77edo: Difference between revisions

Line 5:

With [[3/1|harmonic 3]] less than a cent flat, [[5/1|harmonic 5]] a bit over three cents sharp and [[7/1|7]]'s less flat than that, 77edo represents an excellent tuning choice for both [[valentine]], the {{nowrap|31 & 46}} temperament, and [[starling]], the [[126/125]] [[planar temperament]], giving the [[optimal patent val]] for [[11-limit]] valentine and its [[13-limit]] extensions dwynwen and valentino, as well as 11-limit starling and [[oxpecker]] temperaments. It also gives the optimal patent val for [[grackle]] and various members of the [[unicorn family]], with a [[generator]] of 4\77 instead of the 5\77 (which gives valentine). These are 7-limit [[Unicorn family #Alicorn|alicorn]] and 11- and 13-limit [[Unicorn family #Camahueto|camahueto]].

77et tempers out [[32805/32768]] in the [[5-limit]], [[126/125]], [[1029/1024]] and [[6144/6125]] in the 7-limit, [[121/120]], [[176/175]], [[385/384]] and [[441/440]] in the 11-limit, and [[196/195]], [[351/350]], [[352/351]], [[676/675]] and [[729/728]] in the 13-limit.

77et tempers out [[32805/32768]] in the [[5-limit]], [[126/125]], [[1029/1024]], and [[6144/6125]] in the 7-limit, [[121/120]], [[176/175]], [[385/384]], and [[441/440]] in the 11-limit, and [[196/195]], [[351/350]], [[352/351]], [[676/675]] and [[729/728]] in the 13-limit.

The [[17/1|17]] and [[19/1|19]] are tuned fairly well, making it [[consistent]] to the no-11 [[21-odd-limit]]. The equal temperament tempers out [[256/255]] in the 17-limit; and [[171/170]], [[361/360]], [[513/512]], and [[1216/1215]] in the 19-limit.

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default [[File:77-EDO_Revo_Sagittal.svg]]

</imagemap>

=== Ups and downs notation ===

Using [[Helmholtz–Ellis]] accidentals, 77edo can be notated using [[ups and downs notation]]:

== Regular temperament properties ==

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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~~

!EDO

!~~Octave (cents)~~

~~!Consistent~~

~~!Distinct~~

|-

|[[414zpi]]

! ZPI

|76.9918536925042

! Steps per octave

|15.5860645308353

! Step size (cents)

|8.194847

! Height

|1.311364

! Integral

|17.029289

! Gap

|77edo

! EDO

|1200.12696887432

! Octave (cents)

|10

! Consistent

|10

! Distinct

|-

| [[414zpi]]

| 76.9918536925042

| 15.5860645308353

| 8.194847

| 1.311364

| 17.029289

| 77edo

| 1200.12696887432

| 10

|}

== Music ==

; [[Jake Freivald]]

@@ Line 5: / Line 5: @@
 With [[3/1|harmonic 3]] less than a cent flat, [[5/1|harmonic 5]] a bit over three cents sharp and [[7/1|7]]'s less flat than that, 77edo represents an excellent tuning choice for both [[valentine]], the {{nowrap|31 &amp; 46}} temperament, and [[starling]], the [[126/125]] [[planar temperament]], giving the [[optimal patent val]] for [[11-limit]] valentine and its [[13-limit]] extensions dwynwen and valentino, as well as 11-limit starling and [[oxpecker]] temperaments. It also gives the optimal patent val for [[grackle]] and various members of the [[unicorn family]], with a [[generator]] of 4\77 instead of the 5\77 (which gives valentine). These are 7-limit [[Unicorn family #Alicorn|alicorn]] and 11- and 13-limit [[Unicorn family #Camahueto|camahueto]].
-et tempers out [[32805/32768]] in the [[5-limit]], [[126/125]], [[1029/1024]] and [[6144/6125]] in the 7-limit, [[121/120]], [[176/175]], [[385/384]] and [[441/440]] in the 11-limit, and [[196/195]], [[351/350]], [[352/351]], [[676/675]] and [[729/728]] in the 13-limit.
+et tempers out [[32805/32768]] in the [[5-limit]], [[126/125]], [[1029/1024]], and [[6144/6125]] in the 7-limit, [[121/120]], [[176/175]], [[385/384]], and [[441/440]] in the 11-limit, and [[196/195]], [[351/350]], [[352/351]], [[676/675]] and [[729/728]] in the 13-limit.
 The [[17/1|17]] and [[19/1|19]] are tuned fairly well, making it [[consistent]] to the no-11 [[21-odd-limit]]. The equal temperament tempers out [[256/255]] in the 17-limit; and [[171/170]], [[361/360]], [[513/512]], and [[1216/1215]] in the 19-limit.
@@ Line 209: / Line 209: @@
 default [[File:77-EDO_Revo_Sagittal.svg]]
 </imagemap>
+=== Ups and downs notation ===
+Using [[Helmholtz–Ellis]] accidentals, 77edo can be notated using [[ups and downs notation]]:
+{{Sharpness-sharp7}}
 == Regular temperament properties ==
@@ Line 343: / Line 347: @@
 == 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
-!EDO
-!Octave (cents)
-!Consistent
-!Distinct
 |-
-|[[414zpi]]
+! ZPI
-|76.9918536925042
+! Steps per octave
-|15.5860645308353
+! Step size (cents)
-|8.194847
+! Height
-|1.311364
+! Integral
-|17.029289
+! Gap
-|77edo
+! EDO
-|1200.12696887432
+! Octave (cents)
-|10
+! Consistent
-|10
+! Distinct
+|-
+| [[414zpi]]
+| 76.9918536925042
+| 15.5860645308353
+| 8.194847
+| 1.311364
+| 17.029289
+| 77edo
+| 1200.12696887432
+| 10
+| 10
 |}
 == Music ==
 ; [[Jake Freivald]]