311edo: Difference between revisions

Line 5:

== Theory ==

311edo is [[consistent]] through the [[41-odd-limit]] and nearly distinctly consistent through the [[27-odd-limit]] except for [[25/24]][[~]][[26/25]], [[tempering out]] [[625/624]] ({{S|25}}), and is a [[zeta gap edo]] and a [[zeta peak integer edo]]. This is because all [[harmonic]]s up to the 42nd, and all composite harmonics up to the 80th, have no more than ±25% error. (Prime 73 is also unusually accurate, more so than all smaller primes.) As a result, all ratios among those harmonics are mapped consistently, with errors lower than 1.929{{c}}. 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.

311edo is [[consistent]] through the [[41-odd-limit]] and nearly distinctly consistent through the [[27-odd-limit]] except for [[25/24]][[~]][[26/25]], [[tempering out]] [[625/624]] ({{S|25}}), and is a [[zeta gap edo]] and a [[zeta peak integer edo]]. This is because all [[harmonic]]s up to the 42nd, and all composite harmonics up to the 80th, have no more than ±25% error. Prime 73 is also unusually accurate, more so than all smaller primes. As a result, all ratios among those harmonics are mapped consistently, with errors lower than 1.929{{c}}. 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 comes closest to having a higher consistency is [[1600edo]] ([[37-odd-limit~~]]), almost consistent in the [[43-odd-limit]] except for two intervals ([[39/25]], [[50/39~~]]). The next edo with a truly higher [[consistency limit]] is [[17461edo|17461]] ([[45-odd-limit]]), though one may prefer [[20567edo|20567]] ([[57-odd-limit]]).

The smallest edo that comes closest to having a higher consistency is [[1600edo]] ([[37-odd-limit]] ). The next edo with a truly higher [[consistency limit]] is [[17461edo|17461]] ([[45-odd-limit]]), though one may prefer [[20567edo|20567]] ([[57-odd-limit]]).

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

311edo is also the smallest edo that is [[purely consistent]] on all the first 32 harmonics (in this case, up to the 42nd). The next edo with less maximum relative error is [[16808edo|16808]]. The smallest edo purely consistent on the first 64 harmonics is [[3159811edo|3159811]].

Although it does not do as well as [[270edo]] in the 13-limit, it is still very accurate in the lower limits~~, having around +0.15 cents more of ajusted error than 270edo (0.44c)~~. 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 311edo 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. Their edo sum, [[581edo]], is also a very strong 23-limit temperament.

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]]. Their edo sum, [[581edo]], is also a very strong 23-limit temperament.

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.

311edo 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 ===

Line 22:

=== Subsets and supersets ===

311edo is the 64th [[prime edo]].

311edo is the 64th [[prime edo]], so it does not contain any nontrivial subset edos.

As an interval size measure, one step of 311edo is called ''gene'', named by [[Joseph Monzo]] in 2007 after [[Gene Ward Smith]]<ref>[http://tonalsoft.com/enc/g/gene.aspx Tonalsoft Encyclopedia | ''gene, 311-edo'']</ref>.

== 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 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.

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 5: / Line 5: @@
 == Theory ==
-edo is [[consistent]] through the [[41-odd-limit]] and nearly distinctly consistent through the [[27-odd-limit]] except for [[25/24]][[~]][[26/25]], [[tempering out]] [[625/624]] ({{S|25}}), and is a [[zeta gap edo]] and a [[zeta peak integer edo]]. This is because all [[harmonic]]s up to the 42nd, and all composite harmonics up to the 80th, have no more than ±25% error. (Prime 73 is also unusually accurate, more so than all smaller primes.) As a result, all ratios among those harmonics are mapped consistently, with errors lower than 1.929{{c}}. 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.
+edo is [[consistent]] through the [[41-odd-limit]] and nearly distinctly consistent through the [[27-odd-limit]] except for [[25/24]][[~]][[26/25]], [[tempering out]] [[625/624]] ({{S|25}}), and is a [[zeta gap edo]] and a [[zeta peak integer edo]]. This is because all [[harmonic]]s up to the 42nd, and all composite harmonics up to the 80th, have no more than ±25% error. Prime 73 is also unusually accurate, more so than all smaller primes. As a result, all ratios among those harmonics are mapped consistently, with errors lower than 1.929{{c}}. 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 comes closest to having a higher consistency is [[1600edo]] ([[37-odd-limit]]), almost consistent in the [[43-odd-limit]] except for two intervals ([[39/25]], [[50/39]]). The next edo with a truly higher [[consistency limit]] is [[17461edo|17461]] ([[45-odd-limit]]), though one may prefer [[20567edo|20567]] ([[57-odd-limit]]).
+The smallest edo that comes closest to having a higher consistency is [[1600edo]] ([[37-odd-limit]] ). The next edo with a truly higher [[consistency limit]] is [[17461edo|17461]] ([[45-odd-limit]]), though one may prefer [[20567edo|20567]] ([[57-odd-limit]]).
-It is also the smallest edo that is [[purely consistent]] on all the first 32 harmonics (in this case, up to the 42nd). The next edo with less relative error is [[16808edo|16808]]. The smallest edo purely consistent on the first 64 harmonics is [[3159811edo|3159811]].
+edo is also the smallest edo that is [[purely consistent]] on all the first 32 harmonics (in this case, up to the 42nd). The next edo with less maximum relative error is [[16808edo|16808]]. The smallest edo purely consistent on the first 64 harmonics is [[3159811edo|3159811]].
-Although it does not do as well as [[270edo]] in the 13-limit, it is still very accurate in the lower limits, having around +0.15 cents more of ajusted error than 270edo (0.44c). 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 311edo 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. Their edo sum, [[581edo]], is also a very strong 23-limit temperament.
+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]]. Their edo sum, [[581edo]], is also a very strong 23-limit temperament.
-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.
+edo 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 ===
@@ Line 22: / Line 22: @@
 === Subsets and supersets ===
-edo is the 64th [[prime edo]].
+edo is the 64th [[prime edo]], so it does not contain any nontrivial subset edos.
 As an interval size measure, one step of 311edo is called ''gene'', named by [[Joseph Monzo]] in 2007 after [[Gene Ward Smith]]<ref>[http://tonalsoft.com/enc/g/gene.aspx Tonalsoft Encyclopedia | ''gene, 311-edo'']</ref>.
 == 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 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.
+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}.

311edo: Difference between revisions

FloraC (talk | contribs)