Golden meantone: Difference between revisions

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== Construction ==

Golden meantone is approximated with increasing accuracy by the infinite sequence of temperaments indicated in the table below. In any [[meantone]] temperament the ~~five~~ intervals in the column headings form part of a Fibonacci sequence (in the sense that each adjacent pair sums to the interval to its immediate right) and in these equal temperaments the sizes of these intervals (expressed in step units) are consecutive numbers from the integer Fibonacci sequence 0, 1, 1, 2, 3, 5... Both the rows and the columns of the table form Fibonacci sequences, and ~~because~~ the five intervals ~~sums to an octave~~, the ~~octave cardinalities in the first column~~ are ~~formed by summing~~ the ~~five~~ numbers ~~to their right~~. As the cardinality increases the interval sequence better approximates a geometric progression.

Golden meantone is approximated with increasing accuracy by the infinite sequence of temperaments indicated in the table below. In any [[meantone]] temperament the intervals in the column headings form part of a Fibonacci sequence (in the sense that each adjacent pair sums to the interval to its immediate right) and in these equal temperaments the sizes of these intervals (expressed in step units) are consecutive numbers from the integer Fibonacci sequence 0, 1, 1, 2, 3, 5... Both the rows and the columns of the table form Fibonacci sequences. As the Octave is the Sum of two Fourth intervals and a Tone this can be rearranged as the Sum of the first five intervals in this table, the sequence of EDOs is a Fibonacci like sequence where terms are the sum of 5 consecutive Fibonacci numbers.

As the cardinality increases the interval sequence better approximates a geometric progression. 81edo marks the point at which the series ceases to display audible differences and approximates all of theses intervals within 1 cent. From 131edo we need to use the second best 5th as our generating interval.

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 == Construction ==
-Golden meantone is approximated with increasing accuracy by the infinite sequence of temperaments indicated in the table below. In any [[meantone]] temperament the five intervals in the column headings form part of a Fibonacci sequence (in the sense that each adjacent pair sums to the interval to its immediate right) and in these equal temperaments the sizes of these intervals (expressed in step units) are consecutive numbers from the integer Fibonacci sequence 0, 1, 1, 2, 3, 5... Both the rows and the columns of the table form Fibonacci sequences, and because the five intervals sums to an octave, the octave cardinalities in the first column are formed by summing the five numbers to their right. As the cardinality increases the interval sequence better approximates a geometric progression.
+Golden meantone is approximated with increasing accuracy by the infinite sequence of temperaments indicated in the table below. In any [[meantone]] temperament the intervals in the column headings form part of a Fibonacci sequence (in the sense that each adjacent pair sums to the interval to its immediate right) and in these equal temperaments the sizes of these intervals (expressed in step units) are consecutive numbers from the integer Fibonacci sequence 0, 1, 1, 2, 3, 5... Both the rows and the columns of the table form Fibonacci sequences. As the Octave is the Sum of two Fourth intervals and a Tone this can be rearranged as the Sum of the first five intervals in this table, the sequence of EDOs  is a Fibonacci like sequence where terms are the sum of 5 consecutive Fibonacci numbers.
+As the cardinality increases the interval sequence better approximates a geometric progression. 81edo marks the point at which the series ceases to display audible differences and approximates all of theses intervals within 1 cent. From 131edo we need to use the second best 5th as our generating interval.
 {| class="wikitable"