38edo: Difference between revisions

Since 38 = 2 × 19, it can be thought of as two parallel 19edos. While the halving of the step size lowers consistency and leaves it only mediocre in terms of overall 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 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.

Using the 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 consistent within the 38df val ⟨38 60 88 106 131 140 155 161].

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

[Harmonic series 2-20 in 38df]

Prime harmonics

Approximation of prime harmonics in 38edo
Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.0	-7.2	-7.4	+10.1	-14.5	+12.1	-10.2	-13.3	+3.3	+12.5	-8.2
Error	Relative (%)	+0.0	-22.9	-23.3	+32.1	-45.8	+38.3	-32.4	-42.1	+10.5	+39.7	-25.9
Steps (reduced)		38 (0)	60 (22)	88 (12)	107 (31)	131 (17)	141 (27)	155 (3)	161 (9)	172 (20)	185 (33)	188 (36)

Intervals

Step	Cents	19-odd-limit ratios, in 38df val	Ups and downs notation*
0	0.0		Perfect 1sn	P1	D
1	31.6		Up 1sn	^1	^D
2	63.2		Aug 1sn, dim 2nd	A1, d2	D#
3	94.7	20/19, 19/18, 18/17, 17/16	Upaug 1sn, downminor 2nd	^A1, vm2	^D#, vEb
4	126.3	16/15, 15/14, 14/13, 13/12	Minor 2nd	m2	Eb
5	157.9	12/11, 11/10	Mid 2nd	~2	vE
6	189.5	10/9, 19/17, 9/8	Major 2nd	M2	E
7	221.1	17/15	Upmajor 2nd	^M2	^E
8	252.6	8/7, 15/13, 22/19, 7/6	Aug 2nd, Dim 3rd	A2, d3	E#, Fb
9	284.2	20/17, 13/11, 19/16	Downminor 3rd	vm3	vF
10	315.8	6/5	Minor 3rd	m3	F
11	347.4	17/14, 11/9	Mid 3rd	~3	^F
12	378.9	16/13, 5/4	Major 3rd	M3	F#
13	410.5	24/19, 19/15, 14/11	Upmajor 3rd, Downdim 4th	^M3, vd4	^F#, vGb
14	442.1	9/7, 22/17, 13/10	Aug 3rd, dim 4th	A3, d4	Gb
15	473.7	17/13	Down 4th	v4	vG
16	505.3	4/3	Perfect 4th	P4	G
17	536.8	19/14, 15/11, 26/19, 11/8	Up 4th	^4	^G
18	568.4	18/13, 7/5	Aug 4th	A4	G#
19	600.0	24/17, 17/12	Upaug 4th, downdim 5th	^A4, vd5	^G#, vAb
20	631.6	10/7, 13/9	Dim 5th	d5	Ab
21	663.2	16/11, 19/13, 22/15, 28/19	Down 5th	v5	vA
22	694.7	3/2	Perfect 5th	P5	A
23	726.3	26/17	Up 5th	^5	^A
24	757.9	20/13, 17/11, 14/9	Aug 5th, dim 6th	A5, d6	A#
25	789.5	11/7, 30/19, 19/12	Upaug 5th, downminor 6th	^A5, vm6	^A#, vBb
26	821.1	8/5, 13/8	Minor 6th	m6	Bb
27	852.6	18/11, 28/17	Mid 6th	~6	vB
28	884.2	5/3	Major 6th	M6	B
29	915.8	32/19, 22/13, 17/10	Upmajor 6th	^M6	^B
30	947.4	12/7, 19/11, 26/15, 7/4	Aug 6th, dim 7th	A6, d7	B#, Cb
31	978.9	30/17	Downminor 7th	vm7	vC
32	1010.5	16/9, 34/19, 9/5	Minor 7th	m7	C
33	1042.1	20/11, 11/6	Mid 7th	~7	^C
34	1073.7	24/13, 13/7, 28/15, 15/8	Major 7th	M7	C#
35	1105.3	32/17, 17/9, 36/19, 19/10	Upmajor 7th, Downdim 8ve	^M7, vd8	^C#, vDb
36	1136.8		Aug 7th, dim 8ve	A7, d8	Db
37	1168.4		Down 8ve	v8	vD
38	1200.0		Perfect 8ve	P8	D

* Ups and downs may be substituted with semi-sharps and semi-flats, respectively

Zeta properties

Zeta peak index

Tuning			Strength			Closest EDO		Integer limit
ZPI	Steps per octave	Step size (cents)	Height	Integral	Gap	EDO	Octave (cents)	Consistent	Distinct
166zpi	37.8901105027757	31.6705331305933	5.808723	0.986660	15.046792	38edo	1203.48025896255	6	6

Notation

Sagittal notation

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

Evo flavor

Revo flavor

Evo-SZ flavor

Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is also a Stein-Zimmerman notation.

Instruments

Music

Canon at the Semitone on The Mother's Malison Theme for Cor Anglais and Violin by Claudi Meneghin

@@ 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

38edo: Difference between revisions

Revision as of 01:47, 19 January 2025

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