53edo: Difference between revisions

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53edo is the 16th [[prime edo]], following [[47edo]] and coming before [[59edo]].

Many of its multiples such as [[159edo]], [[212edo]], [[742edo]], [[901edo]] and [[954edo]] have good consistency limits and are each of their own interest. The [[mercator family]] comprises rank-2 temperaments with 1/53-octave periods.

Many of its multiples such as [[159edo]], [[212edo]], [[742edo]], [[901edo]] and the zeta [[954edo]] have good consistency limits and are each of their own interest. The [[mercator family]] comprises rank-2 temperaments with 1/53-octave periods.

== Intervals ==

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=== Sagittal notation ===

53edo can be notated in [[Sagittal notation|Sagittal]] using the [[Sagittal notation#Spartan extension single-shaft|Spartan extension]], with the apotome equal to 5 edosteps and the limma to 4 edosteps. Here is a simplified table:

{| class="wikitable" style="text-align: center;"

!Degree

|'''0'''

| +1

| +2

| +3

| +4

|'''+5'''

|-

!Evo

| rowspan="2" |<big>{{sagittal||//|}}</big>

| rowspan="2" |<big>{{sagittal|/|}}</big>

| rowspan="2" |<big>{{sagittal|//|}}</big>

|<small>{{sagittal|#}}{{sagittal|\\!}}</small>

|<small>{{sagittal|#}}{{sagittal|\!}}</small>

|<big>{{sagittal|#}}</big>

|-

!Revo

|<big>{{sagittal|)||(}}</big>

|<big>{{sagittal|||\}}</big>

|<big>{{sagittal|/||\}}</big>

|}

The following enharmonics from the Spartan set are present (comma tempered out):

* {{sagittal|//|}} = {{Sagittal|/|)}} = {{Sagittal|/|\}} ([[325/324]], [[352/351]])

* {{sagittal|/|}} = {{sagittal||)}} ([[225/224]])

* {{sagittal||(}} = {{sagittal||//|}} ([[5120/5103]])

See [[Sagittal notation#Revo|apotome complements]] for equivalent accidental pairs.

Featured below is the 53edo gamut notated using the best accidental approximants; in this case, pai/pao and phai/phao.

==== Evo flavor ====

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|}

Because the 5th is so accurate, 53edo also offers ~~good approximations for~~ Pythagorean ~~thirds~~. In addition, the 43\53 interval is only 4.8 cents wider than 7/4, so 53edo can also be used for 7-limit harmony, in which it tempers out the [[septimal kleisma]], 225/224.

Because the 5th is so incredibly accurate (next edo with a more accurate fifth is [[200edo]]), 53edo also offers a great approximation to Pythagorean tuning. In addition, the 43\53 interval is only 4.8 cents wider than 7/4, so 53edo can also be used for 7-limit harmony, in which it tempers out the [[septimal kleisma]], 225/224.

=== 15-odd-limit interval mappings ===

@@ Line 28: / Line 28: @@
 edo is the 16th [[prime edo]], following [[47edo]] and coming before [[59edo]].
-Many of its multiples such as [[159edo]], [[212edo]], [[742edo]], [[901edo]] and [[954edo]] have good consistency limits and are each of their own interest. The [[mercator family]] comprises rank-2 temperaments with 1/53-octave periods.
+Many of its multiples such as [[159edo]], [[212edo]], [[742edo]], [[901edo]] and the zeta [[954edo]] have good consistency limits and are each of their own interest. The [[mercator family]] comprises rank-2 temperaments with 1/53-octave periods.
 == Intervals ==
@@ Line 641: / Line 641: @@
 === Sagittal notation ===
+edo can be notated in [[Sagittal notation|Sagittal]] using the [[Sagittal notation#Spartan extension single-shaft|Spartan extension]], with the apotome equal to 5 edosteps and the limma to 4 edosteps. Here is a simplified table:
+{| class="wikitable" style="text-align: center;"
+!Degree
+|'''0'''
+| +1
+| +2
+| +3
+| +4
+|'''+5'''
+|-
+!Evo
+| rowspan="2" |<big>{{sagittal||//|}}</big>
+| rowspan="2" |<big>{{sagittal|/|}}</big>
+| rowspan="2" |<big>{{sagittal|//|}}</big>
+|<small>{{sagittal|#}}{{sagittal|\\!}}</small>
+|<small>{{sagittal|#}}{{sagittal|\!}}</small>
+|<big>{{sagittal|#}}</big>
+|-
+!Revo
+|<big>{{sagittal|)||(}}</big>
+|<big>{{sagittal|||\}}</big>
+|<big>{{sagittal|/||\}}</big>
+|}
+The following enharmonics from the Spartan set are present (comma tempered out):
+* {{sagittal|//|}} = {{Sagittal|/|)}} = {{Sagittal|/|\}} ([[325/324]], [[352/351]])
+* {{sagittal|/|}} = {{sagittal||)}} ([[225/224]])
+* {{sagittal||(}} = {{sagittal||//|}} ([[5120/5103]])
+See [[Sagittal notation#Revo|apotome complements]] for equivalent accidental pairs.
+Featured below is the 53edo gamut notated using the best accidental approximants; in this case, pai/pao and phai/phao.
 ==== Evo flavor ====
 {{Sagittal chart|Evo}}
@@ Line 698: / Line 731: @@
 |}
-Because the 5th is so accurate, 53edo also offers good approximations for Pythagorean thirds. In addition, the 43\53 interval is only 4.8 cents wider than 7/4, so 53edo can also be used for 7-limit harmony, in which it tempers out the [[septimal kleisma]], 225/224.
+Because the 5th is so incredibly accurate (next edo with a more accurate fifth is [[200edo]]), 53edo also offers a great approximation to Pythagorean tuning. In addition, the 43\53 interval is only 4.8 cents wider than 7/4, so 53edo can also be used for 7-limit harmony, in which it tempers out the [[septimal kleisma]], 225/224.
 === 15-odd-limit interval mappings ===