94edo: Difference between revisions

Line 8:

=== As a tuning of other temperaments ===

94edo can also be thought of as the "sum" of [[41edo]] and [[53edo]] {{nowrap|(41 + 53 {{=}} 94)}}, both of which are not only known for their approximation of [[Pythagorean tuning]], but also support a variety of [[Schismatic family|schismatic ~~temperament~~]] ~~known as~~ [[Schismatic family#Cassandra|cassandra]] (which is itself a variety of [[Schismatic family#Garibaldi|garibaldi]]), tempering out [[32805/32768]], [[225/224]], and [[385/384]]. Therefore, 94edo's fifth is the [[mediant]] of these two edos' fifths; it is slightly sharp of just and less accurate than 53edo's fifth, but more accurate than 41edo's, and acts as a generator for a highly optimized and high-prime-limit form of cassandra. Few, if any, edos that support schismatic by [[Val|patent val]] have at least as high of a consistency limit as 94edo while also having a fifth that can stack to reach any interval in it.

94edo can also be thought of as the "sum" of [[41edo]] and [[53edo]] {{nowrap|(41 + 53 {{=}} 94)}}, both of which are not only known for their approximation of [[Pythagorean tuning]], but also support a variety of [[Schismatic family|schismatic temperaments]], like [[Schismatic family#Cassandra|cassandra]] (which is itself a variety of [[Schismatic family#Garibaldi|garibaldi]]), tempering out [[32805/32768]], [[225/224]], and [[385/384]], and tempering together the [[81/80|syntonic]], [[Septimal comma|septimal]], and [[pythagorean comma]] into the same interval. Therefore, 94edo's fifth is the [[mediant]] of these two edos' fifths; it is ever so slightly sharp of just and less accurate than 53edo's fifth, but more accurate than 41edo's, and acts as a generator for a highly optimized and high-prime-limit form of cassandra. Few, if any, edos that support schismatic by [[Val|patent val]] have at least as high of a consistency limit as 94edo while also having a fifth that can stack to reach any interval in it. Other non-garibaldi schismatic notable edos in the patent val are [[118edo|118]], [[159edo|159]], [[171edo|171]], [[224edo|224]], and [[460edo|460]].

The list of 23-limit commas it tempers out is huge, and in lower prime limits, it also tempers out [[3125/3087]], [[4000/3969]], [[5120/5103]] and [[540/539]]. It provides the [[optimal patent val]] for gassormic, the rank-5 temperament tempering out [[275/273]] (despite one edostep being very close in size to this comma), and for a number of other temperaments, such as [[isis]].

Line 56:

| 2

| 25.532

| 81/80, 64/63, (50/49)

| 531441/524288, 81/80, 64/63, (50/49)

|

|{{UDnote|step=2}}

Line 119:

| 9

| 114.894

| 16/15, (15/14)

| 2187/2048, 16/15, (15/14)

|

|{{UDnote|step=9}}

Line 308:

| 30

| 382.979

| 5/4

| 8192/6561, 5/4

|

|{{UDnote|step=30}}

Line 903:

== Notation ==

94edo can be notated in [[Sagittal notation|Sagittal]] using the [[Sagittal_notation#Athenian_extension_single-shaft|Athenian extension]], with the apotome ~~equating~~ to 9 edosteps and the limma to 7 edosteps.

94edo can be notated in [[Sagittal notation|Sagittal]] using the [[Sagittal_notation#Athenian_extension_single-shaft|Athenian extension]], with the apotome equal to 9 edosteps and the limma to 7 edosteps.

Thus, the JI chord 16:17:18:19:20:21:22:23:24:25:26:27:28:30 from D would be written D{{sagittal||//|}}:E{{sagittal|(!!(}}:E{{sagittal||//|}}:F{{sagittal||//|}}:F{{sagittal|||\}}:G{{sagittal|\!}}:G{{sagittal|/|\}}:G{{sagittal|~|||(}}:A{{sagittal||//|}}:A{{sagittal|(|)}}:B{{sagittal|(!)}}:B{{sagittal||//|}}:C{{sagittal|\!}}:C{{sagittal|||\}}. Music that doesn't modulate much in the 2.3.5.7.11.13.19 subgroup can be notated by only using {{sagittal|/|}} {{sagittal|/|\}} / {{sagittal|\!}} {{sagittal|\!/}} and their apotome complements; where naturals are used for 3 and 19, {{sagittal|/|}} / {{sagittal|\!}} for 5 and 7, and {{sagittal|/|\}} / {{sagittal|\!/}} for 11 and 13.

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

!Degree

Line 1,055:

Line 1,057:

| 11\94

| 140.43

| ~~243~~/~~224~~

| 13/12

| [[Tsaharuk]] / [[quanic]]

| [[Tsaharuk]] / [[quanic]] / [[glacier]]

|-

| 1

@@ Line 8: / Line 8: @@
 === As a tuning of other temperaments ===
-edo can also be thought of as the "sum" of [[41edo]] and [[53edo]] {{nowrap|(41 + 53 {{=}} 94)}}, both of which are not only known for their approximation of [[Pythagorean tuning]], but also support a variety of [[Schismatic family|schismatic temperament]] known as [[Schismatic family#Cassandra|cassandra]] (which is itself a variety of [[Schismatic family#Garibaldi|garibaldi]]), tempering out [[32805/32768]], [[225/224]], and [[385/384]]. Therefore, 94edo's fifth is the [[mediant]] of these two edos' fifths; it is slightly sharp of just and less accurate than 53edo's fifth, but more accurate than 41edo's, and acts as a generator for a highly optimized and high-prime-limit form of cassandra. Few, if any, edos that support schismatic by [[Val|patent val]] have at least as high of a consistency limit as 94edo while also having a fifth that can stack to reach any interval in it.
+edo can also be thought of as the "sum" of [[41edo]] and [[53edo]] {{nowrap|(41 + 53 {{=}} 94)}}, both of which are not only known for their approximation of [[Pythagorean tuning]], but also support a variety of [[Schismatic family|schismatic temperaments]], like [[Schismatic family#Cassandra|cassandra]] (which is itself a variety of [[Schismatic family#Garibaldi|garibaldi]]), tempering out [[32805/32768]], [[225/224]], and [[385/384]], and tempering together the [[81/80|syntonic]], [[Septimal comma|septimal]], and [[pythagorean comma]] into the same interval. Therefore, 94edo's fifth is the [[mediant]] of these two edos' fifths; it is ever so slightly sharp of just and less accurate than 53edo's fifth, but more accurate than 41edo's, and acts as a generator for a highly optimized and high-prime-limit form of cassandra. Few, if any, edos that support schismatic by [[Val|patent val]] have at least as high of a consistency limit as 94edo while also having a fifth that can stack to reach any interval in it. Other non-garibaldi schismatic notable edos in the patent val are [[118edo|118]], [[159edo|159]], [[171edo|171]], [[224edo|224]], and [[460edo|460]].
 The list of 23-limit commas it tempers out is huge, and in lower prime limits, it also tempers out [[3125/3087]], [[4000/3969]], [[5120/5103]] and [[540/539]]. It provides the [[optimal patent val]] for gassormic, the rank-5 temperament tempering out [[275/273]] (despite one edostep being very close in size to this comma), and for a number of other temperaments, such as [[isis]].
@@ Line 56: / Line 56: @@
 | 2
 | 25.532
-| 81/80, 64/63, (50/49)
+| 531441/524288, 81/80, 64/63, (50/49)
 |
 |{{UDnote|step=2}}
@@ Line 119: / Line 119: @@
 | 9
 | 114.894
-| 16/15, (15/14)
+| 2187/2048, 16/15, (15/14)
 |
 |{{UDnote|step=9}}
@@ Line 308: / Line 308: @@
 | 30
 | 382.979
-| 5/4
+| 8192/6561, 5/4
 |
 |{{UDnote|step=30}}
@@ Line 903: / Line 903: @@
 == Notation ==
-edo can be notated in [[Sagittal notation|Sagittal]] using the [[Sagittal_notation#Athenian_extension_single-shaft|Athenian extension]], with the apotome equating to 9 edosteps and the limma to 7 edosteps.
+edo can be notated in [[Sagittal notation|Sagittal]] using the [[Sagittal_notation#Athenian_extension_single-shaft|Athenian extension]], with the apotome equal to 9 edosteps and the limma to 7 edosteps.
+Thus, the JI chord 16:17:18:19:20:21:22:23:24:25:26:27:28:30 from D would be written D{{sagittal||//|}}:E{{sagittal|(!!(}}:E{{sagittal||//|}}:F{{sagittal||//|}}:F{{sagittal|||\}}:G{{sagittal|\!}}:G{{sagittal|/|\}}:G{{sagittal|~|||(}}:A{{sagittal||//|}}:A{{sagittal|(|)}}:B{{sagittal|(!)}}:B{{sagittal||//|}}:C{{sagittal|\!}}:C{{sagittal|||\}}. Music that doesn't modulate much in the 2.3.5.7.11.13.19 subgroup can be notated by only using {{sagittal|/|}} {{sagittal|/|\}} / {{sagittal|\!}} {{sagittal|\!/}} and their apotome complements; where naturals are used for 3 and 19, {{sagittal|/|}} / {{sagittal|\!}} for 5 and 7, and {{sagittal|/|\}} / {{sagittal|\!/}} for 11 and 13.
 {| class="wikitable" style="text-align: center;"
 !Degree
@@ Line 1,055: / Line 1,057: @@
 | 11\94
 | 140.43
-| 243/224
+| 13/12
-| [[Tsaharuk]] / [[quanic]]
+| [[Tsaharuk]] / [[quanic]] / [[glacier]]
 |-
 | 1