65edo: Difference between revisions

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65et can be characterized as the temperament which [[tempering out|tempers out]] 32805/32768 ([[schisma]]), 78732/78125 ([[sensipent comma]]), 393216/390625 ([[würschmidt comma]]), and {{monzo| -13 17 -6 }} ([[graviton]]). In the [[7-limit]], there are two different maps; the first is {{val| 65 103 151 '''182''' }} (65), tempering out [[126/125]], [[245/243]] and [[686/675]], so that it [[support]]s [[sensi]], and the second is {{val| 65 103 151 '''183''' }} (65d), tempering out [[225/224]], [[3125/3087]], [[4000/3969]] and [[5120/5103]], so that it supports [[garibaldi]]. In both cases, the tuning privileges the [[5-limit]] over the 7-limit, as the 5-limit of 65 is quite accurate. The same can be said for the two different versions of 7-limit [[würschmidt]] temperament (wurschmidt and worschmidt) these two mappings provide.

65edo approximates the intervals [[3/2]], [[5/4]], [[11/8]], [[19/16]], [[23/16]], [[31/16]] and [[47/32]] well, so that it does a good job representing the 2.3.5.11.19.23.31.47 [[just intonation subgroup]]. To this one may want to add [[17/16]], [[29/16]] and [[43/32]], giving the [[47-limit]] no-7's no-13's no-37's no-41's subgroup 2.3.5.11.17.19.23.29.31.43.47. In this sense it is a tuning of [[schismic]]/[[nestoria]] that focuses on the very primes that [[53edo]] neglects and that instead elegantly connects primes 7, 13, 37 and 41 to nestoria. Also of interest is the [[19-limit]] [[k*N subgroups|2*65 subgroup]] 2.3.5.49.11.91.119.19, on which 65 has the same tuning and commas as the [[zeta]] edo [[130edo]].

65edo approximates the intervals [[3/2]], [[5/4]], [[11/8]], [[19/16]], [[23/16]], [[31/16]] and [[47/32]] well, so that it does a good job representing the 2.3.5.11.19.23.31.47 [[just intonation subgroup]]. To this one may want to add [[17/16]], [[29/16]] and [[43/32]], giving the [[47-limit]] no-7's no-13's no-37's no-41's subgroup 2.3.5.11.17.19.23.29.31.43.47. In this sense it is a tuning of [[schismic]]/[[nestoria]] that focuses on the very primes that [[53edo]] neglects and that instead elegantly connects primes 7, 13, 37, and 41 to nestoria. Also of interest is the [[19-limit]] [[k*N subgroups|2*65 subgroup]] 2.3.5.49.11.91.119.19, on which 65 has the same tuning and commas as the [[zeta]] edo [[130edo]].

=== Prime harmonics ===

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! #

! [[Cent]]s

! Approximate ratios*

! Approximate ratios<ref group="note">{{sg|limit=2.3.5.11.13/7.17.19.23.29.31.47 subgroup}}</ref>

! colspan="2" | [[Ups and downs notation]]

|-

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| D

|}

~~<nowiki />* {{sg|limit=2.3.5.11.13/7.17.19.23.29.31.47 subgroup}}~~

== Notation ==

=== Ups and downs notation ===

Using [[Helmholtz–Ellis]] accidentals, 65edo can also be notated using [[ups and downs notation]]:

===Sagittal notation===

In some cases, certain notes may be best notated using semi- and sesquisharps and flats with arrows:

=== Ivan Wyschnegradsky's notation ===

Since a sharp raises by six steps, Wyschnegradsky accidentals borrowed from [[72edo]] can also be used:

=== Sagittal notation ===

This notation uses the same sagittal sequence as EDOs [[72edo#Sagittal notation|72]] and [[79edo#Sagittal notation|79]].

~~====Evo flavor====~~

==== Evo flavor ====

File:65-EDO_Evo_Sagittal.svg

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</imagemap>

====Revo flavor====

==== Revo flavor ====

File:65-EDO_Revo_Sagittal.svg

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</imagemap>

====Evo-SZ flavor====

==== Evo-SZ flavor ====

File:65-EDO_Evo-SZ_Sagittal.svg

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| 32805/32768, 78732/78125

| {{mapping| 65 103 151 }}

| ~~−0~~.110

| −0.110

| 0.358

| 1.94

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| 243/242, 4000/3993, 5632/5625

| {{mapping| 65 103 151 225 }}

| ~~−0~~.266

| −0.266

| 0.410

| 2.22

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== Zeta properties ==

===Zeta peak index===

=== Zeta peak index ===

{| class="wikitable"

~~! colspan="3" |Tuning~~

~~! colspan="3" |Strength~~

~~! colspan="2" |Closest EDO~~

~~! colspan="2" |Integer limit~~

|-

!ZPI

! colspan="3" | Tuning

!Steps per octave

! colspan="3" | Strength

!Step size (cents)

! colspan="2" | Closest EDO

!Height

! colspan="2" | Integer limit

!Integral

|-

!Gap

! ZPI

!EDO

! Steps per octave

!Octave (cents)

! Step size (cents)

!Consistent

! Height

!Distinct

! Integral

! Gap

! EDO

! Octave (cents)

! Consistent

! Distinct

|-

|[[334zpi]]

| [[334zpi]]

|65.0158450885860

| 65.0158450885860

|18.4570391781413

| 18.4570391781413

|7.813349

| 7.813349

|1.269821

| 1.269821

|16.514861

| 16.514861

|65edo

| 65edo

|1199.70754657919

| 1199.70754657919

|6

| 6

|6

| 6

|}

== Scales ==

* Amulet{{idiosyncratic}}, (approximated from [[25edo]], subset of [[würschmidt]]): 5 3 5 5 3 5 12 5 5 3 5 12 5

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== Instruments ==

[[Lumatone mapping for 65edo]]

== Notes ==

[[Category:Listen]]

@@ Line 5: / Line 5: @@
 et can be characterized as the temperament which [[tempering out|tempers out]] 32805/32768 ([[schisma]]), 78732/78125 ([[sensipent comma]]), 393216/390625 ([[würschmidt comma]]), and {{monzo| -13 17 -6 }} ([[graviton]]). In the [[7-limit]], there are two different maps; the first is {{val| 65 103 151 '''182''' }} (65), tempering out [[126/125]], [[245/243]] and [[686/675]], so that it [[support]]s [[sensi]], and the second is {{val| 65 103 151 '''183''' }} (65d), tempering out [[225/224]], [[3125/3087]], [[4000/3969]] and [[5120/5103]], so that it supports [[garibaldi]]. In both cases, the tuning privileges the [[5-limit]] over the 7-limit, as the 5-limit of 65 is quite accurate. The same can be said for the two different versions of 7-limit [[würschmidt]] temperament (wurschmidt and worschmidt) these two mappings provide.
-edo approximates the intervals [[3/2]], [[5/4]], [[11/8]], [[19/16]], [[23/16]], [[31/16]] and [[47/32]] well, so that it does a good job representing the 2.3.5.11.19.23.31.47 [[just intonation subgroup]]. To this one may want to add [[17/16]], [[29/16]] and [[43/32]], giving the [[47-limit]] no-7's no-13's no-37's no-41's subgroup 2.3.5.11.17.19.23.29.31.43.47. In this sense it is a tuning of [[schismic]]/[[nestoria]] that focuses on the very primes that [[53edo]] neglects and that instead elegantly connects primes 7, 13, 37 and 41 to nestoria. Also of interest is the [[19-limit]] [[k*N subgroups|2*65 subgroup]] 2.3.5.49.11.91.119.19, on which 65 has the same tuning and commas as the [[zeta]] edo [[130edo]].
+edo approximates the intervals [[3/2]], [[5/4]], [[11/8]], [[19/16]], [[23/16]], [[31/16]] and [[47/32]] well, so that it does a good job representing the 2.3.5.11.19.23.31.47 [[just intonation subgroup]]. To this one may want to add [[17/16]], [[29/16]] and [[43/32]], giving the [[47-limit]] no-7's no-13's no-37's no-41's subgroup 2.3.5.11.17.19.23.29.31.43.47. In this sense it is a tuning of [[schismic]]/[[nestoria]] that focuses on the very primes that [[53edo]] neglects and that instead elegantly connects primes 7, 13, 37, and 41 to nestoria. Also of interest is the [[19-limit]] [[k*N subgroups|2*65 subgroup]] 2.3.5.49.11.91.119.19, on which 65 has the same tuning and commas as the [[zeta]] edo [[130edo]].
 === Prime harmonics ===
@@ Line 20: / Line 20: @@
 ! &#35;
 ! [[Cent]]s
-! Approximate ratios*
+! Approximate ratios<ref group="note">{{sg|limit=2.3.5.11.13/7.17.19.23.29.31.47 subgroup}}</ref>
 ! colspan="2" | [[Ups and downs notation]]
 |-
@@ Line 419: / Line 419: @@
 | D
 |}
-<nowiki />* {{sg|limit=2.3.5.11.13/7.17.19.23.29.31.47 subgroup}}
 == Notation ==
+=== Ups and downs notation ===
+Using [[Helmholtz–Ellis]] accidentals, 65edo can also be notated using [[ups and downs notation]]:
+{{Sharpness-sharp6}}
-===Sagittal notation===
+In some cases, certain notes may be best notated using semi- and sesquisharps and flats with arrows:
+{{Sharpness-sharp6-qt}}
+=== Ivan Wyschnegradsky's notation ===
+Since a sharp raises by six steps, Wyschnegradsky accidentals borrowed from [[72edo]] can also be used:
+{{sharpness-sharp6-iw}}
+=== Sagittal notation ===
 This notation uses the same sagittal sequence as EDOs [[72edo#Sagittal notation|72]] and [[79edo#Sagittal notation|79]].
-====Evo flavor====
+==== Evo flavor ====
 <imagemap>
 File:65-EDO_Evo_Sagittal.svg
@@ Line 438: / Line 448: @@
 </imagemap>
-====Revo flavor====
+==== Revo flavor ====
 <imagemap>
 File:65-EDO_Revo_Sagittal.svg
@@ Line 451: / Line 460: @@
 </imagemap>
-====Evo-SZ flavor====
+==== Evo-SZ flavor ====
 <imagemap>
 File:65-EDO_Evo-SZ_Sagittal.svg
@@ Line 486: / Line 494: @@
 | 32805/32768, 78732/78125
 | {{mapping| 65 103 151 }}
-| &minus;0.110
+| −0.110
 | 0.358
 | 1.94
@@ Line 493: / Line 501: @@
 | 243/242, 4000/3993, 5632/5625
 | {{mapping| 65 103 151 225 }}
-| &minus;0.266
+| −0.266
 | 0.410
 | 2.22
@@ Line 577: / Line 585: @@
 == Zeta properties ==
-===Zeta peak index===
+=== Zeta peak index ===
 {| class="wikitable"
-! colspan="3" |Tuning
-! colspan="3" |Strength
-! colspan="2" |Closest EDO
-! colspan="2" |Integer limit
 |-
-!ZPI
+! colspan="3" | Tuning
-!Steps per octave
+! colspan="3" | Strength
-!Step size (cents)
+! colspan="2" | Closest EDO
-!Height
+! colspan="2" | Integer limit
-!Integral
+|-
-!Gap
+! ZPI
-!EDO
+! Steps per octave
-!Octave (cents)
+! Step size (cents)
-!Consistent
+! Height
-!Distinct
+! Integral
+! Gap
+! EDO
+! Octave (cents)
+! Consistent
+! Distinct
 |-
-|[[334zpi]]
+| [[334zpi]]
-|65.0158450885860
+| 65.0158450885860
-|18.4570391781413
+| 18.4570391781413
-|7.813349
+| 7.813349
-|1.269821
+| 1.269821
-|16.514861
+| 16.514861
-|65edo
+| 65edo
-|1199.70754657919
+| 1199.70754657919
-|6
+| 6
-|6
+| 6
 |}
 == Scales ==
 * Amulet{{idiosyncratic}}, (approximated from [[25edo]], subset of [[würschmidt]]): 5 3 5 5 3 5 12 5 5 3 5 12 5
@@ Line 614: / Line 624: @@
 == Instruments ==
 [[Lumatone mapping for 65edo]]
+== Notes ==
+<references group="note" />
 [[Category:Listen]]