1700edo: Difference between revisions

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== Theory ==

1700edo is only [[consistent]] in the [[5-odd-limit]], and there is a large relative delta on the ~~3rd~~ harmonic. ~~From~~ a ~~regular temperament theory perspective~~, ~~its best usage~~ is ~~as a~~ 2.9.11.21.23.31 [[subgroup]] ~~tuning because all other harmonics up to 29th have more than 25% error~~. Nonetheless, it tunes the 323 & 2023 temperament [[leaves]] in the 17-limit on the patent val.

1700edo is only [[consistent]] in the [[5-odd-limit]], and there is a large relative delta on the [[harmonic]] [[3/1|3]]. It has a reasonable approximation to the 2.9.15.21.11.13.17.23 [[subgroup]], or if the harmonic [[5/1|5]] is desired, the 2.9.5.21.11.23 subgroup. Otherwise, it can be considered in the 2.9.21.11.23.31 [[subgroup]] (not including either 5 or 15). Nonetheless, it tunes the {{nowrap|323 & 2023}} temperament [[leaves]] in the 17-limit on the [[patent val]].

~~One step of 1700edo is the [[relative cent]] for [[17edo]]. It has been named '''iota''' by [[Margo Schulter]] and [[George Secor~~]].

=== Odd harmonics ===

=== Subsets and supersets ===

Since 1700 factors into {{factorization|1700}}, 1700edo has subset edos {{EDOs| 2, 4, 5, 10, 17, 20, 25, 34, 50, 68, 85, 100, 170, 340, 425, and 850 }}.

One step of 1700edo is the [[relative cent]] for [[17edo]]. It has been named '''iota''' by [[Margo Schulter]] and [[George Secor]].

== Regular temperament properties ==

=== Rank-2 temperaments===

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

! Periods per 8ve

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

|-

! Periods per 8ve

! Generator*

! Cents*

! Associated ~~Ratio~~

! Associated ratio*

! Temperament

|-

| 17

| 121\1700 (21\1700)

| 121\1700 (21\1700)

| 85.412 (14.824)

| 85.412 (14.824)

| 1024/975 (8192/8125)

| 1024/975 (8192/8125)

| [[Leaves]]

|}

<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 forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|1700}}
+{{ED intro}}
 == Theory ==
-edo is only [[consistent]] in the [[5-odd-limit]], and there is a large relative delta on the 3rd harmonic. From a regular temperament theory perspective, its best usage is as a 2.9.11.21.23.31 [[subgroup]] tuning because all other harmonics up to 29th have more than 25% error. Nonetheless, it tunes the 323 & 2023 temperament [[leaves]] in the 17-limit on the patent val.
+edo is only [[consistent]] in the [[5-odd-limit]], and there is a large relative delta on the [[harmonic]] [[3/1|3]]. It has a reasonable approximation to the 2.9.15.21.11.13.17.23 [[subgroup]], or if the harmonic [[5/1|5]] is desired, the 2.9.5.21.11.23 subgroup. Otherwise, it can be considered in the 2.9.21.11.23.31 [[subgroup]] (not including either 5 or 15). Nonetheless, it tunes the {{nowrap|323 &amp; 2023}} temperament [[leaves]] in the 17-limit on the [[patent val]].
-One step of 1700edo is the [[relative cent]] for [[17edo]]. It has been named '''iota''' by [[Margo Schulter]] and [[George Secor]].
 === Odd harmonics ===
 {{Harmonics in equal|1700}}
+=== Subsets and supersets ===
+Since 1700 factors into {{factorization|1700}}, 1700edo has subset edos {{EDOs| 2, 4, 5, 10, 17, 20, 25, 34, 50, 68, 85, 100, 170, 340, 425, and 850 }}.
+One step of 1700edo is the [[relative cent]] for [[17edo]]. It has been named '''iota''' by [[Margo Schulter]] and [[George Secor]].
 == Regular temperament properties ==
 === Rank-2 temperaments===
 {| class="wikitable center-all left-5"
-! Periods<br>per 8ve
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
+|-
+! Periods<br />per 8ve
 ! Generator*
 ! Cents*
-! Associated<br>Ratio
+! Associated<br />ratio*
 ! Temperament
 |-
 | 17
-| 121\1700<br>(21\1700)
+| 121\1700<br />(21\1700)
-| 85.412<br>(14.824)
+| 85.412<br />(14.824)
-| 1024/975<br>(8192/8125)
+| 1024/975<br />(8192/8125)
 | [[Leaves]]
 |}
-<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 forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct