1665edo: Difference between revisions

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== Theory ==

1665edo is a very strong 5-limit (as well as 2.3.5.11 subgroup) tuning and it is consistent in the 15-odd-limit. In the 5-limit, 1665edo is a tuning for the [[gross]] temperament.

1665edo is a very strong 5-limit (as well as 2.3.5.11 [[subgroup]]) tuning and it is [[consistent]] in the [[15-odd-limit]]. In the 5-limit, 1665edo is a tuning for the [[gross]] temperament.

1665edo provides the optimal patent val for the [[rhodium]] temperament in the 11-limit and also in the 13-limit. In addition, it provides the optimal patent val for [[dzelic]] temperament in the 13-limit.

1665edo provides the [[optimal patent val]] for the [[rhodium]] temperament in the 11-limit and also in the 13-limit. In addition, it provides the optimal patent val for [[dzelic]] temperament in the 13-limit.

1665cc val is a tuning for the [[roentgenium]] temperament, and the patent val tunes the unnamed 111 & 1665 temperament in the 13-limit which has a comma basis {6656/6655, 123201/123200, 250047/250000, 91182091/91125000}.

The 1665cc val is a tuning for the [[roentgenium]] temperament, and the patent val tunes the unnamed 111 & 1665 temperament in the 13-limit which has a comma basis {6656/6655, 123201/123200, 250047/250000, 91182091/91125000}.

=== Prime harmonics ===

=== Subsets and supersets ===

Since 1665 factors into {{factorization|1665}}, 1665edo has subset edos {{EDOs| 3, 5, 9, 15, 37, 45, 111, 185, 333, and 555 }}.

== Regular temperament properties ==

=== 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

|-

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|-

| 37

| 377\1665 (17\1665)

| 377\1665 (17\1665)

| 271.711 (12.252)

| 271.711 (12.252)

| 117/100 (?)

| 117/100 (?)

| [[Dzelic]]

|-

| 45

| 1301\1665 (6\1665)

| 1301\1665 (6\1665)

| 937.657 (4.324)

| 937.657 (4.324)

| 55/32 (?)

| 55/32 (?)

| [[Rhodium]]

|}

<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

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|1665}}
+{{ED intro}}
 == Theory ==
-edo is a very strong 5-limit (as well as 2.3.5.11 subgroup) tuning and it is consistent in the 15-odd-limit. In the 5-limit, 1665edo is a tuning for the [[gross]] temperament.
+edo is a very strong 5-limit (as well as 2.3.5.11 [[subgroup]]) tuning and it is [[consistent]] in the [[15-odd-limit]]. In the 5-limit, 1665edo is a tuning for the [[gross]] temperament.
-edo provides the optimal patent val for the [[rhodium]] temperament in the 11-limit and also in the 13-limit. In addition, it provides the optimal patent val for [[dzelic]] temperament in the 13-limit.
+edo provides the [[optimal patent val]] for the [[rhodium]] temperament in the 11-limit and also in the 13-limit. In addition, it provides the optimal patent val for [[dzelic]] temperament in the 13-limit.
-cc val is a tuning for the [[roentgenium]] temperament, and the patent val tunes the unnamed 111 & 1665 temperament in the 13-limit which has a comma basis {6656/6655, 123201/123200, 250047/250000, 91182091/91125000}.
+The 1665cc val is a tuning for the [[roentgenium]] temperament, and the patent val tunes the unnamed 111 & 1665 temperament in the 13-limit which has a comma basis {6656/6655, 123201/123200, 250047/250000, 91182091/91125000}.
 === Prime harmonics ===
 {{Harmonics in equal|1665}}
+=== Subsets and supersets ===
+Since 1665 factors into {{factorization|1665}}, 1665edo has subset edos {{EDOs| 3, 5, 9, 15, 37, 45, 111, 185, 333, and 555 }}.
 == Regular temperament properties ==
 === 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
 |-
@@ Line 29: / Line 33: @@
 |-
 | 37
-| 377\1665<br>(17\1665)
+| 377\1665<br />(17\1665)
-| 271.711<br>(12.252)
+| 271.711<br />(12.252)
-| 117/100<br>(?)
+| 117/100<br />(?)
 | [[Dzelic]]
 |-
 | 45
-| 1301\1665<br>(6\1665)
+| 1301\1665<br />(6\1665)
-| 937.657<br>(4.324)
+| 937.657<br />(4.324)
-| 55/32<br>(?)
+| 55/32<br />(?)
 | [[Rhodium]]
 |}
-<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