38edo: Difference between revisions

Line 5:

Since {{nowrap|38 {{=}} 2 × 19}}, it can be thought of as two parallel [[19edo]]s. While the halving of the step size lowers [[consistency]] and leaves it only mediocre in terms of overall [[relative interval error|relative error]], the fact that the 3rd & 5th harmonics are flat by almost exactly the same amount, while the 11th is double that means there are quite a few near perfect composite ratios, such as the the [[6/5]] it shares with 19edo, plus [[11/9]], [[15/11]] & [[25/22]], (and their inversions) while a single step nears [[55/54]]; the approximation to [[11/9]] in particular should be noted for forming a 10-strong [[consistent circle]]. This gives several interesting possibilities for unusual near-just chords such as 15:18:22:25:30. It [[tempering out|tempers out]] the same [[5-limit]] commas as 19edo, namely [[81/80]], [[3125/3072]] and [[15625/15552]]. In the [[7-limit]], we can add [[50/49]], and tempering out 81/80 and 50/49 gives [[injera]] temperament, for which 38 is the [[optimal patent val]]. In the [[11-limit]], we can add 121/120 and 176/175.

In [[Warts|38df]], every [[prime interval]] from 3 to 19 is characterized by a flat intonation. Furthermore, the [[mapping]] of all [[19-odd-limit]] intervals in 38df aligns with their closest approximations in 38edo, excepting for 7/4 and 13/8, along with their octave complements 8/7 and 16/13, which are by definition mapped to their secondary optimal steps within 38df. In other words, all 19-odd-limit intervals are [[consistency|consistent]] within the 38df [[val]] {{val| 38 60 88 106 131 140 155 161 }}.

Using the [[Warts|38df]] mapping, every [[prime interval]] from 3 to 19 is characterized by a flat intonation. Furthermore, the [[mapping]] of all [[19-odd-limit]] intervals in 38df aligns with their closest approximations in 38edo, excepting for 7/4 and 13/8, along with their octave complements 8/7 and 16/13, which are by definition mapped to their secondary optimal steps within 38df. In other words, all 19-odd-limit intervals are [[consistency|consistent]] within the 38df [[val]] {{val| 38 60 88 106 131 140 155 161 }}.

The harmonic series from 1 to 20 is approximated within 38df by the sequence: {{nowrap| 38 22 16 12 10 8 8 6 6 5 5 4 4 4 4 3 3 3 3 }}

Line 298:

== 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~~

|-

|[[166zpi]]

! ZPI

|37.8901105027757

! Steps per octave

|31.6705331305933

! Step size (cents)

|5.808723

! Height

|0.986660

! Integral

|15.046792

! Gap

|38edo

! EDO

|1203.48025896255

! Octave (cents)

|6

! Consistent

|6

! Distinct

|-

| [[166zpi]]

| 37.8901105027757

| 31.6705331305933

| 5.808723

| 0.986660

| 15.046792

| 38edo

| 1203.48025896255

| 6

|}

== Notation ==

===Sagittal notation===

=== Sagittal notation ===

This notation uses the same sagittal sequence as EDOs [[17edo#Sagittal notation|17]], [[24edo#Sagittal notation|24]], and [[31edo#Sagittal notation|31]], is a subset of the notation for [[76edo#Sagittal notation|76-EDO]], and is a superset of the notation for [[19edo#Sagittal notation|19-EDO]].

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

==== Evo flavor ====

File:38-EDO_Evo_Sagittal.svg

Line 341:

Line 343:

</imagemap>

====Revo flavor====

==== Revo flavor ====

File:38-EDO_Revo_Sagittal.svg

Line 352:

Line 353:

</imagemap>

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

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

File:38-EDO_Evo-SZ_Sagittal.svg

@@ Line 5: / Line 5: @@
 Since {{nowrap|38 {{=}} 2 × 19}}, it can be thought of as two parallel [[19edo]]s. While the halving of the step size lowers [[consistency]] and leaves it only mediocre in terms of overall [[relative interval error|relative error]], the fact that the 3rd & 5th harmonics are flat by almost exactly the same amount, while the 11th is double that means there are quite a few near perfect composite ratios, such as the the [[6/5]] it shares with 19edo, plus [[11/9]], [[15/11]] & [[25/22]], (and their inversions) while a single step nears [[55/54]]; the approximation to [[11/9]] in particular should be noted for forming a 10-strong [[consistent circle]]. This gives several interesting possibilities for unusual near-just chords such as 15:18:22:25:30. It [[tempering out|tempers out]] the same [[5-limit]] commas as 19edo, namely [[81/80]], [[3125/3072]] and [[15625/15552]]. In the [[7-limit]], we can add [[50/49]], and tempering out 81/80 and 50/49 gives [[injera]] temperament, for which 38 is the [[optimal patent val]]. In the [[11-limit]], we can add 121/120 and 176/175.
-In [[Warts|38df]], every [[prime interval]] from 3 to 19 is characterized by a flat intonation. Furthermore, the [[mapping]] of all [[19-odd-limit]] intervals in 38df aligns with their closest approximations in 38edo, excepting for 7/4 and 13/8, along with their octave complements 8/7 and 16/13, which are by definition mapped to their secondary optimal steps within 38df. In other words, all 19-odd-limit intervals are [[consistency|consistent]] within the 38df [[val]] {{val| 38 60 88 106 131 140 155 161 }}.
+Using the [[Warts|38df]] mapping, every [[prime interval]] from 3 to 19 is characterized by a flat intonation. Furthermore, the [[mapping]] of all [[19-odd-limit]] intervals in 38df aligns with their closest approximations in 38edo, excepting for 7/4 and 13/8, along with their octave complements 8/7 and 16/13, which are by definition mapped to their secondary optimal steps within 38df. In other words, all 19-odd-limit intervals are [[consistency|consistent]] within the 38df [[val]] {{val| 38 60 88 106 131 140 155 161 }}.
 The harmonic series from 1 to 20 is approximated within 38df by the sequence: {{nowrap| 38 22 16 12 10 8 8 6 6 5 5 4 4 4 4 3 3 3 3 }}
@@ Line 298: / Line 298: @@
 == 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
 |-
-|[[166zpi]]
+! ZPI
-|37.8901105027757
+! Steps per octave
-|31.6705331305933
+! Step size (cents)
-|5.808723
+! Height
-|0.986660
+! Integral
-|15.046792
+! Gap
-|38edo
+! EDO
-|1203.48025896255
+! Octave (cents)
-|6
+! Consistent
-|6
+! Distinct
+|-
+| [[166zpi]]
+| 37.8901105027757
+| 31.6705331305933
+| 5.808723
+| 0.986660
+| 15.046792
+| 38edo
+| 1203.48025896255
+| 6
+| 6
 |}
 == Notation ==
-===Sagittal notation===
+=== Sagittal notation ===
 This notation uses the same sagittal sequence as EDOs [[17edo#Sagittal notation|17]], [[24edo#Sagittal notation|24]], and [[31edo#Sagittal notation|31]], is a subset of the notation for [[76edo#Sagittal notation|76-EDO]], and is a superset of the notation for [[19edo#Sagittal notation|19-EDO]].
-====Evo flavor====
+==== Evo flavor ====
 <imagemap>
 File:38-EDO_Evo_Sagittal.svg
@@ Line 341: / Line 343: @@
 </imagemap>
-====Revo flavor====
+==== Revo flavor ====
 <imagemap>
 File:38-EDO_Revo_Sagittal.svg
@@ Line 352: / Line 353: @@
 </imagemap>
-====Evo-SZ flavor====
+==== Evo-SZ flavor ====
 <imagemap>
 File:38-EDO_Evo-SZ_Sagittal.svg