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= (WIP) Relative Errors of Small Edos =
= (WIP) Relative Errors of Small Edos =


The relative error of an interval in an edo is the error approximating JI divided by the size of a single step. The formula for closest mapping: ''error''(''n'', ''r'') = round (''n'' log<sub>2</sub>''r'') - ''n'' log<sub>2</sub>''r'', where ''n'' is the edo number and ''r'' is the frequency ratio. With closest mapping, the relative error ranges from -0.5 to +0.5. With patent val mapping, it can be farther from zero.  
The relative error of an interval in an [[edo]] is the error approximating [[JI]] divided by the size of a single step. The formula for closest mapping: ''error''(''n'', ''r'') = round (''n'' log<sub>2</sub>''r'') - ''n'' log<sub>2</sub>''r'', where ''n'' is the edo number and ''r'' is the frequency ratio. With closest mapping, the relative error ranges from -0.5 to +0.5. With patent val mapping, it can be farther from zero.  


This article contains two lists. The first shows relative errors of the first 9 prime harmonies for edos up to 99. There is no point for showing higher primes because no edo under 99 is consistent up to them. For other intervals, the relative error follows the additive rule (see below), so they can be derived easily. Also by that rule, however, finding large errors with such ''p''/1 harmonies will not suffice that the edo does a poor approximation in the ''p''-limit overall. One must inspect every relevant interval to be sure of that. The second list comes in naturally for showing the root-mean-squared relative errors of a certain JI subgroup, and may be used as the criterion.   
This article contains two lists. The first shows relative errors of the first 9 prime harmonies for edos up to 99. There is no point for showing higher primes because no edo under 99 is consistent up to them. For other intervals, the relative error follows the additive rule (see below), so they can be derived easily. Also by that rule, however, finding large errors with such ''p''/1 harmonies will not suffice that the edo does a poor approximation in the ''p''-limit overall. One must inspect every relevant interval to be sure of that. The second list comes in naturally for showing the root-mean-squared relative errors of a certain JI subgroup, and may be used as the criterion.   
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== List of RMS Relative Errors of JI Subgroup for Small Edos ==
== List of RMS Relative Errors of JI Subgroup for Small Edos ==
Note: octave equivalence is assumed. For example, the 5-odd-limit takes account of 2/1, 3/1, 5/1, 5/3 and their octave inverses. The 9-odd-limit takes account of 3/2 twice (as 3/2 and 9/6).  
Note: octave equivalence is assumed. For example, the 5-odd-limit takes account of 2/1, 3/1, 5/1, 5/3 and their octave inverses. The 9-odd-limit takes account of 3/2 twice as the [[tonality diamond]] suggests.  
{| class="wikitable sortable mw-collapsible"
{| class="wikitable sortable mw-collapsible"
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|
|
|-
|-
|3
|98
|230.7
|478.1
|
|
|
|
|
|
|
|
|
|
|
|-
|99
|62.7
|80.9
|
|
|
|
Retrieved from "https://en.xen.wiki/w/Relative_errors_of_small_EDOs"