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 ===

46edo can be treated as two circles of [[23edo]] separated by an interval of 26.087 cents.

== Intervals ==

! 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>

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

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

| [a, b, 0, 1>

| [a, b>, b &lt; −1

| 32/27, 16/9

| gu

| [a, b, −1>

| 6/5, 9/5

| ilo

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

| 11/9, 11/6

| [a, b, 0, 0, −1>

| 12/11, 18/11

| 5/4, 5/3

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

| 9/8, 27/16

| [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"

| C ^E G

| C^

| C upmajor or C up

|}

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

== Notation ==

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).

Another notation uses [[Alternative symbols for ups and downs notation#Sharp-5|alternative ups and downs]]. Here, this can be done using sharps and flats with arrows, borrowed from extended [[Helmholtz–Ellis notation]]:

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

File:46-EDO_Evo_Sagittal.svg

rect 80 0 300 50 [[Sagittal_notation]]

rect 300 0 511 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]

rect 20 80 120 106 [[81/80]]

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

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

</imagemap>

==== Revo flavor ====

File:46-EDO_Revo_Sagittal.svg

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

rect 20 80 120 106 [[81/80]]

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

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

</imagemap>