Relative interval error: Difference between revisions

Line 17:

== Additivity ==

~~There~~ are two additivities of relative errors.

In indirect mapping, there are two additivities of relative errors.

First, for the same edo, a ratio which is the product of some other ratios have their relative errors additive, that is, if ''r''3 = ''r''1''r''2 for ''n'', then ''e'' (''n'', ''r''3) = ''e'' (''n'', ''r''1) + ''e'' (''n'', ''r''2).

If the ~~error exceeds the range -50% to +50%~~, ~~it indicates that~~ an ~~inconsistency occurs, and there is a discrepancy in patent val mapping and closest mapping, so~~ is the ~~error. The patent val mapping error is unchanged~~, ~~and~~ that ~~of closest mapping~~ is ~~the previous result reduced by an integer to fit it into the range~~.

Second, for the same ratio, an edo which is the sum of some other edos have their relative errors additive, that is, if ''n''3 = ''n''1 + ''n''2 for ''r'', then ''e'' (''n''3, ''r'') = ''e'' (''n''1, ''r'') + ''e'' (''n''2'','' ''r'').

~~For example~~, the ~~errors of 2/1, 3/1 and 5/1 in 19~~-~~edo are 0, -11.43~~% ~~and -11.66~~%, ~~respectively. Since 6/5 = (2/1)(3/1) / (5/1)~~, ~~its error~~ is ~~0 + (-11.43%) - (-11.66%) = 0.23%. That shows 19-edo has fairly flat fifths~~ and ~~major thirds~~, ~~yet they cancel out when it comes to minor thirds and results~~ in ~~a very accurate approximation~~.

In either case, if the error exceeds the range -50% to +50%, it indicates that an inconsistency occurs, and there is a discrepancy in val mapping and direct mapping, so is the error. To find the error in closest mapping, modulo the previous result by 100%.

~~Second, for~~ the ~~same ratio, an edo which~~ is ~~the sum~~ of ~~some other edos have their relative errors additive~~, ~~that is, if ''n''~~3</~~sub> = ''n''~~1</~~sub> + ''n''~~2</~~sub> for ''r'', then ''e''~~ (~~''n''~~3</~~sub>, ''r''~~) ~~= ''e''~~ (~~''n''~~1~~~~, ~~''r''~~) ~~+ ''e''~~ (~~''n''2'','' ''r''~~). ~~This also needs to be reduced by an integer to fit into the range~~. ~~In special, if an~~ edo ~~duplicates itself~~, and ~~if the mappings do not change, then the error also duplicates~~.

An example of the first additivity is shown as follows. The errors of 2/1, 3/1 and 5/1 in 19-edo are 0, -11.43% and -11.66%, respectively. Since 6/5 = (2/1)(3/1) / (5/1), its error is 0 + (-11.43%) - (-11.66%) = 0.23%. That shows 19-edo has fairly flat fifths and major thirds, yet they cancel out when it comes to minor thirds and results in a very accurate approximation.

~~For~~ example, the errors of 3/1 for 26-edo and 27-edo are -20.90% and +20.60%, ~~repectively~~, and their sum -0.30% is the error of 3/1 for 53-edo.

Here is an example for the second additivity. The errors of 3/1 for 26-edo and 27-edo are -20.90% and +20.60%, respectively, and their sum -0.30% is the error of 3/1 for 53-edo.

== See also ==

@@ Line 17: / Line 17: @@
 == Additivity ==
-There are two additivities of relative errors.
+In indirect mapping, there are two additivities of relative errors.
 First, for the same edo, a ratio which is the product of some other ratios have their relative errors additive, that is, if ''r''<sub>3</sub> = ''r''<sub>1</sub>''r''<sub>2</sub> for ''n'', then ''e'' (''n'', ''r''<sub>3</sub>) = ''e'' (''n'', ''r''<sub>1</sub>) + ''e'' (''n'', ''r''<sub>2</sub>).
-If the error exceeds the range -50% to +50%, it indicates that an inconsistency occurs, and there is a discrepancy in patent val mapping and closest mapping, so is the error. The patent val mapping error is unchanged, and that of closest mapping is the previous result reduced by an integer to fit it into the range.
+Second, for the same ratio, an edo which is the sum of some other edos have their relative errors additive, that is, if ''n''<sub>3</sub> = ''n''<sub>1</sub> + ''n''<sub>2</sub> for ''r'', then ''e'' (''n''<sub>3</sub>, ''r'') = ''e'' (''n''<sub>1</sub>, ''r'') + ''e'' (''n''<sub>2</sub>'','' ''r'').
-For example, the errors of 2/1, 3/1 and 5/1 in 19-edo are 0, -11.43% and -11.66%, respectively. Since 6/5 = (2/1)(3/1) / (5/1), its error is 0 + (-11.43%) - (-11.66%) = 0.23%. That shows 19-edo has fairly flat fifths and major thirds, yet they cancel out when it comes to minor thirds and results in a very accurate approximation.
+In either case, if the error exceeds the range -50% to +50%, it indicates that an inconsistency occurs, and there is a discrepancy in val mapping and direct mapping, so is the error. To find the error in closest mapping, modulo the previous result by 100%.
-Second, for the same ratio, an edo which is the sum of some other edos have their relative errors additive, that is, if ''n''<sub>3</sub> = ''n''<sub>1</sub> + ''n''<sub>2</sub> for ''r'', then ''e'' (''n''<sub>3</sub>, ''r'') = ''e'' (''n''<sub>1</sub>, ''r'') + ''e'' (''n''<sub>2</sub>'','' ''r''). This also needs to be reduced by an integer to fit into the range. In special, if an edo duplicates itself, and if the mappings do not change, then the error also duplicates.
+An example of the first additivity is shown as follows. The errors of 2/1, 3/1 and 5/1 in 19-edo are 0, -11.43% and -11.66%, respectively. Since 6/5 = (2/1)(3/1) / (5/1), its error is 0 + (-11.43%) - (-11.66%) = 0.23%. That shows 19-edo has fairly flat fifths and major thirds, yet they cancel out when it comes to minor thirds and results in a very accurate approximation.
-For example, the errors of 3/1 for 26-edo and 27-edo are -20.90% and +20.60%, repectively, and their sum -0.30% is the error of 3/1 for 53-edo.
+Here is an example for the second additivity. The errors of 3/1 for 26-edo and 27-edo are -20.90% and +20.60%, respectively, and their sum -0.30% is the error of 3/1 for 53-edo.
 == See also ==