311edo: Difference between revisions

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== Theory ==

311edo is [[consistent]] through the [[41-odd-limit]] and nearly distinctly consistent through the [[27-odd-limit]] with the single exception of [[25/24]]~[[26/25]], [[tempering out]] [[625/624|S25 (625/624)]], and is a [[zeta gap edo]] and a [[zeta peak integer edo]]. ~~It achieves this since~~ all [[~~harmonic~~]]s up to ~~and including~~ the 42nd, and all composite harmonics up to ~~and including~~ the 80th, are more ~~in-tune~~ than ~~out-of-tune~~ (~~but note prime~~ 73 ''is~~'' tuned accurately~~, ~~in fact~~ more ~~accurately~~ than all ~~prior~~ primes)~~. Thus~~ all ~~the~~ ratios ~~between~~ those harmonics are mapped consistently, ~~and thus~~ with a maximum error of ~1.929~~{{c}}~~. This means 311edo is an ''~~extremely~~'' efficient temperament for approximating the [[harmonic series]] consistently and ''simply'', given how much harmonic content it approximates/represents for its size.

311edo is [[consistent]] through the [[41-odd-limit]] and nearly distinctly consistent through the [[27-odd-limit]] with the single exception of [[25/24]]~[[26/25]], [[tempering out]] [[625/624|S25 (625/624)]], and is a [[zeta gap edo]] and a [[zeta peak integer edo]]. This is because all [[Harmonic|harmonics]] up to the 42nd, and all composite harmonics up to the 80th, are tuned more closely than they are mistuned. (Prime 73 is also unusually accurate, more so than all smaller primes.) As a result, all ratios among those harmonics are mapped consistently, with a maximum error of about 1.929 ¢. This means 311edo is a ''serendipitously'' efficient temperament for approximating the [[harmonic series]] and the [[41-limit]] in general, consistently and ''simply'', given how much harmonic content it approximates/represents for its size. The smallest EDO that has a higher [[consistency limit]] is [[20567edo|20567]], being consistent in the 57-odd-limit.

It also ~~maintains~~ [[~~relative interval error]]s of [[minimal~~ consistent ~~EDOs|no greater than 25%~~]] on all of the first 42 harmonics ~~of the harmonic series~~, ~~and is the smallest EDO~~ to ~~maintain less than 25% relative error on~~ the ~~first 32 harmonics~~. The next ~~edo~~ with less ~~than 25%~~ error ~~on the first 32 harmonics~~ is [[16808edo|16808]]~~, the~~ smallest ~~EDO that approximates the 43rd harmonic while maintaining the same maximum relative errors on the 42nd and lower is~~ [[~~20567edo|20567~~]]~~, and the smallest edo that maintains less than 25% relative error~~ on the first 64 harmonics is [[3159811edo|3159811]].

It is also the smallest EDO that is [[purely consistent]] on all the first 32 harmonics (in this case, up to the 42nd harmonic). The next [[EDO]] with less relative error is [[16808edo|16808]]. The smallest edo [[purely consistent]] on the first 64 harmonics is [[3159811edo|3159811]].

~~It is still very accurate in the lower limits.~~ Although it does not do as well as [[270edo]] in the 13-limit, it ~~makes for an interesting comparison~~. It tempers out the [[amity comma]], 1600000/1594323, the [[lafa comma]], {{monzo| 77 -31 -12 }}, the [[vavoom comma]], {{monzo| -68 18 17 }} in the [[5-limit]]; 2401/2400 ([[breedsma]]), 65625/65536 ([[horwell comma]]), and 33554432/33480783 ([[garischisma]]) in the 7-limit; [[3025/3024]], [[4000/3993]], [[6250/6237]], [[12005/11979]], and [[19712/19683]] in the 11-limit; and 625/624, [[1575/1573]], [[2080/2079]], [[2200/2197]], [[4096/4095]], and [[4225/4224]] in the 13-limit. It allows [[petrmic chords|petrmic]] and [[nicolic chords]] in the 15-odd-limit.

Although it does not do as well as [[270edo]] in the 13-limit, it is still very accurate in the lower limits. It tempers out the [[amity comma]], 1600000/1594323, the [[lafa comma]], {{monzo| 77 -31 -12 }}, the [[vavoom comma]], {{monzo| -68 18 17 }} in the [[5-limit]]; 2401/2400 ([[breedsma]]), 65625/65536 ([[horwell comma]]), and 33554432/33480783 ([[garischisma]]) in the 7-limit; [[3025/3024]], [[4000/3993]], [[6250/6237]], [[12005/11979]], and [[19712/19683]] in the 11-limit; and 625/624, [[1575/1573]], [[2080/2079]], [[2200/2197]], [[4096/4095]], and [[4225/4224]] in the 13-limit. It allows [[petrmic chords|petrmic]] and [[nicolic chords]] in the 15-odd-limit.

Beyond the 13-limit, primes [[17/1|17]] and [[23/1|23]] are 311edo's first notable improvements over 270edo's approximation. It tempers out [[595/594]], [[833/832]], [[1156/1155]], [[1225/1224]], [[1275/1274]], [[2058/2057]], [[2431/2430]] in the 17-limit; [[969/968]], [[1216/1215]], [[1445/1444]], [[1540/1539]], [[1729/1728]] in the 19-limit; and [[760/759]], [[875/874]], [[1105/1104]], [[1197/1196]], [[1288/1287]], [[1496/1495]] in the 23-limit.

It is valuable from a psychoacoustic perspective as its step is also ~~conincidentally close enough to~~ the [[just-noticeable difference]], which only affirms its efficiency of interval representation.

It is valuable from a psychoacoustic perspective as its step is also coincidentally above the melodic [[just-noticeable difference]], which only affirms its efficiency of interval representation.

=== Prime harmonics ===

{{Harmonics in equal|311|columns=12|start=13|prec=3|collapsed=true|title=Approximation of prime harmonics in 311edo (continued)}}

{{Harmonics in equal|311|columns=17|start=14|prec=3|collapsed=true|title=Approximation of prime harmonics in 311edo (continued)}}

=== Subsets and supersets ===

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== Intervals ==

The 41-limit add-73 add-89 add-101 add-109 add-113 123-odd-limit is represented very close to completely [[consistent]]ly, and as aforementioned, the 77-[[odd-limit]] subset of that odd-limit is ~~perfectly~~ consistent, to which a variety of odds can be added that keep ~~perfect~~ consistency, but for comprehensiveness and practical use as a temperament approximating the low-to-mid end of the harmonic series, we consider a larger odd-limit than that which seeks to be more complete.

The 41-limit add-73 add-89 add-101 add-109 add-113 123-odd-limit is represented very close to completely [[consistent]]ly, and as aforementioned, the 77-[[odd-limit]] subset of that odd-limit is purely consistent, to which a variety of odds can be added that keep pure consistency, but for comprehensiveness and practical use as a temperament approximating the low-to-mid end of the harmonic series, we consider a larger odd-limit than that which seeks to be more complete.

There are 884 interval pairs in that [[odd limit]] (the [[41-limit]] add-73 add-89 add-101 add-109 add-113 123-odd-limit), where "pairs" refers to that each interval has an [[octave complement]] with equal and opposite error. That odd limit can be described explicitly as the [[tonality diamond]] of {1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 45, 49, 51, 55, 57, 63, 65, 69, 73, 75, 77, 81, 85, 87, 89, 91, 93, 95, 99, 101, 105, 109, 111, 113, 115, 117, 119, 121, 123}. We can also express that odd-limit as the 123-odd-limit minus only the following twelve prime odds: {43, 47, 53, 59, 61, 67, 71, 79, 83, 97, 103, 107}.

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The below table was generated by a simple Python 3 script to print it in plaintext using [[User: Godtone #My Python 3 code|Godtone's code]] to simplify certain steps.

It should be noted that while almost all intervals shown in the table are intervals of the 123-odd-limit restricted to the aforementioned prime subgroup, the [[square-particular]]s up to [[1681/1680|S41 = (41/40)/(42/41)]] were added manually for completeness and reference in understanding the mapping of the [[41-odd-limit]] by 311edo. ~~Therefore, the very beginning~~ of the table ~~(from 0\311 to 3\311 inclusive) is the only part that~~ is ~~not~~ algorithmically generated.

It should be noted that while almost all intervals shown in the table are intervals of the 123-odd-limit restricted to the aforementioned prime subgroup, the [[square-particular]]s up to [[1681/1680|S41 = (41/40)/(42/41)]] were added manually for completeness and reference in understanding the mapping of the [[41-odd-limit]] by 311edo for the first three edosteps and the unison. The rest of the table is algorithmically generated.

=== Interval table ===

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-{{Infobox ET}}
+{{Infobox ET|Zeta=yes}}
 {{ED intro}}
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 == Theory ==
-edo is [[consistent]] through the [[41-odd-limit]] and nearly distinctly consistent through the [[27-odd-limit]] with the single exception of [[25/24]]~[[26/25]], [[tempering out]] [[625/624|S25 (625/624)]], and is a [[zeta gap edo]] and a [[zeta peak integer edo]]. It achieves this since all [[harmonic]]s up to and including the 42nd, and all composite harmonics up to and including the 80th, are more in-tune than out-of-tune (but note prime 73 ''is'' tuned accurately, in fact more accurately than all prior primes). Thus all the ratios between those harmonics are mapped consistently, and thus with a maximum error of ~1.929{{c}}. This means 311edo is an ''extremely'' efficient temperament for approximating the [[harmonic series]] consistently and ''simply'', given how much harmonic content it approximates/represents for its size.
+edo is [[consistent]] through the [[41-odd-limit]] and nearly distinctly consistent through the [[27-odd-limit]] with the single exception of [[25/24]]~[[26/25]], [[tempering out]] [[625/624|S25 (625/624)]], and is a [[zeta gap edo]] and a [[zeta peak integer edo]]. This is because all [[Harmonic|harmonics]] up to the 42nd, and all composite harmonics up to the 80th, are tuned more closely than they are mistuned. (Prime 73 is also unusually accurate, more so than all smaller primes.) As a result, all ratios among those harmonics are mapped consistently, with a maximum error of about 1.929 ¢. This means 311edo is a ''serendipitously'' efficient temperament for approximating the [[harmonic series]] and the [[41-limit]] in general, consistently and ''simply'', given how much harmonic content it approximates/represents for its size. The smallest EDO that has a higher [[consistency limit]] is [[20567edo|20567]], being consistent in the 57-odd-limit.
-It also maintains [[relative interval error]]s of [[minimal consistent EDOs|no greater than 25%]] on all of the first 42 harmonics of the harmonic series, and is the smallest EDO to maintain less than 25% relative error on the first 32 harmonics. The next edo with less than 25% error on the first 32 harmonics is [[16808edo|16808]], the smallest EDO that approximates the 43rd harmonic while maintaining the same maximum relative errors on the 42nd and lower is [[20567edo|20567]], and the smallest edo that maintains less than 25% relative error on the first 64 harmonics is [[3159811edo|3159811]].
+It is also the smallest EDO that is [[purely consistent]] on all the first 32 harmonics (in this case, up to the 42nd harmonic). The next [[EDO]] with less relative error is [[16808edo|16808]]. The smallest edo [[purely consistent]] on the first 64 harmonics is [[3159811edo|3159811]].
-It is still very accurate in the lower limits. Although it does not do as well as [[270edo]] in the 13-limit, it makes for an interesting comparison. It tempers out the [[amity comma]], 1600000/1594323, the [[lafa comma]], {{monzo| 77 -31 -12 }}, the [[vavoom comma]], {{monzo| -68 18 17 }} in the [[5-limit]]; 2401/2400 ([[breedsma]]), 65625/65536 ([[horwell comma]]), and 33554432/33480783 ([[garischisma]]) in the 7-limit; [[3025/3024]], [[4000/3993]], [[6250/6237]], [[12005/11979]], and [[19712/19683]] in the 11-limit; and 625/624, [[1575/1573]], [[2080/2079]], [[2200/2197]], [[4096/4095]], and [[4225/4224]] in the 13-limit. It allows [[petrmic chords|petrmic]] and [[nicolic chords]] in the 15-odd-limit.
+Although it does not do as well as [[270edo]] in the 13-limit, it is still very accurate in the lower limits. It tempers out the [[amity comma]], 1600000/1594323, the [[lafa comma]], {{monzo| 77 -31 -12 }}, the [[vavoom comma]], {{monzo| -68 18 17 }} in the [[5-limit]]; 2401/2400 ([[breedsma]]), 65625/65536 ([[horwell comma]]), and 33554432/33480783 ([[garischisma]]) in the 7-limit; [[3025/3024]], [[4000/3993]], [[6250/6237]], [[12005/11979]], and [[19712/19683]] in the 11-limit; and 625/624, [[1575/1573]], [[2080/2079]], [[2200/2197]], [[4096/4095]], and [[4225/4224]] in the 13-limit. It allows [[petrmic chords|petrmic]] and [[nicolic chords]] in the 15-odd-limit.
 Beyond the 13-limit, primes [[17/1|17]] and [[23/1|23]] are 311edo's first notable improvements over 270edo's approximation. It tempers out [[595/594]], [[833/832]], [[1156/1155]], [[1225/1224]], [[1275/1274]], [[2058/2057]], [[2431/2430]] in the 17-limit; [[969/968]], [[1216/1215]], [[1445/1444]], [[1540/1539]], [[1729/1728]] in the 19-limit; and [[760/759]], [[875/874]], [[1105/1104]], [[1197/1196]], [[1288/1287]], [[1496/1495]] in the 23-limit.
-It is valuable from a psychoacoustic perspective as its step is also conincidentally close enough to the [[just-noticeable difference]], which only affirms its efficiency of interval representation.
+It is valuable from a psychoacoustic perspective as its step is also coincidentally above the melodic [[just-noticeable difference]], which only affirms its efficiency of interval representation.
 === Prime harmonics ===
-{{Harmonics in equal|311|prec=3|columns=12}}
+{{Harmonics in equal|311|prec=3|columns=13}}
-{{Harmonics in equal|311|columns=12|start=13|prec=3|collapsed=true|title=Approximation of prime harmonics in 311edo (continued)}}
+{{Harmonics in equal|311|columns=17|start=14|prec=3|collapsed=true|title=Approximation of prime harmonics in 311edo (continued)}}
 === Subsets and supersets ===
@@ Line 25: / Line 25: @@
 == Intervals ==
-The 41-limit add-73 add-89 add-101 add-109 add-113 123-odd-limit is represented very close to completely [[consistent]]ly, and as aforementioned, the 77-[[odd-limit]] subset of that odd-limit is perfectly consistent, to which a variety of odds can be added that keep perfect consistency, but for comprehensiveness and practical use as a temperament approximating the low-to-mid end of the harmonic series, we consider a larger odd-limit than that which seeks to be more complete.
+The 41-limit add-73 add-89 add-101 add-109 add-113 123-odd-limit is represented very close to completely [[consistent]]ly, and as aforementioned, the 77-[[odd-limit]] subset of that odd-limit is purely consistent, to which a variety of odds can be added that keep pure consistency, but for comprehensiveness and practical use as a temperament approximating the low-to-mid end of the harmonic series, we consider a larger odd-limit than that which seeks to be more complete.
 There are 884 interval pairs in that [[odd limit]] (the [[41-limit]] add-73 add-89 add-101 add-109 add-113 123-odd-limit), where "pairs" refers to that each interval has an [[octave complement]] with equal and opposite error. That odd limit can be described explicitly as the [[tonality diamond]] of {1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 45, 49, 51, 55, 57, 63, 65, 69, 73, 75, 77, 81, 85, 87, 89, 91, 93, 95, 99, 101, 105, 109, 111, 113, 115, 117, 119, 121, 123}. We can also express that odd-limit as the 123-odd-limit minus only the following twelve prime odds: {43, 47, 53, 59, 61, 67, 71, 79, 83, 97, 103, 107}.
@@ Line 35: / Line 35: @@
 The below table was generated by a simple Python 3 script to print it in plaintext using [[User: Godtone #My Python 3 code|Godtone's code]] to simplify certain steps.
-It should be noted that while almost all intervals shown in the table are intervals of the 123-odd-limit restricted to the aforementioned prime subgroup, the [[square-particular]]s up to [[1681/1680|S41 = (41/40)/(42/41)]] were added manually for completeness and reference in understanding the mapping of the [[41-odd-limit]] by 311edo. Therefore, the very beginning of the table (from 0\311 to 3\311 inclusive) is the only part that is not algorithmically generated.
+It should be noted that while almost all intervals shown in the table are intervals of the 123-odd-limit restricted to the aforementioned prime subgroup, the [[square-particular]]s up to [[1681/1680|S41 = (41/40)/(42/41)]] were added manually for completeness and reference in understanding the mapping of the [[41-odd-limit]] by 311edo for the first three edosteps and the unison. The rest of the table is algorithmically generated.
 === Interval table ===