385edo: Difference between revisions

(5 intermediate revisions by 3 users not shown)

Line 1:

== Theory ==

385edo has a reasonable approximation to the 11-limit, and perhaps beyond. The equal temperament [[tempering out|tempers out]] [[19683/19600]], [[589824/588245]], and [[703125/702464]] in the 7-limit; [[540/539]], [[8019/8000]], 43923/43904, 151263/151250, 160083/160000, 166698/166375, and 172032/171875 in the 11-limit. It [[support]]s [[hemipental]] and provides the [[optimal patent val]] for the 7-limit version thereof. Using the [[patent val]], it tempers out [[1575/1573]], [[1716/1715]], [[2200/2197]], [[4096/4095]], [[6656/~~665~~]] and [[10648/10647]] in the 13-limit; and [[936/935]], [[1275/1274]], 1377/1375, and [[2601/2600]] in the 17-limit.

385edo has a reasonable approximation to the 11-limit, and perhaps beyond. The equal temperament [[tempering out|tempers out]] [[19683/19600]], [[589824/588245]], and [[703125/702464]] in the 7-limit; [[540/539]], [[8019/8000]], 43923/43904, 151263/151250, 160083/160000, 166698/166375, and 172032/171875 in the 11-limit. It [[support]]s [[hemipental]] and provides the [[optimal patent val]] for the 7-limit version thereof. Using the [[patent val]], it tempers out [[1575/1573]], [[1716/1715]], [[2200/2197]], [[4096/4095]], [[6656/6655]], and [[10648/10647]] in the 13-limit; and [[936/935]], [[1275/1274]], 1377/1375, and [[2601/2600]] in the 17-limit.

=== Prime harmonics ===

Line 13:

== Regular temperament properties ==

{| class="wikitable center-4 center-5 center-6"

|-

! rowspan="2" | [[Subgroup]]

! rowspan="2" | [[Comma list~~|Comma List~~]]

! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal 8ve ~~Stretch~~ (¢)

! rowspan="2" | Optimal 8ve stretch (¢)

! colspan="2" | Tuning ~~Error~~

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

! [[TE simple badness|Relative]] (%)

Line 53:

Line 54:

| 540/539, 1575/1573, 2200/2197, 4096/4095, 8019/8000

| {{mapping| 385 610 894 1081 1332 1425 }}

| -0.0394

| −0.0394

| 0.2207

| 7.08

Line 60:

Line 61:

| 540/539, 936/935, 1377/1375, 1575/1573, 2200/2197, 4096/4095

| {{mapping| 385 610 894 1081 1332 1425 1574 }}

| -0.0693

| −0.0693

| 0.2171

| 6.97

Line 67:

Line 68:

=== Rank-2 temperaments ===

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

|+Table of rank-2 temperaments by generator

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

! Periods per 8ve

|-

! Periods per 8ve

! Generator*

! Cents*

! Associated ~~Ratio~~*

! Associated ratio*

! Temperaments

|-

| 1

| 62\385

| 193.~~247~~

| 193.25

| ~~4096~~/~~3645~~

| 262144/234375

| [[Luna]]

|-

| 1

| 162/385

| 504.~~935~~

| 504.94

| 4/3

| [[Countermeantone]]

|-

| 5

| 160\385 (6\385)

| 80\385 (3\385)

| 498.~~701~~ (18.~~701~~)

| 249.35 (9.35)

| 4/3 (81/80)

| 81/70 (176/175)

| [[Pental]] (5-limit)

| [[Hemipental]]

|-

| 5

| 160\385 (6\385)

| 498.70 (18.70)

| 4/3 (81/80)

| [[Pental (temperament)|Pental]] (5-limit)

|}

<nowiki>*~~</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

[[Category:Hemipental]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|385}}
+{{ED intro}}
 == Theory ==
-edo has a reasonable approximation to the 11-limit, and perhaps beyond. The equal temperament [[tempering out|tempers out]] [[19683/19600]], [[589824/588245]], and [[703125/702464]] in the 7-limit; [[540/539]], [[8019/8000]], 43923/43904, 151263/151250, 160083/160000, 166698/166375, and 172032/171875 in the 11-limit. It [[support]]s [[hemipental]] and provides the [[optimal patent val]] for the 7-limit version thereof. Using the [[patent val]], it tempers out [[1575/1573]], [[1716/1715]], [[2200/2197]], [[4096/4095]], [[6656/665]] and [[10648/10647]] in the 13-limit; and [[936/935]], [[1275/1274]], 1377/1375, and [[2601/2600]] in the 17-limit.
+edo has a reasonable approximation to the 11-limit, and perhaps beyond. The equal temperament [[tempering out|tempers out]] [[19683/19600]], [[589824/588245]], and [[703125/702464]] in the 7-limit; [[540/539]], [[8019/8000]], 43923/43904, 151263/151250, 160083/160000, 166698/166375, and 172032/171875 in the 11-limit. It [[support]]s [[hemipental]] and provides the [[optimal patent val]] for the 7-limit version thereof. Using the [[patent val]], it tempers out [[1575/1573]], [[1716/1715]], [[2200/2197]], [[4096/4095]], [[6656/6655]], and [[10648/10647]] in the 13-limit; and [[936/935]], [[1275/1274]], 1377/1375, and [[2601/2600]] in the 17-limit.
 === Prime harmonics ===
@@ Line 13: / Line 13: @@
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
+|-
 ! rowspan="2" | [[Subgroup]]
-! rowspan="2" | [[Comma list|Comma List]]
+! rowspan="2" | [[Comma list]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal<br>8ve Stretch (¢)
+! rowspan="2" | Optimal<br />8ve stretch (¢)
-! colspan="2" | Tuning Error
+! colspan="2" | Tuning error
 |-
 ! [[TE error|Absolute]] (¢)
 ! [[TE simple badness|Relative]] (%)
@@ Line 53: / Line 54: @@
 | 540/539, 1575/1573, 2200/2197, 4096/4095, 8019/8000
 | {{mapping| 385 610 894 1081 1332 1425 }}
-| -0.0394
+| −0.0394
 | 0.2207
 | 7.08
@@ Line 60: / Line 61: @@
 | 540/539, 936/935, 1377/1375, 1575/1573, 2200/2197, 4096/4095
 | {{mapping| 385 610 894 1081 1332 1425 1574 }}
-| -0.0693
+| −0.0693
 | 0.2171
 | 6.97
@@ Line 67: / Line 68: @@
 === Rank-2 temperaments ===
 {| class="wikitable center-all left-5"
-|+Table of rank-2 temperaments by generator
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
-! Periods<br>per 8ve
+|-
+! Periods<br />per 8ve
 ! Generator*
 ! Cents*
-! Associated<br>Ratio*
+! Associated<br />ratio*
 ! Temperaments
 |-
 | 1
 | 62\385
-| 193.247
+| 193.25
-| 4096/3645
+| 262144/234375
 | [[Luna]]
 |-
 | 1
 | 162/385
-| 504.935
+| 504.94
 | 4/3
 | [[Countermeantone]]
 |-
 | 5
-| 160\385<br>(6\385)
+| 80\385<br />(3\385)
-| 498.701<br>(18.701)
+| 249.35<br />(9.35)
-| 4/3<br>(81/80)
+| 81/70<br />(176/175)
-| [[Pental]] (5-limit)
+| [[Hemipental]]
+|-
+| 5
+| 160\385<br />(6\385)
+| 498.70<br />(18.70)
+| 4/3<br />(81/80)
+| [[Pental (temperament)|Pental]] (5-limit)
 |}
-<nowiki>*</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
 [[Category:Hemipental]]