such as 1:1, 9:8, 5:4, 3:2, 7:4, 2:1 , all the pitches are chosen from the harmonic series (divided by powers of 2 to reduce them to the same octave), so all the intervals are related to each other by simple numeric ratios.
The harmonic scale is a "super-just" musical scale allowing extended just intonation, beyond 5-limit to the 19th harmonic (Play ⓘ), and free modulation through the use of synthesizers. Transpositions and tuning tables are controlled by the left hand on the appropriate note on a one-octave keyboard.
The fourth harmonic vibrates at four times the frequency of the fundamental and sounds a perfect fourth above the third harmonic (two octaves above the fundamental). Double the harmonic number means double the frequency (which sounds an octave higher). An illustration in musical notation of the harmonic series (on C) up to the 20th harmonic.
C-D ♯-F ♯-A-C-D ♯ In the minor-thirds tuning, every interval between successive strings is a minor third. In the minor-thirds tuning beginning with C, the open strings contain the notes (C, D ♯, F ♯) of the diminished C chord. [28]
Some other meanings of the term diatonic scale take the extension to harmonic and melodic minor even further, to be even more inclusive. [16] In general, diatonic is most often used inclusively with respect to music that restricts itself to standard uses of traditional major and minor scales. When discussing music that uses a larger variety of ...
Starting from D for example (D-based tuning), six other notes are produced by moving six times a ratio 3:2 up, and the remaining ones by moving the same ratio down: E♭–B♭–F–C–G–D–A–E–B–F♯–C♯–G♯ This succession of eleven 3:2 intervals spans across a wide range of frequency (on a piano keyboard, it encompasses 77 ...