Gral method: Difference between revisions
Expand example, add labeling section |
Add 'Continued Fractions Without Tears' reference, gralculator link |
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The algorithm is a binary search for the number <math>g/h</math>, | The algorithm is a binary search for the number <math>g/h</math>, | ||
starting with the interval <math>[0/1, 1/0]</math> and 'bisecting' each interval <math>[a/c, b/d]</math> with the mediant <math>(a+b)/(c+d)</math>.<ref name="knuth1994concrete"/> | starting with the interval <math>[0/1, 1/0]</math> and 'bisecting' each interval <math>[a/c, b/d]</math> with the mediant <math>(a+b)/(c+d)</math>.<ref name="knuth1994concrete"/><ref name="richards1981continued"/> | ||
Each interval <math>[a/c, b/d]</math> we encounter in the search has <math>ad - bc = -1</math>, | Each interval <math>[a/c, b/d]</math> we encounter in the search has <math>ad - bc = -1</math>, | ||
so <math>(a, b)</math>, <math>(c, d)</math> form a basis, | so <math>(a, b)</math>, <math>(c, d)</math> form a basis, | ||
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<ref name="knuth1994concrete"> | <ref name="knuth1994concrete"> | ||
D.E. Knuth, O. Patashnik, and R.L. Graham, [https://seriouscomputerist.atariverse.com/media/pdf/book/Concrete%20Mathematics.pdf Concrete Mathematics]. Addison-Wesley, 1994. | D.E. Knuth, O. Patashnik, and R.L. Graham, [https://seriouscomputerist.atariverse.com/media/pdf/book/Concrete%20Mathematics.pdf Concrete Mathematics]. Addison-Wesley, 1994. | ||
</ref> | |||
<ref name="richards1981continued"> | |||
Ian Richards, [https://www.microsoft.com/en-us/research/wp-content/uploads/2016/10/cont-frac-wihtout-tears-richards-1.pdf Continued Fractions Without Tears]. Mathematics Magazine 54.4, 1981. | |||
</ref> | </ref> | ||
<ref name="wilson-temperament"> | <ref name="wilson-temperament"> | ||
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</ref> | </ref> | ||
<ref name="ratan2026another"> | <ref name="ratan2026another"> | ||
Naren Ratan, [https://www.xenharmonikon.org/2026/05/19/another-look-at-wilsons-keyboard-mapping-system/ Another look at Wilson's keyboard mapping system] | Naren Ratan, [https://www.xenharmonikon.org/2026/05/19/another-look-at-wilsons-keyboard-mapping-system/ Another look at Wilson's keyboard mapping system]. Xenharmonikon Online, 2026. | ||
</ref> | </ref> | ||
<ref name="wilson-gralspectrum"> | <ref name="wilson-gralspectrum"> | ||
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</ref> | </ref> | ||
</references> | </references> | ||
== External links == | |||
* [https://narenratan.com/tools/gralculator gralculator] - online Gral method calculator | |||
[[Category:Erv Wilson]] | [[Category:Erv Wilson]] | ||
[[Category:MOS scale]] | [[Category:MOS scale]] | ||