46edo: Difference between revisions

| de = 46-EDO

== Theory ==

In the opinion of some, 46edo is the first equal division to deal adequately with the [[13-limit]], though others award that distinction to [[41edo]] or [[53edo]]. 46edo's fifth is slightly sharp of just, which some people (e.g. [[Margo Schulter]]) prefer, sometimes strongly, over both the [[3/2|just fifth]] and fifths of temperaments with flat fifths, such as meantone. Many say that sharp fifths give a characteristic bright sound to 5-limit triads, and consider the sound of meantone triads to be more mellow in comparison. 46edo is not a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak edo]], but it is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta gap edo]]. It is distinctly [[consistent]] in the [[9-odd-limit]], and it is consistent to the [[13-odd-limit]] or the no-15 no-19 [[23-odd-limit]].

46edo is also notable for being the smallest equal division to approximate harmonics 3, 5, 7, 11, and 13 with less than 25% [[relative interval error]].

[[Rank-2 temperament]]s it [[support]]s include [[sensi]], [[valentine]], [[shrutar]], [[rodan]], [[leapday]], and [[unidec]]. The [[11-odd-limit]] [[minimax tuning]] for valentine, (11/7)^1/10, is only 0.01 cents flat of 3\46 octaves.  

[[Magic22 as srutis #Shrutar.5B22.5D_as_srutis|Shrutar22 as srutis]] describes a possible use of 46edo for [[Indian]] music.

=== Prime harmonics ===

{{Harmonics in equal|46|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 46edo (continued)}}

=== Subsets and supersets ===

{| class="wikitable center-all right-2 left-3 left-4 left-7"

! #

! Approximate ratios<ref name="interval ratios" group="note">{{sg|limit=2.3.5.7.11.13.17.23&nbsp;subgroup}} However, ratios of 15 are not included here, as except for 15/8 and 16/15 themselves 46edo has intervals involving the 15th harmonic poorly approximated in general. This is because, while the 3rd and 5th harmonics are sharp and their deviations from just intonation add up, 7, 11, and 13 are all tuned flat, making the difference even larger. This prevents it from being [[consistent]] in the [[15-odd-limit]], as there is a discrepancy approximating [[15/13]] and [[26/15]]&mdash;9\46 is closer to 15/13 by a hair, but 10\46 represents the difference between 46edo's 15/8 and 13/8 and is more likely to appear in chords actually functioning as 15/13.</ref>

! colspan="3" | [[Ups and downs notation]]

([[Enharmonic unisons in ups and downs notation|EUs]]: v⁵A1 and ^^d2)

! colspan="3" | [[SKULO interval names|SKULO notation]] (K or S = 1, U = 2)

! colspan="2" | [[Solfege]]

| vvE

| ruh<ref name="intervals3" group="note">/ʌ/ as in s'''u'''bmajor</ref>

| [[13/11]], [[20/17]]

|-

| [[11/9]], [[17/14]], [[28/23]]

|-

| [[16/13]], [[26/21]], [[21/17]]

| muh<ref name="intervals3" group="note" />

| 391.3

| [[5/4]]

| 16

| 417.4

| [[14/11]], [[23/18]]

| major 3rd

| M3

| F#

| major 3rd

| M3

| F#

| ma

| maa

| 17

| 443.5

| [[9/7]], [[13/10]], [[22/17]]

| upmajor 3rd

| ^M3

| ^F#

| supermajor 3rd

| SM3

| SF#

| mu

| mo

| 18

| 469.6

| [[21/16]], [[17/13]]

| down 4th

| v4

| vG

| sub 4th

| s4

| sG

| fo

| fe

| 19

| [[4/3]]

| perfect 4th

| P4

| G

| perfect 4th

| P4

| G

| fa

| fa

| 20

| 521.7

| [[27/20]], [[23/17]]

| ^4

| ^G

| comma-wide 4th

| K4

| fu

| fih

| 21

| 547.8

| [[11/8]]

| dup 4th

| ^^4

| ^^G

| uber 4th, sub diminished 5th

| U4, sd5

| UG, sAb

| fi/sho

| fu

| 22

| 573.9

| [[7/5]], [[18/13]], [[32/23]]

| dudaug 4th, <br>dim 5th

| vvA4, d5

| vvG#, Ab

| classic augmented 4th, diminished 5th

| kkA4, d5

| kkG#, Ab

| pi/sha

| fi

| 23

| 600.0

| [[17/12]], [[24/17]]

| downaug 4th, updim 5th

| vA4, ^d5

| vG#, ^Ab

| comma-narrow aug 4th, <br>comma-wide dim 5th

| kA4, Kd5

| kG#, KAb

| po/shu

| seh

| 24

| 626.1

| [[10/7]], [[13/9]], [[23/16]]

| aug 4th, dupdim 5th

| A4, ^^d5

| G#, ^^Ab

| augmented 4th, <br>classic diminished 5th

| A4, KKd5

| G#, KKAb

| pa/shi

| se

| 25

| 652.2

| [[16/11]]

| dud 5th

| vv5

| vvA

| super augmented 4th, <br>unter 5th

| SA4, u5

| SG#, uA

| pu/si

| su

| 26

| 678.3

| [[40/27]], [[34/23]]

| down 5th

| vA

| comma-narrow 5th

| kA

| so

| sih

| 27

| 704.3

| [[3/2]]

| perfect 5th

| P5

| A

| perfect 5th

| P5

| A

| sa

| sol

| 28

| 730.4

| [[32/21]], [[26/17]]

| up 5th

| ^5

| ^A

| super 5th

| S5

| SA

| su

| si

| 29

| 756.5

| [[14/9]], [[20/13]], [[17/11]]

| vBb

| subminor 6th

| sm6

| sBb

| flo

| lo

| 30

| 782.6

| [[11/7]]

| minor 6th

| m6

| Bb

| minor 6th

| m6

| Bb

| fla

| leh

| 31

| 808.7

| [[8/5]]

| upminor 6th

| ^m6

| ^Bb

| classic minor 6th

| Km6

| KBb

| flu

| le

| 32

| 834.8

| [[13/8]], [[21/13]], [[34/21]]

| dupminor 6th

| ^^m6

| ^^Bb

| lesser neutral 6th

| n6

| UBb

| fli

| lu<ref name="intervals2" group="note" />

| 33

| [[18/11]], [[28/17]], [[23/14]]

| dudmajor 6th

| vvM6

| vvB

| greater neutral 6th

| N6

| uB

| li

| luh<ref name="intervals3" group="note" />

| 34

| 887.0

| [[5/3]]

| downmajor 6th

| vM6

| vB

| classic major 6th

| kM6

| kB

| lo

| la

| 35

| 913.0

| [[22/13]], [[17/10]]

| major 6th

| M6

| B

| major 6th

| M6

| B

| la

| laa

| 36

| 939.1

| [[12/7]]

| upmajor 6th

| ^M6

| ^B

| supermajor 6th

| SM6

| SB

| lu

| li

| 37

| 965.2

| [[7/4]], [[40/23]]

| downminor 7th

| vm7

| vC

| sm7

| tho

| ta

| 38

| 991.3

| [[16/9]], [[23/13]]

| minor 7th

| m7

| m7

| C

| tha

| teh

| 39

| 1017.4

| [[9/5]]

| upminor 7th

| ^m7

| ^C

| classic/comma-wide minor 7th

| Km7

| KC

| thu

| te

| 40

| 1043.5

| [[11/6]], [[20/11]], [[42/23]]

| dupminor 7th

| ^^m7

| ^^C

| lesser neutral 7th, sub diminished 8ve

| n7, sd8

| UC, sDb

| thi

| tu<ref name="intervals2" group="note" />

| 41

| 1069.6

| [[24/13]], [[13/7]], [[28/15]]

| dudmajor 7th

| vvM7

| vvC#

| greater neutral 7th,

| N7, d8

| uC#, Db

| ti

| tuh<ref name="intervals3" group="note" />

| 42

| 1095.7

| [[15/8]], [[32/17]], [[17/9]]

| downmajor 7th

| vM7

| vC#

| classic major 7th,

comma-wide dim 8ve

| kM7, Kd8

| kC#, KDb

| to

| ti

| 43

| 1121.7

| [[48/25]], [[40/21]], [[21/11]], [[23/12]], [[44/23]]

| major 7th

| M7

| C#

| major 7th, <br>classic diminished 8ve

| M7, KKd8

| C#, KKDb

| ta

| taa

| 44

| 1147.8

| [[27/14]], [[35/18]], [[64/33]]

| upmajor 7th

| ^M7

| ^C#

| supermajor 7th, unter 8ve

| SM7, u8

| SC#, uD

| tu

| to

| 45

| 1173.9

| [[160/81]], [[63/32]], [[96/49]]

| v8

| vD

| comma-narrow 8ve, sub 8ve

| k8/s8

| kD, sD

| da

| 46

| [[2/1]]

| perfect 8ve

| P8

| D

| perfect 8ve

| P8

| D

| da

| do

|}

 

=== Interval quality and chord names in color notation ===

Combining ups and downs notation with [[color notation]], qualities can be loosely associated with colors:

 

{| class="wikitable center-all"

| downminor

| zo

| [a, b, 0, 1>

| 7/6, 7/4

| minor

| fourthward wa

| [a, b>, b &lt; −1

| 32/27, 16/9

| upminor

| gu

| [a, b, −1>

| 6/5, 9/5

| dupminor

| ilo

| [a, b, 0, 0, 1>

| 11/9, 11/6

| dudmajor

| lu

| [a, b, 0, 0, −1>

| 12/11, 18/11

| downmajor

| yo

| [a, b, 1>

| 5/4, 5/3

| major

| fifthward wa

| [a, b>, b &gt; 1

| 9/8, 27/16

| upmajor

| ru

| [a, b, 0, −1>

| 9/7, 12/7

|}

All 46edo chords can be named using ups and downs. Alterations are always enclosed in parentheses, additions never are. Ups or downs immediately after the chord root affect the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13). Here are the zo, gu, ilo, lu, yo and ru triads:

 

{| class="wikitable center-all"

| zo

| 6:7:9

| 0–10–27

| C vEb G

| Cvm

| C downminor

| gu

| 10:12:15

| 0–12–27

| C ^Eb G

| C^m

| C upminor

| ilo

| 18:22:27

| 0–13–27

| C ^^Eb G

| C^^m

| C dupminor

| lu

| 22:27:33

| 0–14–27

| C vvE G

| Cvv

| C dudmajor or C dud

| yo

| 4:5:6

| 0–15–27

| C vE G

| Cv

| C downmajor or C down

| ru

| 14:18:21

| 0–17–27

| C ^E G

| C^

| C upmajor or C up

For a more complete list, see [[Ups and downs notation #Chords and chord progressions]].

 

46edo can be notated with [[ups and downs]], spoken as up, dup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, dupflat etc. Note that dudsharp is equivalent to trup (triple-up) and dupflat is equivalent to trud (triple-down).

 

This notation uses the same sagittal sequence as [[39edo #Sagittal notation|39edo]].

 

 

rect 300 0 557 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]

rect 120 80 240 106 [[33/32]]

default [[File:46-EDO_Revo_Sagittal.svg]]

 

[[File:46ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 15-limit intervals approximated in 46edo]]

{{Q-odd-limit intervals|46|17}}

{| class="wikitable center-1 center-2 center-3"

! [[Interval]] !! [[Closing error|Closing<br />Error]] !! [[Circle #Definitions|Consistency]] !! 1\1 !! 1\2

| [[68/65]] || 25.9% || Normal || [[Valentine]] || [[Semivalentine]]

| [[10/9]] || 36.1% || Normal || [[Mitonic]] || [[Unidec]], [[hendec]]

| [[31/24]] || 70.2% || Weak || [[Sensible]], [[sensi]] add-31 || [[Bison]] add-31, [[bisensi]] add-31

|}

 

{| class="wikitable center-all left-4"

|+ style="font-size: 105%;" | 23-note circles by gen. with related half-octave temperaments

! [[Interval]] !! [[Closing error|Closing<br />Error]] !! [[Circle #Definitions|Consistency]] !! Temperaments

| [[17/16]] || 53.5% || Weak || [[Diaschismic]]

| [[23/21]] || 85.7% || Weak || [[Bison]]

| [[44/39]] || 12.3% || Super-strong || [[Abigail]]

| [[21/17]] || 53.6% || Weak || ?

|-

| [[14/11]] || 10.2% || Super-strong || ?

|-

| [[21/16]] || 107% || Sub-weak || ?

 

{| class="wikitable center-4 center-5 center-6"

| 2.3.5

| 2048/2025, 78732/78125

| {{mapping| 46 73 107 }}

| −1.219

| 0.90

| 3.45

| 2.3.5.7

| 126/125, 245/243, 1029/1024

| {{mapping| 46 73 107 129 }}

| −0.595

| 1.34

| 5.12

| 2.3.5.7.11

| 121/120, 126/125, 176/175, 245/243

| {{mapping| 46 73 107 129 159 }}

| −0.274

| 1.36

| 5.20

| 2.3.5.7.11.13

| 91/90, 121/120, 169/168, 176/175, 245/243

| {{mapping| 46 73 107 129 159 170 }}

| +0.030

| 1.41

| 5.42

| 2.3.5.7.11.13.17

| 91/90, 121/120, 154/153, 169/168, 176/175, 245/243

| {{mapping| 46 73 107 129 159 170 188 }}

| +0.047

| 1.31

| 5.02

| 2.3.5.7.11.13.17.23

| 91/90, 121/120, 154/153, 169/168, 176/175, 208/207, 231/230

| {{mapping| 46 73 107 129 159 170 188 208 }}

| +0.101

| 1.23

| 4.72

* 46et is lower in relative error than any previous equal temperaments in the 17-, 19-, 23-limit, and others. The next equal temperaments doing better in the aforementioned subgroups are 72, 72, 94, respectively. 46et is even more prominent in the no-19 23-limit, and the next equal temperament doing better in this subgroup is 140.

=== Commas ===

This is a partial list of the [[commas]] that 46et [[tempering out|tempers out]] with its [[patent val]], {{val| 24 38 56 67 83 89 }}.  

{| class="commatable wikitable center-1 center-2 right-4 center-5"

| 5

| <abbr title="50331648/48828125">(16 digits)</abbr>

| 52.50

| [[Magus comma]]

| 5

| <abbr title="1990656/1953125">(14 digits)</abbr>

| [[Valentine comma]]

| 5

| [[2048/2025]]

| {{monzo| 11 -4 -2 }}

| 19.55

| 5

| [[78732/78125]]

| {{monzo| 2 9 -7 }}

| 13.40

| Sepgu

| Sensipent comma

| 5

| <abbr title="1600000/1594323">(14 digits)</abbr>

| {{monzo| 9 -13 5 }}

| [[Amity comma]]

| 7

| [[686/675]]

| {{monzo| 1 -3 -2 3 }}

| 27.99

| Trizo-agugu

| Senga

| 7

| [[245/243]]

| {{monzo| 0 -5 1 2 }}

| 14.19

| Zozoyo

| Sensamagic comma

| 7

| [[126/125]]

| {{monzo| 1 2 -3 1 }}

| 13.80

| 7

| [[1029/1024]]

| {{monzo| -10 1 0 3 }}

| 8.43

| 7

| [[5120/5103]]

| {{monzo| 10 -6 1 -1 }}

| 5.76

| 7

| <abbr title="2147483648/2144153025">(20 digits)</abbr>

| {{monzo| 31 -6 -2 -6 }}

| 2.69

| Sasa-tribiru-agugu

| [[Pessoalisma]]

| 7

| [[4375/4374]]

| {{monzo| -1 -7 4 1 }}

| 0.40

| 11

| [[121/120]]

| {{monzo|-3 -1 -1 0 2 }}

| 14.37