11358058edo: Difference between revisions

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{{Infobox ET|~~11358058edo~~|Consistency=35|Distinct consistency=35}}

{{Infobox ET

| Prime factorization = 2 × 5679029

| Consistency = 35

| Distinct consistency = 35

}}

~~11358058edo, or 11358058 equal divisions of the octave, is an equal tuning system with a~~ step size ~~of only about 0.00010565 cents,~~ far beyond the human melodic [[just-noticeable difference]]~~. It~~ has been noted for its highly accurate approximation of the 31-limit.

Although its step size is far beyond the human melodic [[just-noticeable difference]], it has been noted for its highly accurate approximation of the [[31-limit]], and is [[consistent]] up to the [[Odd prime sum limit|36-OPSL]], where it has a lower maximum error (i.e. the error of the least accurate approximation of any interval in the limit from JI) than any smaller [[edo]], meaning it is very likely a [[The Riemann zeta function and tuning|zeta peak]] edo.

While not practical to build an acoustic instrument for, one potential use of this system is in electronic music production, where free modulation between higher-limit JI intervals is desired. Instead of keeping track of the intervals directly, the number of steps to the octave for an interval could simply be added or subtracted from one note to get to the next. However, the consistency of this tuning is limited, and the sequence of intervals may eventually start to deviate from their true JI counterparts.

While not practical to build an acoustic instrument for, one potential use of this system is in electronic music production, where free modulation between higher-limit JI intervals is desired. Instead of keeping track of the intervals directly, the number of steps to the octave for an interval could simply be added or subtracted from one note to get to the next. However, like all other equal temperaments, the consistency of this tuning is limited, and the sequence of intervals may eventually start to deviate from their true JI counterparts.

{{Harmonics in equal

=== Prime harmonics ===

| ~~steps =~~ 11358058

| ~~num~~ = 2

{{Harmonics in equal|11358058|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 11358058edo (continued)}}

| ~~denom = 1~~

| columns = 13

| start = 1

| ~~prec~~ = 7

| title = Approximation of prime harmonics in 11358058edo

~~| intervals = prime~~

}}

@@ Line 1: / Line 1: @@
-{{Infobox ET|11358058edo|Consistency=35|Distinct consistency=35}}
+{{Mathematical interest}}
+{{Infobox ET
+| Prime factorization = 2 × 5679029
+| Consistency = 35
+| Distinct consistency = 35
+}}
+{{ED intro}}
-11358058edo, or 11358058 equal divisions of the octave, is an equal tuning system with a step size of only about 0.00010565 cents, far beyond the human melodic [[just-noticeable difference]]. It has been noted for its highly accurate approximation of the 31-limit.
+Although its step size is far beyond the human melodic [[just-noticeable difference]], it has been noted for its highly accurate approximation of the [[31-limit]], and is [[consistent]] up to the [[Odd prime sum limit|36-OPSL]], where it has a lower maximum error (i.e. the error of the least accurate approximation of any interval in the limit from JI) than any smaller [[edo]], meaning it is very likely a [[The Riemann zeta function and tuning|zeta peak]] edo.
-While not practical to build an acoustic instrument for, one potential use of this system is in electronic music production, where free modulation between higher-limit JI intervals is desired. Instead of keeping track of the intervals directly, the number of steps to the octave for an interval could simply be added or subtracted from one note to get to the next. However, the consistency of this tuning is limited, and the sequence of intervals may eventually start to deviate from their true JI counterparts.
+While not practical to build an acoustic instrument for, one potential use of this system is in electronic music production, where free modulation between higher-limit JI intervals is desired. Instead of keeping track of the intervals directly, the number of steps to the octave for an interval could simply be added or subtracted from one note to get to the next. However, like all other equal temperaments, the consistency of this tuning is limited, and the sequence of intervals may eventually start to deviate from their true JI counterparts.
-{{Harmonics in equal
+=== Prime harmonics ===
-| steps = 11358058
+{{Harmonics in equal|11358058|columns=9}}
-| num = 2
+{{Harmonics in equal|11358058|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 11358058edo (continued)}}
-| denom = 1
-| columns = 13
-| start = 1
-| prec = 7
-| title = Approximation of prime harmonics in 11358058edo
-| intervals = prime
-}}