1665edo: Difference between revisions

(10 intermediate revisions by 4 users not shown)

Line 1:

~~==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 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.

== Theory ==

===Prime harmonics===

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.

==Regular temperament properties==

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.

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

|-

! Generator~~ (Reduced)~~

! Periods per 8ve

! Cents~~ (Reduced)~~

! Generator*

! Associated ~~Ratio~~

! Cents*

! Associated ratio*

! Temperaments

|-

Line 20:

Line 29:

| 127\1665

| 91.531

| {{monzo|9 -32 18}}

| {{monzo| 9 -32 18 }}

| [[Gross]]

|-

| 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 />* [[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 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.
+== Theory ==
-===Prime harmonics===
+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.
-{{harmonics in equal|1665}}
-==Regular temperament properties==
+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.
+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
+|-
-! Generator<br>(Reduced)
+! Periods<br />per 8ve
-! Cents<br>(Reduced)
+! Generator*
-! Associated<br>Ratio
+! Cents*
+! Associated<br />ratio*
 ! Temperaments
 |-
@@ Line 20: / Line 29: @@
 | 127\1665
 | 91.531
-| {{monzo|9 -32 18}}
+| {{monzo| 9 -32 18 }}
 | [[Gross]]
 |-
 | 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 />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct