3395edo: Difference between revisions

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~~The '''3395 equal divisions of the octave''' divides the [[octave]] into 3395 [[equal]] steps of 0.35346 [[cent]]s each.~~

3395edo is an extremely strong 17- and 19-limit system, and a [[The Riemann ~~Zeta~~ function and tuning #Zeta EDO lists|zeta peak, integral and gap edo]]. It has a lower 17-limit [[TE relative error]] than any edo until [[7033edo|7033]], and a lower 19-limit relative error than any edo until [[8269edo|8269]]. Besides, it provides the [[optimal patent val]] for the 13-limit rank-5 temperament tempering out [[6656/6655]], the jacobin comma. A basis for the 17-limit commas is {6656/6655, 12376/12375, 28561/28560, 31213/31212, 37180/37179, 937125/937024}, and for the 19-limit, {6656/6655, 12376/12375, 12636/12635, 13377/13376, 14365/14364, 23409/23408, 28561/28560}.

== Theory ==

3395edo is an extremely strong 17- and 19-limit system, and a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak, integral, and gap edo]]. It has a lower 17-limit [[TE relative error]] than any edo until [[7033edo|7033]], and a lower 19-limit relative error than any edo until [[8269edo|8269]]. Besides, it provides the [[optimal patent val]] for the 13-limit rank-5 temperament tempering out [[6656/6655]], the jacobin comma, and for [[quartismic]], which also tempers out [[123201/123200]]. A basis for the 17-limit commas is {6656/6655, 12376/12375, 28561/28560, 31213/31212, 37180/37179, 937125/937024}, and for the 19-limit, {6656/6655, 12376/12375, 12636/12635, 13377/13376, 14365/14364, 23409/23408, 28561/28560}.

3395 = ~~5 × 7 × 97~~, ~~with~~ subset edos 5, 7, 35, 485, and 679.

=== Prime harmonics ===

=== Subsets and supersets ===

Since 3395 factors into {{factorization|3395}}, 3395edo has subset edos 5, 7, 35, 97, 485, and 679.

== Regular temperament properties ==

3395edo has a lower 17-limit [[TE relative error]] than any edo until [[7033edo|7033]], and a lower 19-limit relative error than any edo until [[8269edo|8269]].

=== Rank-2 temperaments ===

{| class="wikitable center-all left-5"

|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

|-

! Periods per 8ve

! Generator*

! Cents*

! Associated ratio*

! Temperaments

|-

| 1

| 2319\3395

| 819.676

| 55115776/34328125

| [[Genojacobin]]

|-

| 35

| 1409\3395 (51\3395)

| 498.027 (18.026)

| 4/3 (?)

| [[Bromine]]

|-

| 97

| 1409\3395 (9\3395)

| 498.027 (3.181)

| 4/3 (?)

| [[Berkelium]]

|}

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

~~[[Category:Equal divisions of the octave]]~~

[[Category:Jacobin]]

[[Category:Quartismic]]

@@ Line 1: / Line 1: @@
-The '''3395 equal divisions of the octave''' divides the [[octave]] into 3395 [[equal]] steps of 0.35346 [[cent]]s each.
+{{Infobox ET}}
+{{ED intro}}
-edo is an extremely strong 17- and 19-limit system, and a [[The Riemann Zeta function and tuning #Zeta EDO lists|zeta peak, integral and gap edo]]. It has a lower 17-limit [[TE relative error]] than any edo until [[7033edo|7033]], and a lower 19-limit relative error than any edo until [[8269edo|8269]]. Besides, it provides the [[optimal patent val]] for the 13-limit rank-5 temperament tempering out [[6656/6655]], the jacobin comma. A basis for the 17-limit commas is {6656/6655, 12376/12375, 28561/28560, 31213/31212, 37180/37179, 937125/937024}, and for the 19-limit, {6656/6655, 12376/12375, 12636/12635, 13377/13376, 14365/14364, 23409/23408, 28561/28560}.
+== Theory ==
+edo is an extremely strong 17- and 19-limit system, and a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak, integral, and gap edo]]. It has a lower 17-limit [[TE relative error]] than any edo until [[7033edo|7033]], and a lower 19-limit relative error than any edo until [[8269edo|8269]]. Besides, it provides the [[optimal patent val]] for the 13-limit rank-5 temperament tempering out [[6656/6655]], the jacobin comma, and for [[quartismic]], which also tempers out [[123201/123200]]. A basis for the 17-limit commas is {6656/6655, 12376/12375, 28561/28560, 31213/31212, 37180/37179, 937125/937024}, and for the 19-limit, {6656/6655, 12376/12375, 12636/12635, 13377/13376, 14365/14364, 23409/23408, 28561/28560}.
-= 5 × 7 × 97, with subset edos 5, 7, 35, 485, and 679.
+=== Prime harmonics ===
+{{Harmonics in equal|3395|columns=11}}
+=== Subsets and supersets ===
+Since 3395 factors into {{factorization|3395}}, 3395edo has subset edos 5, 7, 35, 97, 485, and 679.
+== Regular temperament properties ==
+edo has a lower 17-limit [[TE relative error]] than any edo until [[7033edo|7033]], and a lower 19-limit relative error than any edo until [[8269edo|8269]].
+=== Rank-2 temperaments ===
+{| class="wikitable center-all left-5"
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
+|-
+! Periods<br />per 8ve
+! Generator*
+! Cents*
+! Associated<br />ratio*
+! Temperaments
+|-
+| 1
+| 2319\3395
+| 819.676
+| 55115776/34328125
+| [[Genojacobin]]
+|-
+| 35
+| 1409\3395<br />(51\3395)
+| 498.027<br />(18.026)
+| 4/3<br />(?)
+| [[Bromine]]
+|-
+| 97
+| 1409\3395<br />(9\3395)
+| 498.027<br />(3.181)
+| 4/3<br />(?)
+| [[Berkelium]]
+|}
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
-[[Category:Equal divisions of the octave]]
 [[Category:Jacobin]]
+[[Category:Quartismic]]