On the numbers 256/243, 25/24, 16/15

Rok, strany: 1998, 145 - 151

The ratios 256/243, 25/24, 16/15 are known as the minor Pythago rean, chromatic, and diatonic semitone, respectively. The main result of this paper is the following statement which has a valuable consequence for musical acoustic theory: {\sl{According to symmetry, all rational triplets $(X₁, X₂, X₃)$ TDS-generating generalized geometrical progressions

$$ \multline <Γ_i>=\big<X₁^{ν_{i, 1}} X₂^{ν_{i, 2}} X₃^{ν_{i, 3}}: ν_{i, 1} + ν_{i, 2} + ν_{i, 3} = i , 0 ≤ ν_{0, ·} ≤ ν_{1, ·} ≤ … ≤ ν_{i, ·} ≤ …\big>_{νi,· \in \Bbb N³} \endmultline $$

with the subsequences

$$ <Γ_{12 l}> = \big<2^l \big> ,   <Γ_{12l + 7}> = \big<3· 2^l-1\big> ,   <Γ_{12l + 4}> = \big<5· 2^l-2\big> $$

are exactly as follows:

$$ (25/24, 135/128, 16/15), (256/243, 135/128,16/15), (25/24, 16/15, 27/25) . $$

}} Thus, not only the diatonic and chromatic but also the minor Pythagorean semitone (together with the diatonic semitone and its complement to the major whole tone) can serve as a basis for the construction of 12-degree diatonic scales.

ISO 690:

Haluška, J. 1998. On the numbers 256/243, 25/24, 16/15. In Tatra Mountains Mathematical Publications, vol. 14, no.1, pp. 145-151. 1210-3195.

APA:

Haluška, J. (1998). On the numbers 256/243, 25/24, 16/15. Tatra Mountains Mathematical Publications, 14(1), 145-151. 1210-3195.