270edo: Difference between revisions

Line 1:

The '''270 equal divisions of the octave''' ('''270edo'''), or the '''270(-tone) equal temperament''' ('''270tet''', '''270et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 270 [[equal]] parts of 4.{{~~overline|4~~}} ~~[[cent]]s each, a size close to [[385/384]], the keenanisma.~~

{{EDO intro}} 270edo's step size is called a '''tredek''' when used as an [[interval size unit]].

270edo's step size is called a '''tredek''' when used as an [[interval size unit]].

== Theory ==

270edo is an extremely strong [[13-limit]] system, [[~~consistency|~~distinctly consistent]] through the [[15-odd-limit]] with all intervals in the 15-odd-limit being approximated with less than 25% relative error with only the exception of [[15/13]] which barely misses (corresponding to the fact of tempering out [[676/675]]). This results in it being a record edo for [[Pepper ambiguity]] in the 11-, 13- and 15-odd-limit. It is [[The Riemann zeta function and tuning #Zeta EDO lists|the 11th zeta gap edo, the 13th zeta integral edo, the 23rd zeta peak edo, and the 18th zeta peak integer edo]], making it a strict zeta edo, and is the first [[Trivial temperament|non-trivial]] edo to be consistent in the 16-[[Odd prime sum limit|odd-prime-sum-limit]].

270edo is an extremely strong [[13-limit]] system, [[distinctly consistent]] through the [[15-odd-limit]] with all intervals in the 15-odd-limit being approximated with less than 25% relative error with only the exception of [[15/13]] which barely misses (corresponding to the fact of tempering out [[676/675]]). This results in it being a record edo for [[Pepper ambiguity]] in the 11-, 13- and 15-odd-limit. It is [[The Riemann zeta function and tuning #Zeta EDO lists|the 11th zeta gap edo, the 13th zeta integral edo, the 23rd zeta peak edo, and the 18th zeta peak integer edo]], making it a strict zeta edo, and is the first [[Trivial temperament|non-trivial]] edo to be consistent in the 16-[[Odd prime sum limit|odd-prime-sum-limit]].

In the [[5-limit]] it tempers out the [[ennealimma]], {{monzo| 1 -27 18 }}, the [[vulture comma]], {{monzo| 24 -21 4 }}, and the [[vishnuzma]] (a.k.a. semisuper comma), {{monzo| 23 6 -14 }}.

Line 51:

Line 49:

| {{monzo| 23 6 -14 }}, {{monzo| 24 -21 4 }}

| {{mapping| 270 428 627 }}

| ~~−0~~.1069

| −0.1069

| 0.0759

| 1.71

Line 58:

Line 56:

| 2401/2400, 4375/4374, 29360128/29296875

| {{mapping| 270 428 627 758 }}

| ~~−0~~.0858

| −0.0858

| 0.0752

| 1.69

Line 65:

Line 63:

| 2401/2400, 3025/3024, 4375/4374, 5632/5625

| {{mapping| 270 428 627 758 934 }}

| ~~−0~~.0567

| −0.0567

| 0.0889

| 2.00

Line 72:

Line 70:

| 676/675, 1001/1000, 1716/1715, 3025/3024, 4096/4095

| {{mapping| 270 428 627 758 934 999 }}

| ~~−0~~.0235

| −0.0235

| 0.1100

| 2.48

Line 79:

Line 77:

| 676/675, 1001/1000, 1216/1215, 1331/1330, 1540/1539, 1729/1728

| {{mapping| 270 428 627 758 934 999 1147 }}

| ~~−0~~.0290

| −0.0290

| 0.1028

| 2.31

Line 86:

Line 84:

| 676/675, 715/714, 936/935, 1001/1000, 1225/1224, 4096/4095

| {{mapping| 270 428 627 758 934 999 1104 }}

| ~~−0~~.0799

| −0.0799

| 0.1718

| 3.86

Line 93:

Line 91:

| 676/675, 715/714, 936/935, 1001/1000, 1216/1215, 1225/1224, 1331/1330

| {{mapping| 270 428 627 758 934 999 1104 1147 }}

| ~~−0~~.0777

| −0.0777

| 0.1608

| 3.62

Line 100:

Line 98:

| 460/459, 529/528, 676/675, 715/714, 736/735, 936/935, 1001/1000, 1216/1215

| {{mapping| 270 428 627 758 934 999 1104 1147 1221 }}

| ~~−0~~.0296

| −0.0296

| 0.2037

| 4.58

Line 274:

Line 272:

| [[Rhodium]]

|}

<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if ~~it is~~ distinct

<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== External links ==

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-The '''270 equal divisions of the octave''' ('''270edo'''), or the '''270(-tone) equal temperament''' ('''270tet''', '''270et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 270 [[equal]] parts of 4.{{overline|4}} [[cent]]s each, a size close to [[385/384]], the keenanisma.
+{{EDO intro}} 270edo's step size is called a '''tredek''' when used as an [[interval size unit]].
-edo's step size is called a '''tredek''' when used as an [[interval size unit]].
 == Theory ==
-edo is an extremely strong [[13-limit]] system, [[consistency|distinctly consistent]] through the [[15-odd-limit]] with all intervals in the 15-odd-limit being approximated with less than 25% relative error with only the exception of [[15/13]] which barely misses (corresponding to the fact of tempering out [[676/675]]). This results in it being a record edo for [[Pepper ambiguity]] in the 11-, 13- and 15-odd-limit. It is [[The Riemann zeta function and tuning #Zeta EDO lists|the 11th zeta gap edo, the 13th zeta integral edo, the 23rd zeta peak edo, and the 18th zeta peak integer edo]], making it a strict zeta edo, and is the first [[Trivial temperament|non-trivial]] edo to be consistent in the 16-[[Odd prime sum limit|odd-prime-sum-limit]].
+edo is an extremely strong [[13-limit]] system, [[distinctly consistent]] through the [[15-odd-limit]] with all intervals in the 15-odd-limit being approximated with less than 25% relative error with only the exception of [[15/13]] which barely misses (corresponding to the fact of tempering out [[676/675]]). This results in it being a record edo for [[Pepper ambiguity]] in the 11-, 13- and 15-odd-limit. It is [[The Riemann zeta function and tuning #Zeta EDO lists|the 11th zeta gap edo, the 13th zeta integral edo, the 23rd zeta peak edo, and the 18th zeta peak integer edo]], making it a strict zeta edo, and is the first [[Trivial temperament|non-trivial]] edo to be consistent in the 16-[[Odd prime sum limit|odd-prime-sum-limit]].
 In the [[5-limit]] it tempers out the [[ennealimma]], {{monzo| 1 -27 18 }}, the [[vulture comma]], {{monzo| 24 -21 4 }}, and the [[vishnuzma]] (a.k.a. semisuper comma), {{monzo| 23 6 -14 }}.
@@ Line 51: / Line 49: @@
 | {{monzo| 23 6 -14 }}, {{monzo| 24 -21 4 }}
 | {{mapping| 270 428 627 }}
-| &minus;0.1069
+| −0.1069
 | 0.0759
 | 1.71
@@ Line 58: / Line 56: @@
 | 2401/2400, 4375/4374, 29360128/29296875
 | {{mapping| 270 428 627 758 }}
-| &minus;0.0858
+| −0.0858
 | 0.0752
 | 1.69
@@ Line 65: / Line 63: @@
 | 2401/2400, 3025/3024, 4375/4374, 5632/5625
 | {{mapping| 270 428 627 758 934 }}
-| &minus;0.0567
+| −0.0567
 | 0.0889
 | 2.00
@@ Line 72: / Line 70: @@
 | 676/675, 1001/1000, 1716/1715, 3025/3024, 4096/4095
 | {{mapping| 270 428 627 758 934 999 }}
-| &minus;0.0235
+| −0.0235
 | 0.1100
 | 2.48
@@ Line 79: / Line 77: @@
 | 676/675, 1001/1000, 1216/1215, 1331/1330, 1540/1539, 1729/1728
 | {{mapping| 270 428 627 758 934 999 1147 }}
-| &minus;0.0290
+| −0.0290
 | 0.1028
 | 2.31
@@ Line 86: / Line 84: @@
 | 676/675, 715/714, 936/935, 1001/1000, 1225/1224, 4096/4095
 | {{mapping| 270 428 627 758 934 999 1104 }}
-| &minus;0.0799
+| −0.0799
 | 0.1718
 | 3.86
@@ Line 93: / Line 91: @@
 | 676/675, 715/714, 936/935, 1001/1000, 1216/1215, 1225/1224, 1331/1330
 | {{mapping| 270 428 627 758 934 999 1104 1147 }}
-| &minus;0.0777
+| −0.0777
 | 0.1608
 | 3.62
@@ Line 100: / Line 98: @@
 | 460/459, 529/528, 676/675, 715/714, 736/735, 936/935, 1001/1000, 1216/1215
 | {{mapping| 270 428 627 758 934 999 1104 1147 1221 }}
-| &minus;0.0296
+| −0.0296
 | 0.2037
 | 4.58
@@ Line 274: / Line 272: @@
 | [[Rhodium]]
 |}
-<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 == External links ==