730edo: Difference between revisions

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== Theory ==

730edo is a very strong 5-limit system, but is also distinctly consistent up to the [[15-odd-limit]]. It tempers out the [[counterschisma]], {{monzo| -69 45 -1 }}, the minortone comma, {{monzo| -16 35 -17 }}, the kwazy comma, {{monzo| -53 10 16 }}, the whoosh comma, {{monzo| 37 25 -33 }}, and the pirate comma, {{monzo| -90 -15 49 }}. In the 7-limit it tempers out [[4375/4374]] and {{monzo| -21 0 3 5 }}, so that it [[support]]s the [[mitonic]] temperament. In the 11-limit, [[3025/3024]] and {{monzo| 4 -3 -6 4 1 }}, so that it supports the [[deca]] temperament. In the 13-limit, [[1001/1000]] and [[4225/4224]], supporting 13-limit deca.

730edo is a very strong 5-limit system, but is also [[Consistency|distinctly consistent]] up to the [[15-odd-limit]]. The equal temperament [[Tempering out|tempers out]] the [[counterschisma]], {{monzo| -69 45 -1 }}, the minortone comma, {{monzo| -16 35 -17 }}, the kwazy comma, {{monzo| -53 10 16 }}, the whoosh comma, {{monzo| 37 25 -33 }}, and the pirate comma, {{monzo| -90 -15 49 }}. In the 7-limit it tempers out [[4375/4374]] and {{monzo| -21 0 3 5 }}, so that it [[support]]s the [[mitonic]] temperament. In the 11-limit, [[3025/3024]] and {{monzo| 4 -3 -6 4 1 }}, so that it supports the [[deca]] temperament. In the 13-limit, [[1001/1000]] and [[4225/4224]], supporting 13-limit deca.

[[W. S. B. Woolhouse]] proposed 730edo<ref name="summary">[https://www.webcitation.org/5zxZzQ3eS A summary of W. S. B. Woolhouse's Essay on musical intervals], 1999 by [[Joseph Monzo]]</ref> as a [[Interval size measure|logarithmic measure of interval size]], sometimes called the '''Woolhouse unit'''. While 730 is divisible by 2, 5, 10, 73, 146 and 365, it is not divisible by 12 and it is also deficient, with [[abundancy index]] of 0.82, which limits its application as an interval size measure.

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=== Prime harmonics ===

=== Subsets and supersets ===

Since 730 factors into 2 × 5 × 73, 730edo has subset edos {{EDOs| 2, 5, 10, 73, 146, and 365 }}. 1460edo, which doubles it, gives alternative approximations to harmonics 7, 11, and 13. [[2190edo]], which triples it, corrects these harmonics to near-just levels of accuracy. [[4380edo]] gives a possible full 31-limit system.

== Intervals ==

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| 2.3

| {{monzo| -1157 730 }}

| [{{~~val~~| 730 1157 }}]

| {{mapping| 730 1157 }}

| +0.0117

| 0.0117

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| 2.3.5

| {{monzo| -53 10 16 }}, {{monzo| -16 35 -17 }}

| [{{~~val~~| 730 1157 1695 }}]

| {{mapping| 730 1157 1695 }}

| +0.0096

| 0.0100

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| 2.3.5.7

| 4375/4374, 2100875/2097152, {{monzo| 12 -3 -14 9 }}

| [{{~~val~~| 730 1157 1695 2049 }}]

| {{mapping| 730 1157 1695 2049 }}

| +0.0612

| 0.0899

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| 2.3.5.7.11

| 3025/3024, 4375/4374, 391314/390625, 2100875/2097152

| [{{~~val~~| 730 1157 1695 2049 2525 }}]

| {{mapping| 730 1157 1695 2049 2525 }}

| +0.0856

| 0.0940

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| 2.3.5.7.11.13

| 1001/1000, 3025/3024, 4225/4224, 4375/4374, 2100875/2097152

| [{{~~val~~| 730 1157 1695 2049 2525 2701 }}]

| {{mapping| 730 1157 1695 2049 2525 2701 }}

| +0.0951

| 0.0884

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|+Table of rank-2 temperaments by generator

! Periods<br>per 8ve

! Generator~~<br>(Reduced)~~

! Generator*

! Cents~~<br>(Reduced)~~

! Cents*

! Associated<br>Ratio

! Temperaments

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| [[Deca]]

|}

<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

== Scales ==

@@ Line 3: / Line 3: @@
 == Theory ==
-edo is a very strong 5-limit system, but is also distinctly consistent up to the [[15-odd-limit]]. It tempers out the [[counterschisma]], {{monzo| -69 45 -1 }}, the minortone comma, {{monzo| -16 35 -17 }}, the kwazy comma, {{monzo| -53 10 16 }}, the whoosh comma, {{monzo| 37 25 -33 }}, and the pirate comma, {{monzo| -90 -15 49 }}. In the 7-limit it tempers out [[4375/4374]] and {{monzo| -21 0 3 5 }}, so that it [[support]]s the [[mitonic]] temperament. In the 11-limit, [[3025/3024]] and {{monzo| 4 -3 -6 4 1 }}, so that it supports the [[deca]] temperament. In the 13-limit, [[1001/1000]] and [[4225/4224]], supporting 13-limit deca.
+edo is a very strong 5-limit system, but is also [[Consistency|distinctly consistent]] up to the [[15-odd-limit]]. The equal temperament [[Tempering out|tempers out]] the [[counterschisma]], {{monzo| -69 45 -1 }}, the minortone comma, {{monzo| -16 35 -17 }}, the kwazy comma, {{monzo| -53 10 16 }}, the whoosh comma, {{monzo| 37 25 -33 }}, and the pirate comma, {{monzo| -90 -15 49 }}. In the 7-limit it tempers out [[4375/4374]] and {{monzo| -21 0 3 5 }}, so that it [[support]]s the [[mitonic]] temperament. In the 11-limit, [[3025/3024]] and {{monzo| 4 -3 -6 4 1 }}, so that it supports the [[deca]] temperament. In the 13-limit, [[1001/1000]] and [[4225/4224]], supporting 13-limit deca.
 [[W. S. B. Woolhouse]] proposed 730edo<ref name="summary">[https://www.webcitation.org/5zxZzQ3eS A summary of W. S. B. Woolhouse's Essay on musical intervals], 1999 by [[Joseph Monzo]]</ref> as a [[Interval size measure|logarithmic measure of interval size]], sometimes called the '''Woolhouse unit'''. While 730 is divisible by 2, 5, 10, 73, 146 and 365, it is not divisible by 12 and it is also deficient, with [[abundancy index]] of 0.82, which limits its application as an interval size measure.
@@ Line 9: / Line 9: @@
 === Prime harmonics ===
 {{Harmonics in equal|730|columns=11}}
+=== Subsets and supersets ===
+Since 730 factors into 2 × 5 × 73, 730edo has subset edos {{EDOs| 2, 5, 10, 73, 146, and 365 }}. 1460edo, which doubles it, gives alternative approximations to harmonics 7, 11, and 13. [[2190edo]], which triples it, corrects these harmonics to near-just levels of accuracy. [[4380edo]] gives a possible full 31-limit system.
 == Intervals ==
@@ Line 50: / Line 53: @@
 | 2.3
 | {{monzo| -1157 730 }}
-| [{{val| 730 1157 }}]
+| {{mapping| 730 1157 }}
 | +0.0117
 | 0.0117
@@ Line 57: / Line 60: @@
 | 2.3.5
 | {{monzo| -53 10 16 }}, {{monzo| -16 35 -17 }}
-| [{{val| 730 1157 1695 }}]
+| {{mapping| 730 1157 1695 }}
 | +0.0096
 | 0.0100
@@ Line 64: / Line 67: @@
 | 2.3.5.7
 | 4375/4374, 2100875/2097152, {{monzo| 12 -3 -14 9 }}
-| [{{val| 730 1157 1695 2049 }}]
+| {{mapping| 730 1157 1695 2049 }}
 | +0.0612
 | 0.0899
@@ Line 71: / Line 74: @@
 | 2.3.5.7.11
 | 3025/3024, 4375/4374, 391314/390625, 2100875/2097152
-| [{{val| 730 1157 1695 2049 2525 }}]
+| {{mapping| 730 1157 1695 2049 2525 }}
 | +0.0856
 | 0.0940
@@ Line 78: / Line 81: @@
 | 2.3.5.7.11.13
 | 1001/1000, 3025/3024, 4225/4224, 4375/4374, 2100875/2097152
-| [{{val| 730 1157 1695 2049 2525 2701 }}]
+| {{mapping| 730 1157 1695 2049 2525 2701 }}
 | +0.0951
 | 0.0884
@@ Line 88: / Line 91: @@
 |+Table of rank-2 temperaments by generator
 ! Periods<br>per 8ve
-! Generator<br>(Reduced)
+! Generator*
-! Cents<br>(Reduced)
+! Cents*
 ! Associated<br>Ratio
 ! Temperaments
@@ Line 135: / Line 138: @@
 | [[Deca]]
 |}
+<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
 == Scales ==