684edo: Difference between revisions

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== Theory ==

684edo divides the steps of [[171edo]] into four. It is [[consistent]] to the 17-odd-limit, tempering out [[2401/2400]], [[3025/3024]], [[4225/4224]], [[4375/4374]], and [[32805/32768]] in the 13-limit; [[1089/1088]], [[1225/1224]], [[1701/1700]], [[2025/2023]], [[2058/2057]], [[2500/2499]], 8624/8619, and 14875/14872 in the 17-limit.

684edo divides the steps of [[171edo]] into four. It is [[consistent]] to the [[17-odd-limit]], [[tempering out]] [[2401/2400]], [[3025/3024]], [[4225/4224]], [[4375/4374]], and [[32805/32768]] in the 13-limit; [[1089/1088]], [[1225/1224]], [[1701/1700]], [[2025/2023]], [[2058/2057]], [[2500/2499]], 8624/8619, and 14875/14872 in the 17-limit.

=== Prime harmonics ===

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| 2.3.5.7.11.13

| 2401/2400, 3025/3024, 4225/4224, 4375/4374, 32805/32768

| [{{~~val~~| 684 1084 1588 1920 2366 2531 }}]

| {{mapping| 684 1084 1588 1920 2366 2531 }}

| +0.0994

| 0.0558

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| 2.3.5.7.11.13.17

| 1089/1088, 1225/1224, 1701/1700, 2025/2023, 4225/4224, 13013/13005

| [{{~~val~~| 684 1084 1588 1920 2366 2531 2796 }}]

| {{mapping| 684 1084 1588 1920 2366 2531 2796 }}

| +0.0744

| 0.0800

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=== Rank-2 temperaments ===

Note: 11-limit temperaments supported by [[342edo|342et]] are not shown.

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

|+Table of rank-2 temperaments by generator

! Periods<br>per 8ve

! Generator~~<br>(Reduced)~~

! Generator*

! Cents~~<br>(Reduced)~~

! Cents*

! Associated<br>Ratio

! Temperaments

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| [[Semihemienneadecal]]

|}

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

@@ Line 3: / Line 3: @@
 == Theory ==
-edo divides the steps of [[171edo]] into four. It is [[consistent]] to the 17-odd-limit, tempering out [[2401/2400]], [[3025/3024]], [[4225/4224]], [[4375/4374]], and [[32805/32768]] in the 13-limit; [[1089/1088]], [[1225/1224]], [[1701/1700]], [[2025/2023]], [[2058/2057]], [[2500/2499]], 8624/8619, and 14875/14872 in the 17-limit.
+edo divides the steps of [[171edo]] into four. It is [[consistent]] to the [[17-odd-limit]], [[tempering out]] [[2401/2400]], [[3025/3024]], [[4225/4224]], [[4375/4374]], and [[32805/32768]] in the 13-limit; [[1089/1088]], [[1225/1224]], [[1701/1700]], [[2025/2023]], [[2058/2057]], [[2500/2499]], 8624/8619, and 14875/14872 in the 17-limit.
 === Prime harmonics ===
@@ Line 24: / Line 24: @@
 | 2.3.5.7.11.13
 | 2401/2400, 3025/3024, 4225/4224, 4375/4374, 32805/32768
-| [{{val| 684 1084 1588 1920 2366 2531 }}]
+| {{mapping| 684 1084 1588 1920 2366 2531 }}
 | +0.0994
 | 0.0558
@@ Line 31: / Line 31: @@
 | 2.3.5.7.11.13.17
 | 1089/1088, 1225/1224, 1701/1700, 2025/2023, 4225/4224, 13013/13005
-| [{{val| 684 1084 1588 1920 2366 2531 2796 }}]
+| {{mapping| 684 1084 1588 1920 2366 2531 2796 }}
 | +0.0744
 | 0.0800
@@ Line 39: / Line 39: @@
 === Rank-2 temperaments ===
+Note: 11-limit temperaments supported by [[342edo|342et]] are not shown.
 {| class="wikitable center-all left-5"
 |+Table of rank-2 temperaments by generator
 ! Periods<br>per 8ve
-! Generator<br>(Reduced)
+! Generator*
-! Cents<br>(Reduced)
+! Cents*
 ! Associated<br>Ratio
 ! Temperaments
@@ Line 59: / Line 61: @@
 | [[Semihemienneadecal]]
 |}
+<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct