Goldonic series: Difference between revisions

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A '''goldonic series''' or '''golden series''' is a series of frequencies that form a [~~https~~:~~//en.wikipedia.org/wiki/~~Geometric_progression geometric progression] whose generating interval is the [~~https://en.wikipedia.org/wiki/Golden_ratio~~ golden ratio] (1.~~61803.~~...).

A '''goldonic series''' or '''golden series''' is a series of frequencies that form a [[Wikipedia:Geometric_progression|geometric progression]] whose generating interval is the [[phi|golden ratio]] (φ = 1.618...).

==Unique properties==

== Unique properties ==

The goldonic series is unique among geometric ~~sequencies~~ because only ''~~~~φ~~~~'' satisfies the equation ~~''x''~~<~~span style="vertical-align: super;"~~>n-1~~ ''~~+ x~~''~~n~~ ''~~= x~~''~~n+1</~~span~~>.

The goldonic series is unique among geometric sequences because only ''φ'' satisfies the equation <math>x^{n-1} + x^n = x^{n+1}</math>.

From an acoustic standpoint, the goldonic series contains some "harmonic-like" characteristics despite being inharmonic. Because each term is equal the difference between the next highest and next lowest terms, a phenomenon similar to modelocking can occur (in which each oscillator becomes entrained to the difference tone generated by its nearest-neighbors).

Also, unlike the harmonic series, the goldonic series can be in theory extended infinitely in ''both'' directions and contains no fundamental.

Also, unlike the [[harmonic series]], the goldonic series can be in theory extended infinitely in ''both'' directions and contains no fundamental.

[[Category:~~golden~~]]

[[Category:~~golden_ratio~~]]

[[Category:Golden ratio]]

[[Category:Nonoctave]]

[[Category:Xenharmonic series]]

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-A '''goldonic series''' or '''golden series''' is a series of frequencies that form a [https://en.wikipedia.org/wiki/Geometric_progression geometric progression] whose generating interval is the [https://en.wikipedia.org/wiki/Golden_ratio golden ratio] (1.61803....).
+A '''goldonic series''' or '''golden series''' is a series of frequencies that form a [[Wikipedia:Geometric_progression|geometric progression]] whose generating interval is the [[phi|golden ratio]] (φ = 1.618...).
-==Unique properties==
+== Unique properties ==
-The goldonic series is unique among geometric sequencies because only ''<span style="background-color: #ffffff; color: #252525; font-family: sans-serif; font-size: 14px;">φ</span>'' satisfies the equation ''x''<span style="vertical-align: super;">n-1</span> ''+ x''<span style="vertical-align: super;">n</span> ''= x''<span style="vertical-align: super;">n+1</span>.
+The goldonic series is unique among geometric sequences because only ''φ'' satisfies the equation <math>x^{n-1} + x^n = x^{n+1}</math>.
 From an acoustic standpoint, the goldonic series contains some "harmonic-like" characteristics despite being inharmonic. Because each term is equal the difference between the next highest and next lowest terms, a phenomenon similar to modelocking can occur (in which each oscillator becomes entrained to the difference tone generated by its nearest-neighbors).
-Also, unlike the harmonic series, the goldonic series can be in theory extended infinitely in ''both'' directions and contains no fundamental.
+Also, unlike the [[harmonic series]], the goldonic series can be in theory extended infinitely in ''both'' directions and contains no fundamental.
-[[Category:golden]]
-[[Category:golden_ratio]]
+[[Category:Golden ratio]]
+[[Category:Nonoctave]]
+[[Category:Xenharmonic series]]