198edo: Difference between revisions

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The '''198 equal divisions of the octave''' ('''~~198edo~~'''), or the '''198(-tone) equal temperament''' ('''~~198tet~~''', '''~~198et~~''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 198 parts of 6.061 [[cent]]s each.

The '''198 equal divisions of the octave''' ('''198EDO'''), or the '''198(-tone) equal temperament''' ('''198TET''', '''198ET''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 198 parts of 6.061 [[cent]]s each.

== Theory ==

~~198edo~~ is contorted in the [[7-limit]], with the same tuning as [[99edo]], but makes for a good 11- and 13-limit system. Like 99, it tempers out [[2401/2400]], [[4375/4374]], [[3136/3125]], [[5120/5103]] and [[6144/6125]] in the 7-limit; in the [[11-limit]] it tempers [[3025/3024]], [[9801/9800]] and [[14641/14580]]; and in the [[13-limit]] [[352/351]], [[676/675]], [[847/845]], [[1001/1000]], [[1716/1715]] and [[2080/2079]].

198EDO is contorted in the [[7-limit]], with the same tuning as [[99edo|99EDO]], but makes for a good 11- and 13-limit system. Like 99, it tempers out [[2401/2400]], [[4375/4374]], [[3136/3125]], [[5120/5103]] and [[6144/6125]] in the 7-limit; in the [[11-limit]] it tempers [[3025/3024]], [[9801/9800]] and [[14641/14580]]; and in the [[13-limit]] [[352/351]], [[676/675]], [[847/845]], [[1001/1000]], [[1716/1715]] and [[2080/2079]].

It is the [[optimal patent val]] for the rank-5 temperament tempering out 352/351, plus other temperaments of lower rank also tempering it out, such as [[Misty family #Hemimist|hemimist]], and [[Hemifamity family #Namaka|namaka]]. It is distinctly [[consistent]] through the [[15-odd-limit]]. It factors into 2 × 32 × 11, and has divisors 2, 3, 6, 9, 11, 18, 22, 33, 66 and 99.

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=== Rank-2 temperaments ===

Note: temperaments supported by ~~99et~~ are not included.

Note: temperaments supported by 99EDO are not included.

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

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! Associated ratio

! Temperaments

|-

| 1

| 23\198

| 139.39

| 13/12

| [[Quasijerome]]

|-

| 1

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|-

| 2

| 47\198 (52\198)

| 52\198 (47\198)

| 315.15 (284.85)

| 6/5 (33/28)

@@ Line 1: / Line 1: @@
-The '''198 equal divisions of the octave''' ('''198edo'''), or the '''198(-tone) equal temperament''' ('''198tet''', '''198et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 198 parts of 6.061 [[cent]]s each.
+The '''198 equal divisions of the octave''' ('''198EDO'''), or the '''198(-tone) equal temperament''' ('''198TET''', '''198ET''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 198 parts of 6.061 [[cent]]s each.
 == Theory ==
-edo is contorted in the [[7-limit]], with the same tuning as [[99edo]], but makes for a good 11- and 13-limit system. Like 99, it tempers out [[2401/2400]], [[4375/4374]], [[3136/3125]], [[5120/5103]] and [[6144/6125]] in the 7-limit; in the [[11-limit]] it tempers [[3025/3024]], [[9801/9800]] and [[14641/14580]]; and in the [[13-limit]] [[352/351]], [[676/675]], [[847/845]], [[1001/1000]], [[1716/1715]] and [[2080/2079]].
+EDO is contorted in the [[7-limit]], with the same tuning as [[99edo|99EDO]], but makes for a good 11- and 13-limit system. Like 99, it tempers out [[2401/2400]], [[4375/4374]], [[3136/3125]], [[5120/5103]] and [[6144/6125]] in the 7-limit; in the [[11-limit]] it tempers [[3025/3024]], [[9801/9800]] and [[14641/14580]]; and in the [[13-limit]] [[352/351]], [[676/675]], [[847/845]], [[1001/1000]], [[1716/1715]] and [[2080/2079]].
 It is the [[optimal patent val]] for the rank-5 temperament tempering out 352/351, plus other temperaments of lower rank also tempering it out, such as [[Misty family #Hemimist|hemimist]], and [[Hemifamity family #Namaka|namaka]]. It is distinctly [[consistent]] through the [[15-odd-limit]]. It factors into 2 × 3<sup>2</sup> × 11, and has divisors 2, 3, 6, 9, 11, 18, 22, 33, 66 and 99.
@@ Line 39: / Line 39: @@
 === Rank-2 temperaments ===
-Note: temperaments supported by 99et are not included.
+Note: temperaments supported by 99EDO are not included.
 {| class="wikitable center-all left-5"
@@ Line 48: / Line 48: @@
 ! Associated<br>ratio
 ! Temperaments
+|-
+| 1
+| 23\198
+| 139.39
+| 13/12
+| [[Quasijerome]]
 |-
 | 1
@@ Line 74: / Line 80: @@
 |-
 | 2
-| 47\198<br>(52\198)
+| 52\198<br>(47\198)
 | 315.15<br>(284.85)
 | 6/5<br>(33/28)