Logarithmic approximants: Difference between revisions
Cleanup formatting and fixing some errors. |
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:<math> | :<math> | ||
\begin{ | \begin{aligned} | ||
v &= \frac{n-d}{n+d} \\ | v &= \frac{n-d}{n+d} \\ | ||
&= \text{(frequency difference) / (frequency sum)} \\ | &= \text{(frequency difference) / (frequency sum)} \\ | ||
&= \frac{1}{2} \text{(frequency difference) / (mean frequency)} | &= \frac{1}{2} \text{(frequency difference) / (mean frequency)} | ||
\end{ | \end{aligned} | ||
</math> | </math> | ||
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== Properties == | == Properties == | ||
When ''r'' is small, ''v'' provides an approximate relative measure of the logarithmic size of the interval. This approximation was exploited by Joseph Sauveur in 1701 and later by Euler and others. | When ''r'' is small, ''v'' provides an approximate relative measure of the logarithmic size of the interval, specifically <math>\frac{r-1}{r+1} \approx \frac{1}{2} \ln{r}</math>. This approximation was exploited by Joseph Sauveur in 1701 and later by Euler and others. | ||
Noting that the exact size (in dineper units) of the interval with frequency ratio ''r'' is | Noting that the exact size (in dineper units) of the interval with frequency ratio ''r'' is | ||
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<math> | <math> | ||
\begin{ | \begin{aligned} | ||
v[-J] &= -v[J] \\ | v[-J] &= -v[J] \\ | ||
v[J_1 +J_2] &= \frac{v_1+v_2}{1+v_1 v_2} | v[J_1 +J_2] &= \frac{v_1+v_2}{1+v_1 v_2} | ||
\end{ | \end{aligned} | ||
</math> | </math> | ||
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:<math> | :<math> | ||
\begin{ | \begin{aligned} | ||
q(r) &= \frac{1}{2} \left( r^{\frac{1}{2}} - r^{-\frac{1}{2}} \right) \\ | q(r) &= \frac{1}{2} \left( r^{\frac{1}{2}} - r^{-\frac{1}{2}} \right) \\ | ||
&= \frac{1}{2} \left( e^{J} - e^{-J} \right) \\ | &= \frac{1}{2} \left( e^{J} - e^{-J} \right) \\ | ||
&= \sinh{J} \\ | &= \sinh{J} \\ | ||
&= J + \frac{1}{3!} J^3 + \frac{1}{5!} J^5 + \ldots | &= J + \frac{1}{3!} J^3 + \frac{1}{5!} J^5 + \ldots | ||
\end{ | \end{aligned} | ||
</math> | </math> | ||
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:<math> | :<math> | ||
\begin{ | \begin{aligned} | ||
v &= \tanh{J}, \; q = \sinh{J}, \; \frac{q}{v} = \cosh{J} \\ | v &= \tanh{J}, \; q = \sinh{J}, \; \frac{q}{v} = \cosh{J} \\ | ||
\sqrt{r} &= e^J = q(\frac{1}{v} + 1) \\ | \sqrt{r} &= e^J = q(\frac{1}{v} + 1) \\ | ||
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\frac {q[J_2 + J_1]}{q[J_2 - J_1]} &= \frac{v_2+v_1}{v_2-v_1} \\ | \frac {q[J_2 + J_1]}{q[J_2 - J_1]} &= \frac{v_2+v_1}{v_2-v_1} \\ | ||
q[J_2 + J_1] q[J_2 - J_1] &= q_2^2 - q_1^2 \\ | q[J_2 + J_1] q[J_2 - J_1] &= q_2^2 - q_1^2 \\ | ||
\end{ | \end{aligned} | ||
</math> | </math> | ||
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:<math> | :<math> | ||
\begin{ | \begin{aligned} | ||
\frac{\text{octave}}{\text{large tone}} &\approx \frac{q[F+f]}{q[F-f]} \\ | \frac{\text{octave}}{\text{large tone}} &\approx \frac{q[F+f]}{q[F-f]} \\ | ||
&= \frac{v[F] + v[f]}{v[F] - v[f]} \\ | &= \frac{v[F] + v[f]}{v[F] - v[f]} \\ | ||
&= \frac{1/5 + 1/7}{1/5 - 1/7} = 6 | &= \frac{1/5 + 1/7}{1/5 - 1/7} = 6 | ||
\end{ | \end{aligned} | ||
</math> | </math> | ||
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:<math> | :<math> | ||
\begin{ | \begin{aligned} | ||
b(J_1,J_2) &\approx \left( \frac{1}{3} \left(q_2^2 - q_1^2\right) - \frac{2}{15} \left( q_2^4 - q_1^4 \right) \right) b_m \\ | b(J_1,J_2) &\approx \left( \frac{1}{3} \left(q_2^2 - q_1^2\right) - \frac{2}{15} \left( q_2^4 - q_1^4 \right) \right) b_m \\ | ||
&= \frac{1}{3} \left( q_2^2 - q_1^2 \right) \left( 1 - \frac{2}{5} \left(q_1^2 + q_2^2\right) \right) b_m | &= \frac{1}{3} \left( q_2^2 - q_1^2 \right) \left( 1 - \frac{2}{5} \left(q_1^2 + q_2^2\right) \right) b_m | ||
\end{ | \end{aligned} | ||
</math> | </math> | ||
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This article is based on original research by [[Martin_Gough|Martin Gough]]. See [[:File:Bimod_Approx_2014-6-8.pdf|this paper]] for a fuller account of bimodular approximants. | This article is based on original research by [[Martin_Gough|Martin Gough]]. See [[:File:Bimod_Approx_2014-6-8.pdf|this paper]] for a fuller account of bimodular approximants. | ||
Argent tuning is based on the continued fraction convergents of <math>\sqrt{2}</math>, which have been known since ancient times. | |||
This application to tuning appears to have been first made by [[Erv Wilson]], who described it under the name '2-zig/2-zag' in a [http://anaphoria.com/meruthree.pdf note] dated December 1996, with a comment claiming it as his answer to [[Joseph Yasser]] "at about 1950". | |||
The same construction was later arrived at independently by [[Graham Breed]] and Paul Hahn, who described it in posts ([https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_12592.html#12599 #12599], [https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_12637.html#12670 #12670]) to the Yahoo tuning list on 10 and 12 August 2000. | |||
Thanks to [[Gene_Ward_Smith|Gene Ward Smith]] for the Gelfond-Schneider result. | Thanks to [[Gene_Ward_Smith|Gene Ward Smith]] for the Gelfond-Schneider result. | ||
[[Category:Essays]] | [[Category:Essays]] | ||