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 [[zeta peak edo]], but it is a [[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 edo to approximate odd harmonics 1–13 with less than 25% [[relative interval error|relative error]].

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

{{Harmonics in equal|46|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 46edo (continued)}}

=== As a tuning of other temperaments ===

46edo is a superb tuning for [[sensi]] and [[leapday]]. It also [[support]]s but tunes less optimally [[valentine]], [[shrutar]], [[rodan]], and [[unidec]]. The [[11-odd-limit]] [[minimax tuning]] for valentine, (11/7)^1/10, is only 0.01 cents flat of 3\46, but [[77edo]] has a lower average error overall.  

=== Subsets and supersets ===

46edo can be treated as two circles of [[23edo]] separated by an interval of 26.087 cents. Supersets [[414edo]] and [[460edo]] are also notable for their high consistency.

! Approximate ratios<ref name="interval ratios" group="note">{{safesubst:Subgroup note|limit=2.3.5.7.11.13.17.23-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>

! [[Kite's ups and downs notation|Ups and downs notation]]

|-

| [[1/1]]

| {{UDnote|step=0}}

|-

| 1

| 26.1

| [[49/48]], [[64/63]], [[81/80]]

| {{UDnote|step=1}}

| {{UDnote|step=2}}

| 3

| 78.3

| [[21/20]], [[22/21]], [[24/23]], [[23/22]], [[25/24]]

| {{UDnote|step=3}}

| [[16/15]], [[17/16]], [[18/17]]

| {{UDnote|step=4}}

|-

| 5

| 130.4

| [[13/12]], [[14/13]], [[15/14]]

| {{UDnote|step=5}}

| 156.5

| [[11/10]], [[12/11]], [[23/21]]

| {{UDnote|step=6}}

|-

| 182.6

| [[10/9]]

| {{UDnote|step=7}}

| 208.7

| [[9/8]]

| {{UDnote|step=8}}

| [[8/7]], [[23/20]]

| {{UDnote|step=9}}

|-

| 10

| 260.9

| [[7/6]]

| {{UDnote|step=10}}

| 287.0

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

| {{UDnote|step=11}}

| {{UDnote|step=12}}

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

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

| {{UDnote|step=14}}

| {{UDnote|step=15}}

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

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

| {{UDnote|step=17}}

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

| {{UDnote|step=18}}

| 495.7

| [[4/3]]

| {{UDnote|step=19}}

| 521.7

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

| {{UDnote|step=20}}

| {{UDnote|step=21}}

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

| 600.0

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

| {{UDnote|step=23}}

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

| {{UDnote|step=24}}

| {{UDnote|step=25}}

|-

| 678.3

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

| {{UDnote|step=26}}

| 704.3

| [[3/2]]

| {{UDnote|step=27}}

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

| {{UDnote|step=28}}

|-

| 29

| 756.5

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

| {{UDnote|step=29}}

| 782.6

| [[11/7]]

| {{UDnote|step=30}}

| 808.7

| [[8/5]]

| {{UDnote|step=31}}

|-

| 32

| 834.8

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

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

| {{UDnote|step=33}}

|-

| {{UDnote|step=34}}

|-

| 913.0

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

|-

| {{UDnote|step=36}}

| {{UDnote|step=37}}

| 991.3

| {{UDnote|step=38}}

|-

| 1017.4

| {{UDnote|step=39}}

|-

| 1043.5

| {{UDnote|step=40}}

|-

| {{UDnote|step=41}}

|-

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

|-

| {{UDnote|step=43}}

| 1147.8

| {{UDnote|step=44}}

| 1173.9

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

|-

| [[2/1]]

=== Proposed interval names and solfèges ===

{| class="wikitable center-all right-2 left-2 left-3 left-6 mw-collapsible mw-collapsed"

|+ style="white-space: nowrap;" | Table of proposed interval names and solfèges

|-

! Cents

! colspan="3" | [[Kite's ups and downs notation|Ups and downs notation]]<br>([[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" | [[Solfège]]s

| 0

| 0.0

| perfect unison

| P1

| D

| Perfect unison

| P1

| D

| da

| do

|-

| 1

| up unison

| ^1

| ^D

| comma-wide unison, <br>super unison

| K1, S1

| KD, SD

| du

| di

|-

| downminor 2nd

| vm2

| vEb

| sm2, U1

| sEb, UD

| fro

| ro

|-

| 3

| 78.3

| minor 2nd

| m2

| Eb

| Eb, kkD#

| fra

| rih

|-

| upminor 2nd

| ^m2

| KEb, kD#

| fru

| ra

|-

| 5

| 130.4

| dupminor 2nd

| ^^m2

| n2, A1

| UEb, D#

| fri

| ru<ref name="intervals2" group="note">/u/ as in s'''u'''praminor</ref>

|-

| 6

| vvM2

| vvE

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

|-

| 7

| 182.6

| downmajor 2nd

| vM2

| vE

| classic/comma-narrow major 2nd

| kM2

| kE

| ro

| reh

|-

| 8

| M2

| E

| 9

| 234.8

| SE

| ru

| ri

|-

| 10

| 260.9

| downminor 3rd

| vm3

| vF

| subminor 3rd

| sF

| no

|-

| 287.0

| minor 3rd

| F

| na

| meh

|-

| 12

| 313.0

| upminor 3rd

| ^m3

| ^F

| classic minor 3rd

| Km3

| KF

| nu

| me

|-

| 13

| ^^F

| lesser neutral 3rd

|-

| 14

| 365.2

| dudmajor 3rd

| vvM3

| vvF#

| greater neutral 3rd

| N3

| uF#

| mi

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

|-

| 15

| vM3

| vF#

|-

| F#

| major 3rd

| mo

|-

| 18

| 469.6

| down 4th

| v4

| vG

| sub 4th

| s4

| sG

| fo

| fe

|-

| 19

| 495.7

| P4

| G

| fa

| fa

|-

| 20

| 521.7

| up 4th

| ^4

| ^G

| comma-wide 4th

| K4

| KG

| fu

| fih

|-

| 21

| 547.8

| ^^4

| ^^G

| UG, sAb

| fi/sho

| fu

|-

| 22

| 573.9

| dudaug 4th, <br>dim 5th

| vvA4, d5

| vvG#, Ab

| classic augmented 4th, diminished 5th

| kkA4, d5

| kkG#, Ab

| pi/sha

| fi

|-

| 23

| 600.0

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

| kA4, Kd5

| kG#, KAb

| 626.1

| aug 4th, dupdim 5th

| A4, ^^d5

| G#, ^^Ab

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

| A4, KKd5

| G#, KKAb

| pa/shi

| se

|-

| 25

| 652.2

| dud 5th

| vv5

| vvA

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

| SA4, u5

| SG#, uA

| pu/si

| su

|-

| 26

| 678.3

| down 5th

| v5

| vA

| comma-narrow 5th

| k5

| so

| sih

|-

| P5

| A

| ^5

| super 5th

| S5

| SA

| su

| si

|-

| 29

| 756.5

| downminor 6th

| vm6

| vBb

| subminor 6th

| sm6

| sBb

| flo

| lo

|-

| 782.6

| minor 6th

| leh

|-

| 31

| 808.7

| upminor 6th

| ^m6

| ^Bb

| classic minor 6th

| Km6

| KBb

| flu

| le

|-

| 32

| 834.8

| ^^m6

| ^^Bb

| lesser neutral 6th

| UBb

| fli

| vvM6

| vvB

| greater neutral 6th

| N6

| uB

| li

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

|-

| 34

| 887.0

| downmajor 6th

| vB

| classic major 6th

| kM6

| lo

| la

| M6

| B

| major 6th

| M6

| B

| la

| 36

| 939.1

| upmajor 6th

| ^M6

| supermajor 6th

| SM6

| SB

| lu

|-

| downminor 7th

| vm7

|-

| 38

| 991.3

| minor 7th

| m7

| C

| minor 7th

| m7

| C

| tha

| teh

|-

| 39

| 1017.4

| upminor 7th

| ^m7

| classic/comma-wide minor 7th

| KC

| thu

| 1043.5

| ^^m7

| ^^C

| lesser neutral 7th, sub diminished 8ve

| n7, sd8

| UC, sDb

| thi

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

|-

| 41

| 1069.6

| dudmajor 7th

| vvM7

| vvC#

| greater neutral 7th,

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

|-

comma-wide dim 8ve

| kC#, KDb

| to

| 1121.7

| M7, KKd8

| C#, KKDb

| ta

| taa

|-

| 44

| 1147.8

| supermajor 7th, unter 8ve

| SM7, u8

| SC#, uD

| tu

| to

|-

| 45

| v8

| comma-narrow 8ve, sub 8ve

| k8/s8

| kD, sD

| 1200.0

| perfect 8ve

| P8

| D

| perfect 8ve

| P8

| D

| da

| do

|}

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

| 7/6, 7/4

|-

| 32/27, 16/9

| 11/9, 11/6

| 12/11, 18/11

| 5/4, 5/3

| {{Monzo| ''a'' ''b'' }}, ''b'' > 1

| 9/8, 27/16

| 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:

| 6:7:9

| 10:12:15

| 18:22:27

| 22:27:33

| 4:5:6

| ru

| 14:18:21

|}

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

== Notation ==

[[Stein–Zimmermann–Gould notation]] uses sharps and flats with arrows:

=== Kite's ups and downs notation ===

46edo can also be notated with [[Kite's ups and downs notation|Kite's ups and downs]], spoken as up, dup, dudsharp, downsharp, sharp, upsharp, etc., and likewise 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]].

==== Evo flavor ====

{{Sagittal chart|Evo}}

==== Revo flavor ====