49edo: Difference between revisions

← 48edo 49edo 50edo →
Prime factorization	72
Step size	24.4898 ¢
Fifth	29\49 (710.204 ¢)
Semitones (A1:m2)	7:2 (171.4 ¢ : 48.98 ¢)
Dual sharp fifth	29\49 (710.204 ¢)
Dual flat fifth	28\49 (685.714 ¢) (→ 4\7)
Dual major 2nd	8\49 (195.918 ¢)
Consistency limit	7
Distinct consistency limit	7

← Older edit

VisualWikitext

Latest revision as of 10:43, 25 April 2026

49 equal divisions of the octave (abbreviated 49edo or 49ed2), also called 49-tone equal temperament (49tet) or 49 equal temperament (49et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 49 equal parts of about 24.5 ¢ each. Each step represents a frequency ratio of 2^1/49, or the 49th root of 2.

Theory

49edo is very much on the sharp side of things, with sharp tunings of harmonics 3, 5, 7, and 11. It is the optimal patent val for superpyth temperament in the 7- and 11-limit, archytas (7-limit), and ares (11-limit) planar temperaments, being almost exactly equal to 3⁄10-comma superpyth. It tempers out 64/63, 245/243, and 3125/3087 in the 7-limit, and 100/99, 540/539, and 1375/1372 in the 11-limit. In the 13-limit, its patent val ⟨49 78 114 138 170 181], has a rather flat (by relative error) harmonic 13, which leads to inconsistent mappings; but using the 49f val ⟨49 78 114 138 170 182] improves 13-limit consistency, and in this val it tempers out 364/363 and 847/845.

Harmonics

Approximation of odd harmonics in 49edo
Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	Absolute (¢)	+8.2	+5.5	+10.8	-8.0	+11.9	-7.9	-10.7	-7.0	-3.6	-5.5	+8.5
Error	Relative (%)	+33.7	+22.6	+44.0	-32.6	+48.8	-32.2	-43.8	-28.6	-14.8	-22.4	+34.5
Steps (reduced)		78 (29)	114 (16)	138 (40)	155 (8)	170 (23)	181 (34)	191 (44)	200 (4)	208 (12)	215 (19)	222 (26)

Subsets and supersets

Since 49 factors into primes as 7², 49edo contains 7edo as its only nontrivial subset. 49edo is the first square edo with a non-enfactored diatonic fifth. Doubling it produces 98edo, a respectable (if overly complex) meantone tuning.

Intervals

#	Cents	Approximate ratios*	Ups and downs notation
0	0.000	1/1	D
1	24.490	50/49	^D, vE♭
2	48.980	28/27, 36/35, 49/48, 81/80	^^D, E♭
3	73.469	22/21, 25/24, 33/32	^³D, ^E♭
4	97.959	16/15, 21/20	v³D♯, ^^E♭
5	122.449	15/14	vvD♯, ^³E♭
6	146.939	12/11	vD♯, v³E
7	171.429	10/9, 11/10	D♯, vvE
8	195.918	28/25	^D♯, vE
9	220.408	8/7, 9/8, 25/22	E
10	244.898	125/108, 144/125	^E, vF
11	269.388	7/6	F
12	293.878	25/21, 33/28	^F, vG♭
13	318.367	6/5	^^F, G♭
14	342.857	11/9	^³F, ^G♭
15	367.347	27/22	v³F♯, ^^G♭
16	391.837	5/4	vvF♯, ^³G♭
17	416.327	14/11	vF♯, v³G
18	440.816	9/7	F♯, vvG
19	465.306	125/96, 162/125	^F♯, vG
20	489.796	4/3, 21/16	G
21	514.286	75/56	^G, vA♭
22	538.776	15/11, 27/20	^^G, A♭
23	563.265	11/8	^³G, ^A♭
24	587.755	7/5	v³G♯, ^^A♭
25	612.245	10/7	vvG♯, ^³A♭
26	636.735	16/11	vG♯, v³A
27	661.244	22/15, 40/27	G♯, vvA
28	685.714	112/75	^G♯, vA
29	710.204	3/2, 32/21	A
30	734.694	125/81, 192/125	^A, vB♭
31	759.184	14/9	^^A, B♭
32	783.673	11/7	^³A, ^B♭
33	808.163	8/5	v³A♯, ^^B♭
34	832.653	44/27	vvA♯, ^³B♭
35	857.143	18/11	vA♯, v³B
36	881.633	5/3	A♯, vvB
37	906.122	42/25, 56/33	^A♯, vB
38	930.612	12/7	B
39	955.102	125/72, 216/125	^B, vC
40	979.592	7/4, 16/9, 44/25	C
41	1004.082	25/14	^C, vD♭
42	1028.571	9/5, 20/11	^^C, D♭
43	1053.061	11/6	^³C, ^D♭
44	1077.551	28/15	v³C♯, ^^D♭
45	1102.041	15/8, 40/21	vvC♯, ^³D♭
46	1126.531	21/11, 48/25, 64/33	vC♯, v³D
47	1151.020	27/14, 35/18, 96/49, 160/81	C♯, vvD
48	1175.510	49/25	^C♯, vD
49	1200.000	2/1	D

* Based on 49edo's 11-limit patent val ⟨49 78 114 138 170] mapping

Notation

Ups and downs notation

49edo can be notated using ups and downs. Trup is equivalent to quudsharp, trudsharp is equivalent to quup, etc.

Step offset	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
Sharp symbol
Flat symbol

Alternatively, sharps and flats with arrows borrowed from Helmholtz–Ellis notation can be used:

Step offset	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17
Sharp symbol
Flat symbol

Sagittal notation

Evo flavor

Revo flavor

Approximation to JI

alt : Your browser has no SVG support. — Selected 19-limit intervals approximated in 49edo

Interval mappings

The following tables show how 15-odd-limit intervals are represented in 49edo. Prime harmonics are in bold; inconsistent intervals are in italics.

15-odd-limit intervals in 49edo (direct approximation, even if inconsistent)
Interval and complement	Error (abs, ¢)	Error (rel, %)
1/1, 2/1	0.000	0.0
13/9, 18/13	0.117	0.5
11/7, 14/11	1.181	4.8
15/11, 22/15	1.825	7.5
7/6, 12/7	2.517	10.3
5/3, 6/5	2.726	11.1
15/13, 26/15	2.843	11.6
15/14, 28/15	3.006	12.3
11/6, 12/11	3.698	15.1
11/9, 18/11	4.551	18.6
13/11, 22/13	4.668	19.1
7/5, 10/7	5.243	21.4
5/4, 8/5	5.523	22.6
9/7, 14/9	5.732	23.4
13/7, 14/13	5.849	23.9
11/10, 20/11	6.424	26.2
13/8, 16/13	7.875	32.2
9/8, 16/9	7.992	32.6
3/2, 4/3	8.249	33.7
13/12, 24/13	8.366	34.2
15/8, 16/15	10.718	43.8
7/4, 8/7	10.766	44.0
9/5, 10/9	10.975	44.8
13/10, 20/13	11.092	45.3
11/8, 16/11	11.947	48.8

15-odd-limit intervals in 49edo (patent val mapping)
Interval and complement	Error (abs, ¢)	Error (rel, %)
1/1, 2/1	0.000	0.0
11/7, 14/11	1.181	4.8
15/11, 22/15	1.825	7.5
7/6, 12/7	2.517	10.3
5/3, 6/5	2.726	11.1
15/14, 28/15	3.006	12.3
11/6, 12/11	3.698	15.1
11/9, 18/11	4.551	18.6
7/5, 10/7	5.243	21.4
5/4, 8/5	5.523	22.6
9/7, 14/9	5.732	23.4
11/10, 20/11	6.424	26.2
13/8, 16/13	7.875	32.2
3/2, 4/3	8.249	33.7
7/4, 8/7	10.766	44.0
9/5, 10/9	10.975	44.8
11/8, 16/11	11.947	48.8
13/10, 20/13	13.398	54.7
15/8, 16/15	13.772	56.2
13/12, 24/13	16.124	65.8
9/8, 16/9	16.498	67.4
13/7, 14/13	18.641	76.1
13/11, 22/13	19.822	80.9
15/13, 26/15	21.647	88.4
13/9, 18/13	24.373	99.5

15-odd-limit intervals by 49f val mapping
Interval and complement	Error (abs, ¢)	Error (rel, %)
1/1, 2/1	0.000	0.0
13/9, 18/13	0.117	0.5
11/7, 14/11	1.181	4.8
15/11, 22/15	1.825	7.5
7/6, 12/7	2.517	10.3
5/3, 6/5	2.726	11.1
15/13, 26/15	2.843	11.6
15/14, 28/15	3.006	12.3
11/6, 12/11	3.698	15.1
11/9, 18/11	4.551	18.6
13/11, 22/13	4.668	19.1
7/5, 10/7	5.243	21.4
5/4, 8/5	5.523	22.6
9/7, 14/9	5.732	23.4
13/7, 14/13	5.849	23.9
11/10, 20/11	6.424	26.2
3/2, 4/3	8.249	33.7
13/12, 24/13	8.366	34.2
7/4, 8/7	10.766	44.0
9/5, 10/9	10.975	44.8
13/10, 20/13	11.092	45.3
11/8, 16/11	11.947	48.8
15/8, 16/15	13.772	56.2
9/8, 16/9	16.498	67.4
13/8, 16/13	16.615	67.8

Zeta peaks

The strongest local zeta peak around 49edo is its second closest, 49.141 edo. One step is 24.419 cents, and two steps, 48.839 cents, is a good generator for Triple BP.

Approximation to irrational intervals

Acoustic ϕ and ϕ^ϕ⁻¹

49edo has a very close approximation of both acoustic phi and ϕ^{ϕ^-1}, a kind of logarithmic phi that divides acoustic phi logarithmically by phi (instead of dividing 2/1).

The phith root of phi (ϕ^{ϕ^-1}) has interesting applications as Metallic MOS, and in particular the fractal-like possibilities of self-similar subdivision of musical scales within acoustic phi.

Direct approximation
Interval	Error (abs, ¢)	#\49
ϕ / ϕ^ϕ⁻¹ = ϕ^{(2 − ϕ)}	0.155	13
ϕ	−0.437	34
ϕ^ϕ⁻¹	−0.592	21

Not until 592 do we find a better edo in terms of relative error on these two intervals (but whose edo-steps upon which these intervals are mapped are not based on the Fibonacci sequence, unlike 49edo).

Music

Sevish - Star Nursery uses a scale based on acoustic phi and ϕ^ϕ⁻¹. 49edo provides a suitable approximation for this scale with the mode: 5 3 5 5 3 5 3 5

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Absolute (¢)	Relative (%)
2.3	[78 -49⟩	[⟨49 78]]	−2.60	2.60	10.62
2.3.5	15625/15552, 20480/19683	[⟨49 78 114]]	−2.53	2.12	8.69
2.3.5.7	64/63, 245/243, 3125/3087	[⟨49 78 114 138]]	−2.85	1.92	7.87
2.3.5.7.11	64/63, 100/99, 245/243, 1331/1323	[⟨49 78 114 138 170]]	−2.97	1.74	7.11

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods per 8ve	Generator*	Cents*	Associated ratio*	Temperament
1	1\49	24.5	99/98	Sengagen
1	4\49	98.0	16/15	Passion
1	6\49	146.9	12/11	Bohpier
1	8\49	195.9	28/25	Didacus
1	11\49	269.4	7/6	Infraorwell
1	12\49	293.9	25/21	Kleiboh
1	13\49	318.4	6/5	Catalan
1	16\49	391.8	5/4	Magus
1	17\49	416.3	14/11	Sqrtphi
1	18\49	440.8	9/7	Clyde
1	19\49	465.3	55/36	Semisept
1	20\49	489.8	4/3	Superpyth
7	20\49 (1\49)	489.8 (24.5)	4/3 (250/243)	Sevond (49)
7	20\49 (1\49)	489.8 (24.5)	4/3 (25/24)	Seville (49c)

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

Octave stretch or compression

49edo's primes 3, 5, 7 and 11 are all tuned sharp, so 49edo can benefit from octave shrinking. Some compressed-octave tunings of 49edo include (least to most compression): 176ed12, 114ed5, 233zpi, 127ed6, 138ed7 and 78edt.

Nonoctave temperament

The TE-optimized triple Bohlen–Pierce scale is obtained by taking every second degree of 49edo with the octave compressed by 3.861 cents to 1196.139 cents. It realizes the Tenney–Euclidean regular temperament on the 3.5.7.11.13 subgroup mapped as [⟨78 114 138 170 182]]. Under this compression, the primes map to the 49fgh val in the 23-limit.

Scales

MOS scales

Main article: List of MOS scales in 49edo

Bohpier[8]: 6 6 6 6 7 6 6 6
Catalan[7]: 3 10 3 10 3 10 10 (vaugely diminished-like)
Catalan[11]: 3 7 3 3 7 3 3 7 3 3 7
Catalan[19]: 3 3 1 3 3 3 3 1 3 3 3 1 3 3 3 3 1 3 3
Clyde[5]: 5 13 5 13 13 (mysterious, adventurous)
Didacus[6]: 8 8 8 8 8 9 (like the whole tone scale)
Didacus[13]: 1 7 1 7 1 7 1 7 1 7 1 7 1
Infraorwell[5]: 11 11 5 11 11
Infraorwell[22]: 1 4 1 4 1 1 4 1 4 1 1 4 1 4 1 4 1 1 4 1 4 1
Kleiboh[5]: 12 12 12 12 1
Kleiboh[13]: 1 10 1 1 10 1 1 1 10 1 1 10 1
Magus[7]: 1 15 1 15 1 15 1 (vaguely augmented-like)
Passion[12]: 4 4 4 4 4 4 5 4 4 4 4 4 (like 12edo)
Passion[23]: 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1
Sevond[21]/Seville[21]: 1 5 1 1 5 1 1 5 1 1 5 1 1 5 1 1 5 1 1 5 1
Superpyth[5]: 11 9 9 11 9 (in between minor pentatonic and equipentatonic)
Superpyth[7]: 9 2 9 9 9 2 9 (Dorian mode; rotate for other modes)
Superpyth[12]: 2 7 2 7 2 2 7 2 7 2 7 2 (same melodic shape as 12edo but much more xenharmonic harmonies)
Superpyth[27]: 2 2 1 2 2 2 2 2 1 2 2 2 2 1 2 2 2 2 1 2 2 2 2 2 1 2 2

Other scales

The 6ed7/3+7edo scale (non-octave-repeating)

Instruments

Lumatone

Lumatone mapping for 49edo

Skip fretting

Skip fretting system 49 3 7 is a skip fretting system for 49edo. All examples are for 5-string bass.

Harmonics

1/1: string 2 open

2/1: not easily accessible

3/2: string 4 fret 5 and string 1 fret 12

5/4: string 3 fret 3

7/4: string 3 fret 11

11/8: string 3 fret 5

Music

Modern renderings

The Cure

Boys Don't Cry (1979) – Lumatone cover by YoVariable (2025)

21st century

Bryan Deister

Deltarune – Man (cover) (2023)
microtonal improv in 49edo (2024)
weathergirl - FLAVOR FOLEY (microtonal cover in 49edo) (2025)
I'm Your Captain Now (The Ancients) - The Recovery System (microtonal cover in 49edo) (2026)
49edo improv (2026)
49edo riff (2026)
49edo groove (2026)
49edo prelude (2026)

Mercury Amalgam

Wrong Generation (2022 demo version)

Cam Taylor

49-equal: 7-equal meets superpyth (2023)

49edo: Difference between revisions

Latest revision as of 10:43, 25 April 2026

Contents

Theory

Harmonics

Subsets and supersets

Intervals

Notation

Ups and downs notation

Sagittal notation

Evo flavor

Revo flavor

Approximation to JI

Interval mappings

Zeta peaks

Approximation to irrational intervals

Acoustic ϕ and ϕ^ϕ⁻¹

Music

Regular temperament properties

Rank-2 temperaments

Octave stretch or compression

Nonoctave temperament

Scales

MOS scales

Other scales

Instruments

Lumatone

Skip fretting

Music

Modern renderings

21st century

Navigation menu

@@ Line 1: / Line 1: @@
-{{Infobox ET
+{{Infobox ET}}
-| Prime factorization = 7<sup>2</sup>
+{{ED intro}}
-| Step size = 24.490¢
-| Fifth = 29\49 = 710.2¢
-| Major 2nd = 9\49 = 220.4¢
-| Minor 2nd = 2\49 = 49.0¢
-| Augmented 1sn = 7\49 = 171.4¢ (&rarr;[[7edo|1\7]])
-}}
-'''49-EDO''', or '''49 equal temperament''' divides the octave into 49 equal parts of 24.490 [[cent]]s each.
 == Theory ==
-edo is very much on the sharp side of things, with sharp tunings of harmonics 3 (it is the first square equal division with a "real" 3 of step coprime to its cardinality), 5, 7, and 11. It is the [[optimal patent val]] for [[superpyth]] temperament in the 7 and 11 limits, [[Archytas family #Archytas|archytas]] ([[7-limit]]) and [[Archytas family #Ares|ares]] ([[11-limit]]) planar temperaments and almost identical to the e-based analog of [[Lucy tuning]]. It [[tempering out|tempers out]] [[64/63]], [[245/243]] and [[3125/3087]] in the 7-limit, and [[100/99]] and [[1375/1372]] in the 11-limit.
+edo is very much on the sharp side of things, with sharp tunings of [[harmonic]]s [[3/1|3]], [[5/1|5]], [[7/1|7]], and [[11/1|11]]. It is the [[optimal patent val]] for [[superpyth]] temperament in the 7- and 11-limit, [[Archytas family #Archytas|archytas]] ([[7-limit]]), and [[Archytas family #Ares|ares]] ([[11-limit]]) planar temperaments, being almost exactly equal to {{frac|3|10}}-comma superpyth. It [[tempering out|tempers out]] [[64/63]], [[245/243]], and [[3125/3087]] in the 7-limit, and [[100/99]], [[540/539]], and [[1375/1372]] in the 11-limit. In the 13-limit, its [[patent val]] {{val| 49 78 114 138 170 181 }}, has a rather flat (by relative error) harmonic [[13/1|13]], which leads to inconsistent mappings; but using the 49f val {{val| 49 78 114 138 170 182 }} improves 13-limit consistency, and in this val it tempers out [[364/363]] and [[847/845]].
-{{primes in edo|49}}
+=== Harmonics ===
+{{Harmonics in equal|49}}
+=== Subsets and supersets ===
+Since 49 factors into primes as 7<sup>2</sup>, 49edo contains [[7edo]] as its only nontrivial subset. 49edo is the first square edo with a [[enfactoring|non-enfactored]] diatonic fifth. Doubling it produces [[98edo]], a respectable (if overly complex) [[meantone]] tuning.
 == Intervals ==
 {| class="wikitable center-all right-2 left-3"
-! #
+|-
+! &#35;
 ! Cents
-! Approximate Ratios
+! Approximate ratios*
+! [[Ups and downs notation]]
 |-
 | 0
 | 0.000
 | [[1/1]]
+| {{UDnote|step=0}}
 |-
 | 1
 | 24.490
 | [[50/49]]
+| {{UDnote|step=1}}
 |-
 | 2
 | 48.980
-| [[81/80]], [[28/27]], [[36/35]], [[49/48]]
+| ''[[28/27]]'', [[36/35]], ''[[49/48]]'', ''[[81/80]]''
+| {{UDnote|step=2}}
 |-
 | 3
 | 73.469
-| [[25/24]], [[22/21]], [[33/32]]
+| [[22/21]], [[25/24]], ''[[33/32]]''
+| {{UDnote|step=3}}
 |-
 | 4
 | 97.959
-| [[16/15]], [[21/20]]
+| ''[[16/15]]'', [[21/20]]
+| {{UDnote|step=4}}
 |-
 | 5
 | 122.449
 | [[15/14]]
+| {{UDnote|step=5}}
 |-
 | 6
 | 146.939
 | [[12/11]]
+| {{UDnote|step=6}}
 |-
 | 7
 | 171.429
 | [[10/9]], [[11/10]]
+| {{UDnote|step=7}}
 |-
 | 8
 | 195.918
-|
+| [[28/25]]
+| {{UDnote|step=8}}
 |-
 | 9
 | 220.408
-| [[9/8]], [[8/7]]
+| [[8/7]], ''[[9/8]]'', [[25/22]]
+| {{UDnote|step=9}}
 |-
 | 10
 | 244.898
-|
+| 125/108, 144/125
+| {{UDnote|step=10}}
 |-
 | 11
 | 269.388
 | [[7/6]]
+| {{UDnote|step=11}}
 |-
 | 12
 | 293.878
-|
+| [[25/21]], [[33/28]]
+| {{UDnote|step=12}}
 |-
 | 13
 | 318.367
 | [[6/5]]
+| {{UDnote|step=13}}
 |-
 | 14
 | 342.857
 | [[11/9]]
+| {{UDnote|step=14}}
 |-
 | 15
 | 367.347
 | [[27/22]]
+| {{UDnote|step=15}}
 |-
 | 16
 | 391.837
 | [[5/4]]
+| {{UDnote|step=16}}
 |-
 | 17
 | 416.327
 | [[14/11]]
+| {{UDnote|step=17}}
 |-
 | 18
 | 440.816
 | [[9/7]]
+| {{UDnote|step=18}}
 |-
 | 19
 | 465.306
-|
+| 125/96, ''162/125''
+| {{UDnote|step=19}}
 |-
 | 20
 | 489.796
-| [[4/3]], [[21/16]]
+| [[4/3]], ''[[21/16]]''
+| {{UDnote|step=20}}
 |-
 | 21
 | 514.286
-|
+| [[75/56]]
+| {{UDnote|step=21}}
 |-
 | 22
 | 538.776
-| [[27/20]], [[15/11]]
+| [[15/11]], ''[[27/20]]''
+| {{UDnote|step=22}}
 |-
 | 23
 | 563.265
 | [[11/8]]
+| {{UDnote|step=23}}
 |-
 | 24
 | 587.755
 | [[7/5]]
+| {{UDnote|step=24}}
 |-
 | 25
 | 612.245
 | [[10/7]]
+| {{UDnote|step=25}}
 |-
 | 26
 | 636.735
 | [[16/11]]
+| {{UDnote|step=26}}
 |-
 | 27
 | 661.244
-| [[40/27]], [[22/15]]
+| [[22/15]], ''[[40/27]]''
+| {{UDnote|step=27}}
 |-
 | 28
 | 685.714
-|
+| [[112/75]]
+| {{UDnote|step=28}}
 |-
 | 29
 | 710.204
-| [[3/2]], [[32/21]]
+| [[3/2]], ''[[32/21]]''
+| {{UDnote|step=29}}
 |-
 | 30
 | 734.694
-|
+| ''125/81'', 192/125
+| {{UDnote|step=30}}
 |-
 | 31
 | 759.184
 | [[14/9]]
+| {{UDnote|step=31}}
 |-
 | 32
 | 783.673
 | [[11/7]]
+| {{UDnote|step=32}}
 |-
 | 33
 | 808.163
 | [[8/5]]
+| {{UDnote|step=33}}
 |-
 | 34
 | 832.653
 | [[44/27]]
+| {{UDnote|step=34}}
 |-
 | 35
 | 857.143
 | [[18/11]]
+| {{UDnote|step=35}}
 |-
 | 36
 | 881.633
 | [[5/3]]
+| {{UDnote|step=36}}
 |-
 | 37
 | 906.122
-|
+| [[42/25]], [[56/33]]
+| {{UDnote|step=37}}
 |-
 | 38
 | 930.612
 | [[12/7]]
+| {{UDnote|step=38}}
 |-
 | 39
 | 955.102
-|
+| 125/72, 216/125
+| {{UDnote|step=39}}
 |-
 | 40
 | 979.592
-| [[16/9]], [[7/4]]
+| [[7/4]], ''[[16/9]]'', [[44/25]]
+| {{UDnote|step=40}}
 |-
 | 41
 | 1004.082
-|
+| [[25/14]]
+| {{UDnote|step=41}}
 |-
 | 42
 | 1028.571
 | [[9/5]], [[20/11]]
+| {{UDnote|step=42}}
 |-
 | 43
 | 1053.061
 | [[11/6]]
+| {{UDnote|step=43}}
 |-
 | 44
 | 1077.551
 | [[28/15]]
+| {{UDnote|step=44}}
 |-
 | 45
 | 1102.041
-| [[15/8]], [[40/21]]
+| ''[[15/8]]'', [[40/21]]
+| {{UDnote|step=45}}
 |-
 | 46
 | 1126.531
-| [[48/25]], [[21/11]], [[64/33]]
+| [[21/11]], [[48/25]], ''[[64/33]]''
+| {{UDnote|step=46}}
 |-
 | 47
 | 1151.020
-| [[160/81]], [[27/14]], [[35/18]], [[96/49]]
+| ''[[27/14]]'', [[35/18]], ''[[96/49]]'', ''[[160/81]]''
+| {{UDnote|step=47}}
 |-
 | 48
 | 1175.510
 | [[49/25]]
+| {{UDnote|step=48}}
 |-
 | 49
 | 1200.000
 | [[2/1]]
+| {{UDnote|step=49}}
 |}
+<nowiki />* Based on 49edo's 11-limit patent val {{val| 49 78 114 138 170 }} mapping
+== Notation ==
+=== Ups and downs notation ===
+edo can be notated using [[ups and downs notation|ups and downs]]. Trup is equivalent to quudsharp, trudsharp is equivalent to quup, etc.
+{{Ups and downs sharpness}}
+Alternatively, sharps and flats with arrows borrowed from [[Helmholtz–Ellis notation]] can be used:
+{{Sharpness-sharp7}}
+=== Sagittal notation ===
+==== Evo flavor ====
+<imagemap>
+File:49-EDO_Evo_Sagittal.svg
+desc none
+rect 80 0 300 50 [[Sagittal_notation]]
+rect 300 0 589 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
+rect 20 80 140 106 [[513/512]]
+rect 140 80 240 106 [[81/80]]
+rect 240 80 360 106 [[33/32]]
+default [[File:49-EDO_Evo_Sagittal.svg]]
+</imagemap>
+==== Revo flavor ====
+<imagemap>
+File:49-EDO_Revo_Sagittal.svg
+desc none
+rect 80 0 300 50 [[Sagittal_notation]]
+rect 300 0 534 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
+rect 20 80 140 106 [[513/512]]
+rect 140 80 240 106 [[81/80]]
+rect 240 80 360 106 [[33/32]]
+default [[File:49-EDO_Revo_Sagittal.svg]]
+</imagemap>
+== Approximation to JI ==
+[[File:49ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 19-limit intervals approximated in 49edo]]
+=== Interval mappings ===
+{{Q-odd-limit intervals|49}}
+{{Q-odd-limit intervals|49.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 49f val mapping}}
+=== Zeta peaks ===
+The strongest [[The Riemann zeta function and tuning|local zeta peak]] around 49edo is its second closest, 49.141 edo. One step is 24.419 cents, and two steps, 48.839 cents, is a good generator for [[Triple BP]].
+== Approximation to irrational intervals ==
+=== Acoustic ϕ and ϕ<sup>ϕ<sup>−1</sup></sup> ===
+edo has a very close approximation of both [[acoustic phi]] and [[phith root of phi|ϕ<sup>ϕ<sup>-1</sup></sup>]], a kind of logarithmic phi that divides [[acoustic phi]] logarithmically by phi ([[Logarithmic phi|instead of dividing 2/1]]).
+The  [[phith root of phi|phith root of phi (ϕ<sup>ϕ<sup>-1</sup></sup>)]] has interesting applications as [[Metallic MOS]], and in particular the fractal-like possibilities of self-similar subdivision of musical scales within [[acoustic phi]].
-== Just approximation ==
-=== Temperament measures ===
-The following table shows [[TE temperament measures]] (RMS normalized by the rank) of 49et.
 {| class="wikitable center-all"
-! colspan="2" |
+|+ style="font-size: 105%;" | Direct approximation
-! 3-limit
+|-
-! 5-limit
+! Interval
-! 7-limit
+! Error (abs, [[Cent|¢]])
-! 11-limit
+! #\49
+|-
+| {{nowrap|ϕ / ϕ<sup>ϕ<sup>−1</sup></sup> {{=}} ϕ<sup>(2 − ϕ)</sup>}}
+| 0.155
+| 13
 |-
-! colspan="2" |Octave stretch (¢)
+| ϕ
-| -2.60
+| −0.437
-| -2.53
+| 34
-| -2.85
-| -2.97
 |-
-! rowspan="2" |Error
+| ϕ<sup>ϕ<sup>−1</sup></sup>
-! [[TE error|absolute]] (¢)
+| −0.592
+| 21
+|}
+Not until [[592edo|592]] do we find a better edo in terms of relative error on these two intervals (but whose edo-steps upon which these intervals are mapped are not based on the Fibonacci sequence, unlike 49edo).
+=== Music ===
+* [https://www.youtube.com/watch?v=vZyAm-D3nlk&ab_channel=Sevish Sevish - Star Nursery] uses a scale based on [[acoustic phi]] and ϕ<sup>ϕ<sup>−1</sup></sup>. 49edo provides a suitable approximation for this scale with the mode: 5 3 5 5 3 5 3 5
+== Regular temperament properties ==
+{| class="wikitable center-4 center-5 center-6"
+|-
+! rowspan="2" | [[Subgroup]]
+! rowspan="2" | [[Comma list]]
+! rowspan="2" | [[Mapping]]
+! rowspan="2" | Optimal<br>8ve stretch (¢)
+! colspan="2" | Tuning error
+|-
+! [[TE error|Absolute]] (¢)
+! [[TE simple badness|Relative]] (%)
+|-
+| 2.3
+| {{monzo| 78 -49 }}
+| {{mapping| 49 78 }}
+| −2.60
 | 2.60
+| 10.62
+|-
+| 2.3.5
+| 15625/15552, 20480/19683
+| {{mapping| 49 78 114 }}
+| −2.53
 | 2.12
+| 8.69
+|-
+| 2.3.5.7
+| 64/63, 245/243, 3125/3087
+| {{mapping| 49 78 114 138 }}
+| −2.85
 | 1.92
+| 7.87
+|-
+| 2.3.5.7.11
+| 64/63, 100/99, 245/243, 1331/1323
+| {{mapping| 49 78 114 138 170 }}
+| −2.97
 | 1.74
-|-
-! [[TE simple badness|relative]] (%)
-| 10.62
-| 8.69
-| 7.87
 | 7.11
 |}
-== Rank-2 temperaments ==
+=== Rank-2 temperaments ===
-{| class="wikitable center-all left-3"
+{| class="wikitable center-all left-5"
-! Periods<br>per octave
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
-! Generator
+|-
-! Temperaments
+! Periods<br>per 8ve
+! Generator*
+! Cents*
+! Associated<br>ratio*
+! Temperament
 |-
 | 1
 | 1\49
+| 24.5
+| 99/98
 | [[Sengagen]]
 |-
 | 1
 | 4\49
+| 98.0
+| 16/15
 | [[Passion]]
 |-
 | 1
 | 6\49
+| 146.9
+| 12/11
 | [[Bohpier]]
+|-
+| 1
+| 8\49
+| 195.9
+| 28/25
+| [[Didacus]]
 |-
 | 1
 | 11\49
+| 269.4
+| 7/6
 | [[Infraorwell]]
+|-
+| 1
+| 12\49
+| 293.9
+| 25/21
+| [[Kleiboh]]
 |-
 | 1
 | 13\49
+| 318.4
+| 6/5
 | [[Catalan]]
 |-
 | 1
 | 16\49
+| 391.8
+| 5/4
 | [[Magus]]
+|-
+| 1
+| 17\49
+| 416.3
+| 14/11
+| [[Sqrtphi]]
 |-
 | 1
 | 18\49
+| 440.8
+| 9/7
 | [[Clyde]]
 |-
 | 1
 | 19\49
+| 465.3
+| 55/36
 | [[Semisept]]
 |-
 | 1
 | 20\49
+| 489.8
+| 4/3
 | [[Superpyth]]
 |-
-| 7
+| rowspan="2" | 7
-| 20\49
+| rowspan="2" | 20\49<br />(1\49)
-| [[Sevond]]/[[seville]]
+| rowspan="2" | 489.8<br />(24.5)
+| 4/3<br />(250/243)
+| [[Sevond]] (49)
+|-
+| 4/3<br />(25/24)
+| style="text-align: left;" | [[Seville]] (49c)
 |}
+<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
+== Octave stretch or compression ==
+edo's [[prime]]s 3, 5, 7 and 11 are all tuned sharp, so 49edo can benefit from [[octave shrinking]]. Some compressed-octave tunings of 49edo include (least to most compression): [[ed12|176ed12]], [[ed5|114ed5]], [[zpi|233zpi]], [[ed6|127ed6]], [[ed7|138ed7]] and [[78edt]].
+=== Nonoctave temperament ===
+The TE-optimized [[Triple BP|triple Bohlen–Pierce scale]] is obtained by taking every second degree of 49edo with the octave compressed by 3.861 cents to 1196.139 cents. It realizes the Tenney–Euclidean regular temperament on the 3.5.7.11.13 subgroup mapped as [⟨78 114 138 170 182]]. Under this compression, the primes map to the 49fgh val in the 23-limit.
+== Scales ==
+=== MOS scales ===
+{{main|List of MOS scales in 49edo}}
+* Bohpier[8]: 6 6 6 6 7 6 6 6
+* Catalan[7]: 3 10 3 10 3 10 10 (vaugely diminished-like)
+* Catalan[11]: 3 7 3 3 7 3 3 7 3 3 7
+* Catalan[19]: 3 3 1 3 3 3 3 1 3 3 3 1 3 3 3 3 1 3 3
+* Clyde[5]: 5 13 5 13 13 (mysterious, adventurous)
+* Didacus[6]: 8 8 8 8 8 9 (like the whole tone scale)
+* Didacus[13]: 1 7 1 7 1 7 1 7 1 7 1 7 1
+* Infraorwell[5]: 11 11 5 11 11
+* Infraorwell[22]: 1 4 1 4 1 1 4 1 4 1 1 4 1 4 1 4 1 1 4 1 4 1
+* Kleiboh[5]: 12 12 12 12 1
+* Kleiboh[13]: 1 10 1 1 10 1 1 1 10 1 1 10 1
+* Magus[7]: 1 15 1 15 1 15 1 (vaguely augmented-like)
+* Passion[12]: 4 4 4 4 4 4 5 4 4 4 4 4 (like [[12edo]])
+* Passion[23]: 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1
+* Sevond[21]/Seville[21]: 1 5 1 1 5 1 1 5 1 1 5 1 1 5 1 1 5 1 1 5 1
+* Superpyth[5]: 11 9 9 11 9 (in between minor pentatonic and [[equipentatonic]])
+* Superpyth[7]: 9 2 9 9 9 2 9 (Dorian mode; rotate for other modes)
+* Superpyth[12]: 2 7 2 7 2 2 7 2 7 2 7 2 (same melodic shape as [[12edo]] but much more [[xenharmonic]] harmonies)
+* Superpyth[27]: 2 2 1 2 2 2 2 2 1 2 2 2 2 1 2 2 2 2 1 2 2 2 2 2 1 2 2
+=== Other scales ===
+* [[6ed7/3#6ed7/3+7edo scale|The 6ed7/3+7edo scale]] ''(non-octave-repeating)''
+== Instruments ==
+=== Lumatone ===
+* [[Lumatone mapping for 49edo]]
+=== Skip fretting ===
+'''Skip fretting system 49 3 7''' is a [[skip fretting]] system for [[49edo]]. All examples are for 5-string bass.
+; Harmonics
+/1: string 2 open
+/1: not easily accessible
+/2: string 4 fret 5 and string 1 fret 12
+/4: string 3 fret 3
+/4: string 3 fret 11
+/8: string 3 fret 5
+== Music ==
+=== Modern renderings ===
+; {{W|The Cure}}
+* [https://www.youtube.com/watch?v=GHslu-ZWspk ''Boys Don't Cry''] (1979) – Lumatone cover by [[YoVariable]] (2025)
+=== 21st century ===
+; [[Bryan Deister]]
+* [https://www.youtube.com/watch?v=7pK-JcIrd18 Deltarune – ''Man'' (cover)] (2023)
+* [https://www.youtube.com/shorts/V8t7MyP2Nuo ''microtonal improv in 49edo''] (2024)
+* [https://www.youtube.com/shorts/zb1Z6o-Uvuw ''weathergirl - FLAVOR FOLEY (microtonal cover in 49edo)''] (2025)
+* [https://www.youtube.com/shorts/73PfAAWubVs ''I'm Your Captain Now (The Ancients) - The Recovery System (microtonal cover in 49edo)''] (2026) {{todo|research|comment=Identify the original composers.}}
+* [https://www.youtube.com/shorts/34w7euOF-Ss ''49edo improv''] (2026)
+* [https://www.youtube.com/shorts/_yNrDI6nS1I ''49edo riff''] (2026)
+* [https://www.youtube.com/shorts/BcBtD3nuEQs ''49edo groove''] (2026)
+* [https://www.youtube.com/shorts/VmUIxWb8NCY ''49edo prelude''] (2026)
+; [[Mercury Amalgam]]
+* [https://www.youtube.com/watch?v=c_kzhcMMHWM&pp=ygUFNDllZG8%3D ''Wrong Generation''] (2022 demo version)
+; [[Cam Taylor]]
+* [https://www.youtube.com/watch?v=fns6688IRpg ''49-equal: 7-equal meets superpyth''] (2023)
-[[Category:Theory]]
-[[Category:Equal divisions of the octave]]
-[[Category:49edo]]
-[[Category:Superpyth]]
 [[Category:Archytas]]
 [[Category:Ares]]
+[[Category:Listen]]
+[[Category:Superpyth]]

49edo: Difference between revisions

Latest revision as of 10:43, 25 April 2026

Theory

Harmonics

Subsets and supersets

Intervals

Notation

Ups and downs notation

Sagittal notation

Evo flavor

Revo flavor

Approximation to JI

Interval mappings

Zeta peaks

Approximation to irrational intervals

Acoustic ϕ and ϕϕ−1

Music

Regular temperament properties

Rank-2 temperaments

Octave stretch or compression

Nonoctave temperament

Scales

MOS scales

Other scales

Instruments

Lumatone

Skip fretting

Music

Modern renderings

21st century

Navigation menu

Search

Acoustic ϕ and ϕ^ϕ⁻¹