Non radical intervals with musical significance
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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.
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^{1/b} [[math]], where a and b are integers. What follows is a list of musically significant non-radical intervals.
|| **Ratio** || **Cents** || **Name** || **Musical Significance** ||
|| [[math]]
2^{1/\phi}
\approx
1.5348
[[math]] || 741.64 || || "Logarithmic phi" which divides the octave into two parts, one being Phi times larger than the other in cents. ||
|| [[math]]
\dfrac{\sqrt{5}+1}{2}
\approx
1.6180
[[math]] || 833.09 || [[math]]
\text{Phi } (\phi)
[[math]] || "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 [[https://en.wikipedia.org/wiki/Diophantine_approximation#General_upper_bound|Dirichlet's Approximation Theorem]]. ||
|| [[math]]
e \approx 2.7183
[[math]] || 1731.23 || "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. ||
|| [[math]]
e^{2\pi} \approx 535.4917
[[math]] || 10877.66 || || The zeta function has units that are given as divisions of the interval
[[math]]
e^{2\pi}
[[math]] ||Original HTML content:
<html><head><title>Non radical intervals with musical significance</title></head><body>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.<br />
<br />
There are many non-radical intervals which have musical significance. By <strong>non-radical</strong> is meant a number that cannot be written in the form <a class="wiki_link" href="/math">math</a> a^{1/b} <a class="wiki_link" href="/math">math</a>, where a and b are integers. What follows is a list of musically significant non-radical intervals.<br />
<br />
<table class="wiki_table">
<tr>
<td><strong>Ratio</strong><br />
</td>
<td><strong>Cents</strong><br />
</td>
<td><strong>Name</strong><br />
</td>
<td><strong>Musical Significance</strong><br />
</td>
</tr>
<tr>
<td><!-- ws:start:WikiTextMathRule:0:
[[math]]<br/>
2^{1/\phi}<br />
\approx<br />
1.5348<br/>[[math]]
--><script type="math/tex">2^{1/\phi}
\approx
1.5348</script><!-- ws:end:WikiTextMathRule:0 --><br />
</td>
<td>741.64<br />
</td>
<td><br />
</td>
<td>"Logarithmic phi" which divides the octave into two parts, one being Phi times larger than the other in cents.<br />
</td>
</tr>
<tr>
<td><!-- ws:start:WikiTextMathRule:1:
[[math]]<br/>
\dfrac{\sqrt{5}+1}{2}<br />
\approx<br />
1.6180<br/>[[math]]
--><script type="math/tex">\dfrac{\sqrt{5}+1}{2}
\approx
1.6180</script><!-- ws:end:WikiTextMathRule:1 --><br />
</td>
<td>833.09<br />
</td>
<td><!-- ws:start:WikiTextMathRule:2:
[[math]]<br/>
\text{Phi } (\phi)<br/>[[math]]
--><script type="math/tex">\text{Phi } (\phi)</script><!-- ws:end:WikiTextMathRule:2 --><br />
</td>
<td>"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 <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Diophantine_approximation#General_upper_bound" rel="nofollow">Dirichlet's Approximation Theorem</a>.<br />
</td>
</tr>
<tr>
<td><!-- ws:start:WikiTextMathRule:3:
[[math]]<br/>
e \approx 2.7183<br/>[[math]]
--><script type="math/tex">e \approx 2.7183</script><!-- ws:end:WikiTextMathRule:3 --><br />
</td>
<td>1731.23<br />
</td>
<td>"e-tave"<br />
</td>
<td>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.<br />
</td>
</tr>
<tr>
<td><!-- ws:start:WikiTextMathRule:4:
[[math]]<br/>
e^{2\pi} \approx 535.4917<br/>[[math]]
--><script type="math/tex">e^{2\pi} \approx 535.4917</script><!-- ws:end:WikiTextMathRule:4 --><br />
</td>
<td>10877.66<br />
</td>
<td><br />
</td>
<td>The zeta function has units that are given as divisions of the interval<br />
<!-- ws:start:WikiTextMathRule:5:
[[math]]<br/>
e^{2\pi}<br/>[[math]]
--><script type="math/tex">e^{2\pi}</script><!-- ws:end:WikiTextMathRule:5 --><br />
</td>
</tr>
</table>
</body></html>