Non radical intervals with musical significance: Difference between revisions
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Musicians are typically interested in musical intervals that are rational (i.e. justly intoned ratios), or equal divisions of these intervals. However, not all intervals fit into this box. | Musicians are typically interested in musical intervals that are rational (i.e. justly intoned ratios), or equal divisions of these intervals. However, not all intervals fit into this box. | ||
There are many non-radical intervals which have musical significance. By '''non-radical''' is meant a number that cannot be written in the form | There are many non-radical intervals which have musical significance. By '''non-radical''' is meant a number that cannot be written in the form <math>(a/b)^c</math>, where <math>a</math> and <math>b</math> are integers and <math>c</math> is rational. What follows is a list of musically significant non-radical intervals. | ||
{| class="wikitable" | {| class="wikitable" | ||
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| | '''Ratio''' | | | '''Ratio''' | ||
| | '''Cents''' | | | '''Cents''' | ||
| | '''Name''' | | | '''Name(s)''' | ||
| | '''Musical | | | '''Musical significance''' | ||
|- | |- | ||
| | <math>2^{1/\phi} | | | <math>2^{1/\phi} \approx 1.5348</math> | ||
\approx | | | 741.641 | ||
1.5348</math> | | | [[Logarithmic phi]] | ||
| | 741. | | | Divides the octave into two parts, one being phi times larger than the other in cents. | ||
| | | |||
| | | |||
|- | |- | ||
| | <math>\dfrac{\sqrt{5}+1}{2} | | | <math>\phi = \dfrac{\sqrt{5}+1}{2} \approx 1.6180</math> | ||
\approx | | | 833.090 | ||
1.6180</math> | | | [[Acoustic phi]]<br>[[Golden ratio]]<br>Linear phi | ||
| | 833. | | | "Linear phi," the unique interval whose continued fraction approximations converge more slowly than any other number. This is due to the continued fraction representation only containing 1's, as well as a general consequence of [[Wikipedia:Diophantine_approximation#General_upper_bound|Dirichlet's approximation theorem]]. | ||
| | < | |||
| | "Linear phi," the unique interval whose continued fraction approximations converge more slowly than any other number. This is due to the continued fraction representation only containing 1's, as well as a general consequence of [ | |||
|- | |- | ||
| | <math>e \approx 2.7183</math> | | | <math>e \approx 2.7183</math> | ||
| | 1731.23 | | | 1731.23 | ||
| | "e-tave" | | | [[Natave]]<br>[[Neper]]<br>"e-tave" | ||
| | In Gene's black magic formulas, it is mathematically more "natural" to consider the number of divisions to the "e-tave" rather than the octave. | | | In Gene's black magic formulas, it is mathematically more "natural" to consider the number of divisions to the "e-tave" rather than the octave. | ||
|- | |- | ||
| | <math>e^{2\pi} \approx 535.4917</math> | | | <math>e^{2\pi} \approx 535.4917</math> | ||
| | 10877.66 | | | 10877.66 | ||
| | | | | [[Zetave]] | ||
| | The zeta function has units that are given as divisions of the interval | | | The [[zeta]] function has units that are given as divisions of the interval <math>e^{2\pi}</math>. | ||
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[[Category:Lists of intervals]] | |||
[[Category:Math]] | |||