Xenharmonic Wiki

46edo: Difference between revisions

← Older editNewer edit →
VisualWikitext
Yourmusic Productions (talk | contribs)
autopatrolled, emailconfirmed
891 edits
m Add link to name
General cleanup; relative errors
Line 1: Line 1:
__FORCETOC__
__FORCETOC__
=Theory=
 
The 46 equal temperament, often abbreviated <b>46-tET</b>, <b>46-EDO</b>, or <b>46-ET</b>, is the scale derived by dividing the [[Octave|octave]] into 46 equally-sized steps. Each step represents a frequency ratio of 26.087 [[cent|cent]]s, an interval close in size to [[66/65|66/65]], the interval from [[13/11|13/11]] to [[6/5|6/5]].
== Theory ==
The 46 equal temperament, often abbreviated <b>46-tET</b>, <b>46-EDO</b>, or <b>46-ET</b>, is the scale derived by dividing the [[octave]] into 46 equally-sized steps. Each step represents a frequency ratio of 26.087 [[cent|cents]], an interval close in size to [[66/65]], the interval from [[13/11]] to [[6/5]].


46et tempers out 507/500, 91/90, 686/675, 2048/2025, 121/120, 245/243, 126/125, 169/168, 176/175, 896/891, 196/195, 1029/1024, 5120/5103, 385/384, and 441/440 among other intervals, with various consequences. [[Rank_two_temperaments|Rank two temperaments]] it supports include sensi, valentine, shrutar, rodan, leapday and unidec. The [[11-limit|11-limit]] [[Target_tunings|minimax]] tuning for [[Starling_family|valentine temperament]], (11/7)^(1/10), is only 0.01 cents flat of 3/46 octaves. In the opinion of some, 46et is the first equal division to deal adequately with the [[13-limit|13-limit]], though others award that distinction to [[41edo|41edo]]. In fact, while 41 is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta integral edo]] but not a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta gap edo]], 46 is zeta gap but not zeta integral.
46et tempers out 507/500, 91/90, 686/675, 2048/2025, 121/120, 245/243, 126/125, 169/168, 176/175, 896/891, 196/195, 1029/1024, 5120/5103, 385/384, and 441/440 among other intervals, with various consequences. [[Rank_two_temperaments|Rank two temperaments]] it supports include sensi, valentine, shrutar, rodan, leapday and unidec. The [[11-limit|11-limit]] [[Target_tunings|minimax]] tuning for [[Starling_family|valentine temperament]], (11/7)^(1/10), is only 0.01 cents flat of 3/46 octaves. In the opinion of some, 46et is the first equal division to deal adequately with the [[13-limit|13-limit]], though others award that distinction to [[41edo|41edo]]. In fact, while 41 is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta integral edo]] but not a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta gap edo]], 46 is zeta gap but not zeta integral.


The fifth of 46 equal is 2.39 cents sharp, which some people (eg, [https://en.xen.wiki/w/Margo_Schulter Margo Schulter]) prefer, sometimes strongly, over both the [https://en.xen.wiki/w/3/2 just fifth] and fifths of temperaments with flat fifths, such as meantone. It gives a characteristic bright sound to triads, distinct from the mellowness of a meantone triad.
The fifth of 46 equal is 2.39 cents sharp, which some people (eg, [https://en.xen.wiki/w/Margo_Schulter Margo Schulter]) prefer, sometimes strongly, over both the [[3/2|just fifth]] and fifths of temperaments with flat fifths, such as meantone. It gives a characteristic bright sound to triads, distinct from the mellowness of a meantone triad.


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


[[Magic22_as_srutis#shrutar22assrutis|Shrutar22 as srutis]] describes a possible use of 46edo for [[Indian|Indian]] music.
[[Magic22_as_srutis#shrutar22assrutis|Shrutar22 as srutis]] describes a possible use of 46edo for [[Indian]] music.


=Intervals=
== Intervals ==


{| class="wikitable"
{| class="wikitable center-all right-2 left-3 left-4"
|-
|-
! |
!  
! | Cents
! Cents
! | Approximate Ratios<nowiki>*</nowiki>
! Approximate Ratios<nowiki>*</nowiki>
! colspan="3" | [[Ups_and_Downs_Notation|Ups and Downs Notation]]
! colspan="3" | [[Ups and Downs Notation]]
! | Solfege
! Solfege
|-
|-
| style="text-align:center;" | 0
| 0
| 0.000
| 0.000
| | [[1/1]]
| [[1/1]]
| style="text-align:center;" | perfect unison
| perfect unison
| style="text-align:center;" | P1
| P1
| style="text-align:center;" | D
| D
| style="text-align:center;" | do
| do
|-
|-
| style="text-align:center;" | 1
| 1
| | 26.087
| 26.087
| |[[81/80]], [[64/63]]
| [[81/80]], [[64/63]]
| style="text-align:center;" | up unison
| up unison
| style="text-align:center;" | ^1
| ^1
| style="text-align:center;" | ^D
| ^D
| style="text-align:center;" | di
| di
|-
|-
| style="text-align:center;" | 2
| 2
| | 52.174
| 52.174
| | [[28/27]], [[33/32]]
| [[28/27]], [[33/32]]
| style="text-align:center;" | downminor 2nd
| downminor 2nd
| style="text-align:center;" | vm2
| vm2
| style="text-align:center;" | vEb
| vEb
| style="text-align:center;" | ro
| ro
|-
|-
| style="text-align:center;" | 3
| 3
| | 78.261
| 78.261
| | [[25/24]], [[21/20]], [[22/21]], [[24/23]], [[23/22]]
| [[25/24]], [[21/20]], [[22/21]], [[24/23]], [[23/22]]
| style="text-align:center;" | minor 2nd
| minor 2nd
| style="text-align:center;" | m2
| m2
| style="text-align:center;" | Eb
| Eb
| style="text-align:center;" | rih
| rih
|-
|-
| style="text-align:center;" | 4
| 4
| | 104.348
| 104.348
| | [[16/15]], [[17/16]], [[18/17]]
| [[16/15]], [[17/16]], [[18/17]]
| style="text-align:center;" | upminor 2nd
| upminor 2nd
| style="text-align:center;" | ^m2
| ^m2
| style="text-align:center;" | ^Eb
| ^Eb
| style="text-align:center;" | ra
| ra
|-
|-
| style="text-align:center;" | 5
| 5
| | 130.435
| 130.435
| | [[13/12]], [[14/13]], [[15/14]]
| [[13/12]], [[14/13]], [[15/14]]
| style="text-align:center;" | downmid 2nd
| downmid 2nd
| style="text-align:center;" | v~2
| v~2
| style="text-align:center;" | ^^Eb
| ^^Eb
| style="text-align:center;" | ru (as in supraminor)
| ru (as in supraminor)
|-
|-
| style="text-align:center;" | 6
| 6
| | 156.522
| 156.522
| | [[12/11]], [[11/10]], [[23/21]]
| [[12/11]], [[11/10]], [[23/21]]
| style="text-align:center;" | upmid 2nd
| upmid 2nd
| style="text-align:center;" | ^~2
| ^~2
| style="text-align:center;" | vvE
| vvE
| style="text-align:center;" | ruh (as in submajor)
| ruh (as in submajor)
|-
|-
| style="text-align:center;" | 7
| 7
| | 182.609
| 182.609
| | [[10/9]]
| [[10/9]]
| style="text-align:center;" | downmajor 2nd
| downmajor 2nd
| style="text-align:center;" | vM2
| vM2
| style="text-align:center;" | vE
| vE
| style="text-align:center;" | reh
| reh
|-
|-
| style="text-align:center;" | 8
| 8
| | 208.696
| 208.696
| | [[9/8]]
| [[9/8]]
| style="text-align:center;" | major 2nd
| major 2nd
| style="text-align:center;" | M2
| M2
| style="text-align:center;" | E
| E
| style="text-align:center;" | re
| re
|-
|-
| style="text-align:center;" | 9
| 9
| | 234.783
| 234.783
| | [[8/7]], [[23/20]]
| [[8/7]], [[23/20]]
| style="text-align:center;" | upmajor 2nd
| upmajor 2nd
| style="text-align:center;" | ^M2
| ^M2
| style="text-align:center;" | ^E
| ^E
| style="text-align:center;" | ri
| ri
|-
|-
| style="text-align:center;" | 10
| 10
| | 260.87
| 260.870
| | [[7/6]]
| [[7/6]]
| style="text-align:center;" | downminor 3rd
| downminor 3rd
| style="text-align:center;" | vm3
| vm3
| style="text-align:center;" | vF
| vF
| style="text-align:center;" | ma
| ma
|-
|-
| style="text-align:center;" | 11
| 11
| | 286.957
| 286.957
| | [[13/11]], [[20/17]]
| [[13/11]], [[20/17]]
| style="text-align:center;" | minor 3rd
| minor 3rd
| style="text-align:center;" | m3
| m3
| style="text-align:center;" | F
| F
| style="text-align:center;" | meh
| meh
|-
|-
| style="text-align:center;" | 12
| 12
| | 313.043
| 313.043
| | [[6/5]]
| [[6/5]]
| style="text-align:center;" | upminor 3rd
| upminor 3rd
| style="text-align:center;" | ^m3
| ^m3
| style="text-align:center;" | ^F
| ^F
| style="text-align:center;" | me
| me
|-
|-
| style="text-align:center;" | 13
| 13
| | 339.13
| 339.130
| | [[11/9]], [[17/14]]
| [[11/9]], [[17/14]]
| style="text-align:center;" | downmid 3rd
| downmid 3rd
| style="text-align:center;" | v~3
| v~3
| style="text-align:center;" | ^^F
| ^^F
| style="text-align:center;" | mu
| mu
|-
|-
| style="text-align:center;" | 14
| 14
| | 365.217
| 365.217
| | [[16/13]], [[26/21]], [[21/17]]
| [[16/13]], [[26/21]], [[21/17]]
| style="text-align:center;" | upmid 3rd
| upmid 3rd
| style="text-align:center;" | ^~3
| ^~3
| style="text-align:center;" | vvF#
| vvF#
| style="text-align:center;" | muh
| muh
|-
|-
| style="text-align:center;" | 15
| 15
| | 391.304
| 391.304
| | [[5/4]]
| [[5/4]]
| style="text-align:center;" | downmajor 3rd
| downmajor 3rd
| style="text-align:center;" | vM3
| vM3
| style="text-align:center;" | vF#
| vF#
| style="text-align:center;" | mi
| mi
|-
|-
| style="text-align:center;" | 16
| 16
| | 417.391
| 417.391
| | [[14/11]], [[23/18]]
| [[14/11]], [[23/18]]
| style="text-align:center;" | major 3rd
| major 3rd
| style="text-align:center;" | M3
| M3
| style="text-align:center;" | F#
| F#
| style="text-align:center;" | maa
| maa
|-
|-
| style="text-align:center;" | 17
| 17
| | 443.478
| 443.478
| | [[9/7]], [[13/10]], [[22/17]]
| [[9/7]], [[13/10]], [[22/17]]
| style="text-align:center;" | upmajor 3rd
| upmajor 3rd
| style="text-align:center;" | ^M3
| ^M3
| style="text-align:center;" | ^F#
| ^F#
| style="text-align:center;" | mo
| mo
|-
|-
| style="text-align:center;" | 18
| 18
| | 469.565
| 469.565
| | [[21/16]], [[17/13]]
| [[21/16]], [[17/13]]
| style="text-align:center;" | down 4th
| down 4th
| style="text-align:center;" | v4
| v4
| style="text-align:center;" | vG
| vG
| style="text-align:center;" | fe
| fe
|-
|-
| style="text-align:center;" | 19
| 19
| | 495.652
| 495.652
| | [[4/3]]
| [[4/3]]
| style="text-align:center;" | perfect 4th
| perfect 4th
| style="text-align:center;" | P4
| P4
| style="text-align:center;" | G
| G
| style="text-align:center;" | fa
| fa
|-
|-
| style="text-align:center;" | 20
| 20
| | 521.739
| 521.739
| | [[23/17]]
| [[23/17]]
| style="text-align:center;" | up 4th
| up 4th
| style="text-align:center;" | ^4
| ^4
| style="text-align:center;" | ^G
| ^G
| style="text-align:center;" | fih
| fih
|-
|-
| style="text-align:center;" | 21
| 21
| | 547.826
| 547.826
| | [[11/8]]
| [[11/8]]
| style="text-align:center;" | downmid 4th
| downmid 4th
| style="text-align:center;" | v~4
| v~4
| style="text-align:center;" | ^^G
| ^^G
| style="text-align:center;" | fu
| fu
|-
|-
| style="text-align:center;" | 22
| 22
| | 573.913
| 573.913
| | [[7/5]], [[18/13]], [[32/23]]
| [[7/5]], [[18/13]], [[32/23]]
| style="text-align:center;" | upmid 4th, dim 5th
| upmid 4th, dim 5th
| style="text-align:center;" | ^~4, d5
| ^~4, d5
| style="text-align:center;" | vvG#, Ab
| vvG#, Ab
| style="text-align:center;" | fi
| fi
|-
|-
| style="text-align:center;" | 23
| 23
| | 600
| 600.000
| | [[17/12]], [[24/17]]
| [[17/12]], [[24/17]]
| style="text-align:center;" | downaug 4th, updim 5th
| downaug 4th, updim 5th
| style="text-align:center;" | vA4, ^d5
| vA4, ^d5
| style="text-align:center;" | vG#, ^Ab
| vG#, ^Ab
| style="text-align:center;" | seh
| seh
|-
|-
| style="text-align:center;" | 24
| 24
| | 626.087
| 626.087
| | [[10/7]], [[13/9]], [[23/16]]
| [[10/7]], [[13/9]], [[23/16]]
| style="text-align:center;" | aug 4th, downmid 5th
| aug 4th, downmid 5th
| style="text-align:center;" | A4, v~5
| A4, v~5
| style="text-align:center;" | G#, ^^Ab
| G#, ^^Ab
| style="text-align:center;" | se
| se
|-
|-
| style="text-align:center;" | 25
| 25
| | 652.174
| 652.174
| | [[16/11]]
| [[16/11]]
| style="text-align:center;" | double-down 5th
| double-down 5th
| style="text-align:center;" | ^~5
| ^~5
| style="text-align:center;" | vvA
| vvA
| style="text-align:center;" | su
| su
|-
|-
| style="text-align:center;" | 26
| 26
| | 678.261
| 678.261
| | [[34/23]]
| [[34/23]]
| style="text-align:center;" | down 5th
| down 5th
| style="text-align:center;" | v5
| v5
| style="text-align:center;" | vA
| vA
| style="text-align:center;" | sih
| sih
|-
|-
| style="text-align:center;" | 27
| 27
| | 704.348
| 704.348
| | [[3/2]]
| [[3/2]]
| style="text-align:center;" | perfect 5th
| perfect 5th
| style="text-align:center;" | P5
| P5
| style="text-align:center;" | A
| A
| style="text-align:center;" | sol
| sol
|-
|-
| style="text-align:center;" | 28
| 28
| | 730.435
| 730.435
| | [[32/21]], [[26/17]]
| [[32/21]], [[26/17]]
| style="text-align:center;" | up 5th
| up 5th
| style="text-align:center;" | ^5
| ^5
| style="text-align:center;" | ^A
| ^A
| style="text-align:center;" | si
| si
|-
|-
| style="text-align:center;" | 29
| 29
| | 756.522
| 756.522
| | [[14/9]], [[20/13]], [[17/11]]
| [[14/9]], [[20/13]], [[17/11]]
| style="text-align:center;" | downminor 6th
| downminor 6th
| style="text-align:center;" | vm6
| vm6
| style="text-align:center;" | vBb
| vBb
| style="text-align:center;" | lo
| lo
|-
|-
| style="text-align:center;" | 30
| 30
| | 782.609
| 782.609
| | [[11/7]]
| [[11/7]]
| style="text-align:center;" | minor 6th
| minor 6th
| style="text-align:center;" | m6
| m6
| style="text-align:center;" | Bb
| Bb
| style="text-align:center;" | leh
| leh
|-
|-
| style="text-align:center;" | 31
| 31
| | 808.696
| 808.696
| | [[8/5]]
| [[8/5]]
| style="text-align:center;" | upminor 6th
| upminor 6th
| style="text-align:center;" | ^m6
| ^m6
| style="text-align:center;" | ^Bb
| ^Bb
| style="text-align:center;" | le
| le
|-
|-
| style="text-align:center;" | 32
| 32
| | 834.783
| 834.783
| | [[13/8]], [[21/13]], [[34/21]]
| [[13/8]], [[21/13]], [[34/21]]
| style="text-align:center;" | downmid 6th
| downmid 6th
| style="text-align:center;" | v~6
| v~6
| style="text-align:center;" | ^^Bb
| ^^Bb
| style="text-align:center;" | lu
| lu
|-
|-
| style="text-align:center;" | 33
| 33
| | 860.87
| 860.870
| | [[18/11]], [[28/17]], [[23/14]]
| [[18/11]], [[28/17]], [[23/14]]
| style="text-align:center;" | upmid 6th
| upmid 6th
| style="text-align:center;" | ^~6
| ^~6
| style="text-align:center;" | vvB
| vvB
| style="text-align:center;" | luh
| luh
|-
|-
| style="text-align:center;" | 34
| 34
| | 886.957
| 886.957
| | [[5/3]]
| [[5/3]]
| style="text-align:center;" | downmajor 6th
| downmajor 6th
| style="text-align:center;" | vM6
| vM6
| style="text-align:center;" | vB
| vB
| style="text-align:center;" | la
| la
|-
|-
| style="text-align:center;" | 35
| 35
| | 913.043
| 913.043
| | [[22/13]], [[17/10]]
| [[22/13]], [[17/10]]
| style="text-align:center;" | major 6th
| major 6th
| style="text-align:center;" | M6
| M6
| style="text-align:center;" | B
| B
| style="text-align:center;" | laa
| laa
|-
|-
| style="text-align:center;" | 36
| 36
| | 939.13
| 939.130
| | [[12/7]]
| [[12/7]]
| style="text-align:center;" | upmajor 6th
| upmajor 6th
| style="text-align:center;" | ^M6
| ^M6
| style="text-align:center;" | ^B
| ^B
| style="text-align:center;" | li
| li
|-
|-
| style="text-align:center;" | 37
| 37
| | 965.217
| 965.217
| | [[7/4]], [[40/23]]
| [[7/4]], [[40/23]]
| style="text-align:center;" | downminor 7th
| downminor 7th
| style="text-align:center;" | vm7
| vm7
| style="text-align:center;" | vC
| vC
| style="text-align:center;" | ta
| ta
|-
|-
| style="text-align:center;" | 38
| 38
| | 991.304
| 991.304
| | [[16/9]], [[23/13]]
| [[16/9]], [[23/13]]
| style="text-align:center;" | minor 7th
| minor 7th
| style="text-align:center;" | m7
| m7
| style="text-align:center;" | C
| C
| style="text-align:center;" | teh
| teh
|-
|-
| style="text-align:center;" | 39
| 39
| | 1017.391
| 1017.391
| | [[9/5]]
| [[9/5]]
| style="text-align:center;" | upminor 7th
| upminor 7th
| style="text-align:center;" | ^m7
| ^m7
| style="text-align:center;" | ^C
| ^C
| style="text-align:center;" | te
| te
|-
|-
| style="text-align:center;" | 40
| 40
| | 1043.478
| 1043.478
| | [[11/6]], [[20/11]], [[42/23]]
| [[11/6]], [[20/11]], [[42/23]]
| style="text-align:center;" | downmid 7th
| downmid 7th
| style="text-align:center;" | v~7
| v~7
| style="text-align:center;" | ^^C
| ^^C
| style="text-align:center;" | tu
| tu
|-
|-
| style="text-align:center;" | 41
| 41
| | 1069.565
| 1069.565
| | [[24/13]], [[13/7]], [[28/15]]
| [[24/13]], [[13/7]], [[28/15]]
| style="text-align:center;" | upmid 7th
| upmid 7th
| style="text-align:center;" | ^~7
| ^~7
| style="text-align:center;" | vvC#
| vvC#
| style="text-align:center;" | tuh
| tuh
|-
|-
| style="text-align:center;" | 42
| 42
| | 1095.652
| 1095.652
| | [[15/8]], [[32/17]], [[17/9]]
| [[15/8]], [[32/17]], [[17/9]]
| style="text-align:center;" | downmajor 7th
| downmajor 7th
| style="text-align:center;" | vM7
| vM7
| style="text-align:center;" | vC#
| vC#
| style="text-align:center;" | ti
| ti
|-
|-
| style="text-align:center;" | 43
| 43
| | 1121.739
| 1121.739
| | [[48/25]], [[40/21]], [[21/11]], [[23/12]], [[44/23]]
| [[48/25]], [[40/21]], [[21/11]], [[23/12]], [[44/23]]
| style="text-align:center;" | major 7th
| major 7th
| style="text-align:center;" | M7
| M7
| style="text-align:center;" | C#
| C#
| style="text-align:center;" | taa
| taa
|-
|-
| style="text-align:center;" | 44
| 44
| | 1147.826
| 1147.826
| |[[27/14]], [[64/33]]
| [[27/14]], [[64/33]]
| style="text-align:center;" | upmajor 7th
| upmajor 7th
| style="text-align:center;" | ^M7
| ^M7
| style="text-align:center;" | ^C#
| ^C#
| style="text-align:center;" | to
| to
|-
|-
| style="text-align:center;" | 45
| 45
| | 1173.913
| 1173.913
| | [[160/81]], [[63/32]]
| [[160/81]], [[63/32]]
| style="text-align:center;" | down 8ve
| down 8ve
| style="text-align:center;" | v8
| v8
| style="text-align:center;" | vD
| vD
| style="text-align:center;" | da
| da
|-
|-
| style="text-align:center;" | 46
| 46
| | 1200.000
| 1200.000
| | [[2/1]]
| [[2/1]]
| style="text-align:center;" | perfect 8ve
| perfect 8ve
| style="text-align:center;" | P8
| P8
| style="text-align:center;" | D
| D
| style="text-align:center;" | do
| do
|}
|}
<nowiki>*</nowiki> Based on treating 46-edo as a 2.3.5.7.11.13.17.23 subgroup, without ratios of 15 (except the superparticulars). 46-edo has the 15th harmony poorly approximated in general, because, while both the 3rd and 5th harmonies are sharp by a fair amount and they add up, all the other primes are flat, making the difference even larger, to the extent that it is not [[consistent]] in the [[15-odd-limit]]. This can be demonstrated with the discrepancy approximating [[15/13]] (and its inversion [[26/15]]). 9\46edo is closer to 15/13 by a hair; 10\46edo represents the difference between, for instance, 46edo's 15/8 and 13/8, and is more likely to appear in chords actually functioning as 15/13.  
<nowiki>*</nowiki> Based on treating 46-edo as a 2.3.5.7.11.13.17.23 subgroup, without ratios of 15 (except the superparticulars). 46-edo has the 15th harmony poorly approximated in general, because, while both the 3rd and 5th harmonies are sharp by a fair amount and they add up, all the other primes are flat, making the difference even larger, to the extent that it is not [[consistent]] in the [[15-odd-limit]]. This can be demonstrated with the discrepancy approximating [[15/13]] (and its inversion [[26/15]]). 9\46edo is closer to 15/13 by a hair; 10\46edo represents the difference between, for instance, 46edo's 15/8 and 13/8, and is more likely to appear in chords actually functioning as 15/13.  
Line 401: Line 402:
Combining ups and downs notation with [[Kite's_color_notation|color notation]], qualities can be loosely associated with colors:
Combining ups and downs notation with [[Kite's_color_notation|color notation]], qualities can be loosely associated with colors:


{| class="wikitable"
{| class="wikitable center-all"
|-
|-
! | quality
! quality
! | color
! color
! | monzo format
! monzo format
! | examples
! examples
|-
|-
| style="text-align:center;" | downminor
| downminor
| style="text-align:center;" | zo
| zo
| style="text-align:center;" | {a, b, 0, 1}
| {a, b, 0, 1}
| style="text-align:center;" | 7/6, 7/4
| 7/6, 7/4
|-
|-
| style="text-align:center;" | minor
| minor
| style="text-align:center;" | fourthward wa
| fourthward wa
| style="text-align:center;" | {a, b}, b &lt; -1
| {a, b}, b &lt; -1
| style="text-align:center;" | 32/27, 16/9
| 32/27, 16/9
|-
|-
| style="text-align:center;" | upminor
| upminor
| style="text-align:center;" | gu
| gu
| style="text-align:center;" | {a, b, -1}
| {a, b, -1}
| style="text-align:center;" | 6/5, 9/5
| 6/5, 9/5
|-
|-
| style="text-align:center;" | downmid
| downmid
| style="text-align:center;" | ilo
| ilo
| style="text-align:center;" | {a, b, 0, 0, 1}
| {a, b, 0, 0, 1}
| style="text-align:center;" | 11/9, 11/6
| 11/9, 11/6
|-
|-
| style="text-align:center;" | upmid
| upmid
| style="text-align:center;" | lu
| lu
| style="text-align:center;" | {a, b, 0, 0, -1}
| {a, b, 0, 0, -1}
| style="text-align:center;" | 12/11, 18/11
| 12/11, 18/11
|-
|-
| style="text-align:center;" | downmajor
| downmajor
| style="text-align:center;" | yo
| yo
| style="text-align:center;" | {a, b, 1}
| {a, b, 1}
| style="text-align:center;" | 5/4, 5/3
| 5/4, 5/3
|-
|-
| style="text-align:center;" | major
| major
| style="text-align:center;" | fifthward wa
| fifthward wa
| style="text-align:center;" | {a, b}, b &gt; 1
| {a, b}, b &gt; 1
| style="text-align:center;" | 9/8, 27/16
| 9/8, 27/16
|-
|-
| style="text-align:center;" | upmajor
| upmajor
| style="text-align:center;" | ru
| ru
| style="text-align:center;" | {a, b, 0, -1}
| {a, b, 0, -1}
| style="text-align:center;" | 9/7, 12/7
| 9/7, 12/7
|}
|}
All 46edo chords can be named using ups and downs. Alterations are always enclosed in parentheses, additions never are. An up, down or mid immediately after the chord root affects 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:
All 46edo chords can be named using ups and downs. Alterations are always enclosed in parentheses, additions never are. An up, down or mid immediately after the chord root affects 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"
{| class="wikitable center-all"
|-
|-
! | color of the 3rd
! color of the 3rd
! | JI chord
! JI chord
! | notes as edosteps
! notes as edosteps
! | notes of C chord
! notes of C chord
! | written name
! written name
! | spoken name
! spoken name
|-
|-
| style="text-align:center;" | zo
| zo
| style="text-align:center;" | 6:7:9
| 6:7:9
| style="text-align:center;" | 0-10-27
| 0-10-27
| style="text-align:center;" | C vEb G
| C vEb G
| style="text-align:center;" | Cvm
| Cvm
| style="text-align:center;" | C downminor
| C downminor
|-
|-
| style="text-align:center;" | gu
| gu
| style="text-align:center;" | 10:12:15
| 10:12:15
| style="text-align:center;" | 0-12-27
| 0-12-27
| style="text-align:center;" | C ^Eb G
| C ^Eb G
| style="text-align:center;" | C^m
| C^m
| style="text-align:center;" | C upminor
| C upminor
|-
|-
| style="text-align:center;" | ilo
| ilo
| style="text-align:center;" | 18:22:27
| 18:22:27
| style="text-align:center;" | 0-13-27
| 0-13-27
| style="text-align:center;" | C ^^Eb G
| C ^^Eb G
| style="text-align:center;" | Cv~
| Cv~
| style="text-align:center;" | C downmid
| C downmid
|-
|-
| style="text-align:center;" |lu
| lu
| style="text-align:center;" |22:27:33
| 22:27:33
| style="text-align:center;" |0-14-27
| 0-14-27
| style="text-align:center;" |C vvE G
| C vvE G
| style="text-align:center;" |C^~
| C^~
| style="text-align:center;" |C upmid
| C upmid
|-
|-
| style="text-align:center;" | yo
| yo
| style="text-align:center;" | 4:5:6
| 4:5:6
| style="text-align:center;" | 0-15-27
| 0-15-27
| style="text-align:center;" | C vE G
| C vE G
| style="text-align:center;" | Cv
| Cv
| style="text-align:center;" | C downmajor or C down
| C downmajor or C down
|-
|-
| style="text-align:center;" | ru
| ru
| style="text-align:center;" | 14:18:21
| 14:18:21
| style="text-align:center;" | 0-17-27
| 0-17-27
| style="text-align:center;" | C ^E G
| C ^E G
| style="text-align:center;" | C^
| C^
| style="text-align:center;" | C upmajor or C up
| C upmajor or C up
|}
|}
For a more complete list, see [[Ups and Downs Notation#Chords and Chord Progressions|Ups and Downs Notation - Chords and Chord Progressions]].
For a more complete list, see [[Ups and Downs Notation#Chords and Chord Progressions|Ups and Downs Notation - Chords and Chord Progressions]].


==Selected just intervals by error==
=== Selected just intervals by error ===
{| class="wikitable"
{| class="wikitable center-all"
!
! colspan="2" |
!prime 2
! prime 2
!prime 3
! prime 3
!prime 5
! prime 5
!prime 7
! prime 7
!prime 11
! prime 11
!prime 13
! prime 13
!prime 17
! prime 17
!prime 19
! prime 19
!prime 23
! prime 23
|-
|-
! |Error (¢)
! rowspan="2"|Error
|0
! absolute (¢)
| 0.0
| +2.4
| +2.4
| +5.0
| +5.0
Line 527: Line 529:
| -2.1
| -2.1
|-
|-
! |Fifthspan
! relative (%)
|0
| 0.0
| +9.2
| +19.1
| -13.8
| -13.4
| -22.0
| -2.3
| -40.5
| -8.4
|-
! colspan="2"|Fifthspan
| 0
| +1
| +1
| +21
| +21
Line 540: Line 553:
The following table shows how [[Just-24|some prominent just intervals]] are represented in 46edo (ordered by absolute error).
The following table shows how [[Just-24|some prominent just intervals]] are represented in 46edo (ordered by absolute error).


{| class="wikitable"
{| class="wikitable center-all"
|-
|-
! | Interval, complement
! Interval, complement
! | Error (abs., in [[cent|cents]])
! Error (abs., in [[cent|cents]])
|-
|-
| style="text-align:center;" | [[14/11|14/11]], [[11/7|11/7]]
| [[14/11]], [[11/7]]
| style="text-align:center;" | 0.117
| 0.117
|-
|-
| style="text-align:center;" | [[10/9|10/9]], [[9/5|9/5]]
| [[10/9]], [[9/5]]
| style="text-align:center;" | 0.205
| 0.205
|-
|-
| style="text-align:center;" | [[14/13|14/13]], [[13/7|13/7]]
| [[14/13]], [[13/7]]
| style="text-align:center;" | 2.137
| 2.137
|-
|-
| style="text-align:center;" | [[13/11|13/11]], [[22/13|22/13]]
| [[13/11]], [[22/13]]
| style="text-align:center;" | 2.253
| 2.253
|-
|-
| style="text-align:center;" | '''[[4/3|4/3]], [[3/2|3/2]]'''
| '''[[4/3]], [[3/2]]'''
| style="text-align:center;" | '''2.393'''
| '''2.393'''
|-
|-
| style="text-align:center;" | [[6/5|6/5]], [[5/3|5/3]]
| [[6/5]], [[5/3]]
| style="text-align:center;" | 2.598
| 2.598
|-
|-
| style="text-align:center;" | '''[[11/8|11/8]], [[16/11|16/11]]'''
| '''[[11/8]], [[16/11]]'''
| style="text-align:center;" | '''3.492'''
| '''3.492'''
|-
|-
| style="text-align:center;" | '''[[8/7|8/7]], [[7/4|7/4]]'''
| '''[[8/7]], [[7/4]]'''
| style="text-align:center;" | '''3.609'''
| '''3.609'''
|-
|-
| style="text-align:center;" | [[9/8|9/8]], [[16/9|16/9]]
| [[9/8]], [[16/9]]
| style="text-align:center;" | 4.786
| 4.786
|-
|-
| style="text-align:center;" | '''[[5/4|5/4]], [[8/5|8/5]]'''
| '''[[5/4]], [[8/5]]'''
| style="text-align:center;" | '''4.991'''
| '''4.991'''
|-
|-
| style="text-align:center;" | '''[[16/13|16/13]], [[13/8|13/8]]'''
| '''[[16/13]], [[13/8]]'''
| style="text-align:center;" | '''5.745'''
| '''5.745'''
|-
|-
| style="text-align:center;" | [[12/11|12/11]], [[11/6|11/6]]
| [[12/11]], [[11/6]]
| style="text-align:center;" | 5.885
| 5.885
|-
|-
| style="text-align:center;" | [[7/6|7/6]], [[12/7|12/7]]
| [[7/6]], [[12/7]]
| style="text-align:center;" | 6.001
| 6.001
|-
|-
| style="text-align:center;" | [[16/15|16/15]], [[15/8|15/8]]
| [[16/15]], [[15/8]]
| style="text-align:center;" | 7.383
| 7.383
|-
|-
| style="text-align:center;" | [[13/12|13/12]], [[24/13|24/13]]
| [[13/12]], [[24/13]]
| style="text-align:center;" | 8.138
| 8.138
|-
|-
| style="text-align:center;" | [[11/9|11/9]], [[18/11|18/11]]
| [[11/9]], [[18/11]]
| style="text-align:center;" | 8.278
| 8.278
|-
|-
| style="text-align:center;" | [[9/7|9/7]], [[14/9|14/9]]
| [[9/7]], [[14/9]]
| style="text-align:center;" | 8.394
| 8.394
|-
|-
| style="text-align:center;" | [[11/10|11/10]], [[20/11|20/11]]
| [[11/10]], [[20/11]]
| style="text-align:center;" | 8.482
| 8.482
|-
|-
| style="text-align:center;" | [[7/5|7/5]], [[10/7|10/7]]
| [[7/5]], [[10/7]]
| style="text-align:center;" | 8.599
| 8.599
|-
|-
| style="text-align:center;" | [[18/13|18/13]], [[13/9|13/9]]
| [[18/13]], [[13/9]]
| style="text-align:center;" | 10.531
| 10.531
|-
|-
| style="text-align:center;" | [[13/10|13/10]], [[20/13|20/13]]
| [[13/10]], [[20/13]]
| style="text-align:center;" | 10.736
| 10.736
|-
|-
| style="text-align:center;" | [[15/11|15/11]], [[22/15|22/15]]
| [[15/11]], [[22/15]]
| style="text-align:center;" | 10.875
| 10.875
|-
|-
| style="text-align:center;" | [[15/14|15/14]], [[28/15|28/15]]
| [[15/14]], [[28/15]]
| style="text-align:center;" | 10.992
| 10.992
|-
|-
| style="text-align:center;" | 15/13, 26/15
| ''[[15/13]], [[26/15]]''
| style="text-align:center;" | 12.958
| ''12.958''
|}
|}


=Linear temperaments=
== Linear temperaments ==


{| class="wikitable"
{| class="wikitable"
|-
|-
! | Periods
! Periods <br>per octave
 
! Generator
per octave
! Cents
! | Generator
! Temperaments
! | Cents
! MOS/DE Scales available
! | Temperaments
! L:s
! | MOS/DE Scales available
! | L:s
|-
|-
| | 1
| 1
| | 1\46
| 1\46
| | 26.087
| 26.087
| |  
|  
| |  
|  
| |  
|  
|-
|-
| | 1
| 1
| | 3\46
| 3\46
| | 78.261
| 78.261
| | [[Valentine|Valentine]]
| [[Valentine]]
| | 1L 14s (15-tone)
| 1L 14s (15-tone)


15L 1s (16-tone)
15L 1s (16-tone)


16L 15s (31-tone)
16L 15s (31-tone)
| | 4:3 ~ [[Maximal_evenness|quasi-equal]]
| 4:3 ~ [[Maximal_evenness|quasi-equal]]


3:1
3:1
Line 653: Line 664:
2:1 ~ QE
2:1 ~ QE
|-
|-
| | 1
| 1
| | 5\46
| 5\46
| | 130.435
| 130.435
| | [[Twothirdtonic|Twothirdtonic]]
| [[Twothirdtonic]]
| | [[1L_8s|1L 8s]] (9-tone)
| [[1L_8s|1L 8s]] (9-tone)


[[9L_1s|9L 1s]] (10-tone)
[[9L_1s]] (10-tone)


9L 10s (19-tone)
9L 10s (19-tone)
Line 666: Line 677:


9L 28s (37-tone)
9L 28s (37-tone)
| | 6:5 ~ QE
| 6:5 ~ QE


5:1
5:1
Line 676: Line 687:
2:1 ~ QE
2:1 ~ QE
|-
|-
| | 1
| 1
| | 7\46
| 7\46
| | 182.609
| 182.609
| | [[Minortone|Minortone]]
| [[Minortone]]
| | [[1L_5s|1L 5s]] (6-tone)
| [[1L 5s]] (6-tone)


[[6L_1s|6L 1s]] (7-tone)
[[6L 1s]] (7-tone)


7L 6s (13-tone)
7L 6s (13-tone)
Line 689: Line 700:


13L 20s (33-tone)
13L 20s (33-tone)
| | 11:7
| 11:7


7:4
7:4
Line 699: Line 710:
2:1 ~ QE
2:1 ~ QE
|-
|-
| | 1
| 1
| | 9\46
| 9\46
| | 234.783
| 234.783
| | [[Rodan|Rodan]]
| [[Rodan]]
| | [[1L_4s|1L 4s]] (5-tone)
| [[1L 4s]] (5-tone)


[[1L_5s|1L 5s]] (6-tone)
[[1L 5s]] (6-tone)


[[5L_6s|5L 6s]] (11-tone)
[[5L 6s]] (11-tone)


5L 11s (16-tone)
5L 11s (16-tone)
Line 720: Line 731:


5L 36s (41-tone)
5L 36s (41-tone)
| | 10:9 ~QE
| 10:9 ~QE


9:1
9:1
Line 738: Line 749:
2:1 ~ QE
2:1 ~ QE
|-
|-
| | 1
| 1
| | 11\46
| 11\46
| | 286.957
| 286.957
| |  
|  
| | [[4L_1s|4L 1s]] (5-tone)
| [[4L 1s]] (5-tone)


[[4L_5s|4L 5s]] (9-tone)
[[4L 5s]] (9-tone)


4L 9s (13-tone)
4L 9s (13-tone)
Line 753: Line 764:


21L 4s (25-tone)
21L 4s (25-tone)
| | 11:2
| 11:2


9:2
9:2
Line 765: Line 776:
2:1 ~ QE
2:1 ~ QE
|-
|-
| | 1
| 1
| | 13\46
| 13\46
| | 339.130
| 339.130
| | [[Amity|Amity]]/[[Hitchcock|hitchcock]]
| [[Amity]]/[[Hitchcock|hitchcock]]
| | [[4L_3s|4L 3s]] (7-tone)
| [[4L 3s]] (7-tone)


[[7L_4s|7L 4s]] (11-tone)
[[7L 4s]] (11-tone)


7L 11s (18-tone)
7L 11s (18-tone)
Line 780: Line 791:


7L 32s (39-tone)
7L 32s (39-tone)
| | 7:6 ~ QE
| 7:6 ~ QE


6:1
6:1
Line 792: Line 803:
2:1 ~ QE
2:1 ~ QE
|-
|-
| | 1
| 1
| | 15\46
| 15\46
| | 391.304
| 391.304
| | [[amigo|Amigo]]
| [[Amigo]]
| | [[1L_2s|1L 2s]] (3-tone)
| [[1L 2s]] (3-tone)


[[3L_1s|3L 1s]] (4-tone)
[[3L 1s]] (4-tone)


[[3L_4s|3L 4s]] (7-tone)
[[3L 4s]] (7-tone)


[[3L_7s|3L 7s]] (10-tone)
[[3L 7s]] (10-tone)


3L 10s (13-tone)
3L 10s (13-tone)
Line 825: Line 836:


3L 39s (42-tone)
3L 39s (42-tone)
| | 16:15 ~ QE
| 16:15 ~ QE


15:1
15:1
Line 855: Line 866:
2:1 ~ QE
2:1 ~ QE
|-
|-
| | 1
| 1
| | 17\46
| 17\46
| | 443.478
| 443.478
| | [[Sensi|Sensi]]
| [[Sensi]]
| | [[3L_2s|3L 2s]] (5-tone)
| [[3L 2s]] (5-tone)


[[3L_5s|3L 5s]] (8-tone)
[[3L 5s]] (8-tone)


[[8L_3s|8L 3s]] (11-tone)
[[8L 3s]] (11-tone)


8L 11s (19-tone)
8L 11s (19-tone)


19L 8s (27-tone)
19L 8s (27-tone)
| | 12:5
| 12:5


7:5
7:5
Line 878: Line 889:
2:1
2:1
|-
|-
| | 1
| 1
| | 19\46
| 19\46
| | 495.652
| 495.652
| | [[Leapday|Leapday]]
| [[Leapday]]
| | [[2L_3s|2L 3s]] (5-tone)
| [[2L 3s]] (5-tone)


[[5L_2s|5L 2s]] (7-tone)
[[5L 2s]] (7-tone)


[[5L_7s|5L 7s]] (12-tone)
[[5L 7s]] (12-tone)


12L 5s (17-tone)
12L 5s (17-tone)


17L 12s (29-tone)
17L 12s (29-tone)
| | 11:8
| 11:8


8:3
8:3
Line 901: Line 912:
2:1 ~ QE
2:1 ~ QE
|-
|-
| | 1
| 1
| | 21\46
| 21\46
| | 547.826
| 547.826
| | [[Heinz|Heinz]]
| [[Heinz]]
| | [[2L_3s|2L 3s]] (5-tone)
| [[2L 3s]] (5-tone)


[[2L_5s|2L 5s]] (7-tone)
[[2L 5s]] (7-tone)


[[2L_7s|2L 7s]] (9-tone)
[[2L 7s]] (9-tone)


[[2L_9s|2L 9s]] (11-tone)
[[2L 9s]] (11-tone)


11L 2s (13-tone)
11L 2s (13-tone)
Line 918: Line 929:


11L 24s (35-tone)
11L 24s (35-tone)
| | 17:4
| 17:4


13:4
13:4
Line 932: Line 943:
2:1 ~ QE
2:1 ~ QE
|-
|-
| | 2
| 2
| | 1\46
| 1\46
| | 26.087
| 26.087
| | [[Ketchup|Ketchup]]
| [[Ketchup]]
| |  
|  
| |  
|  
|-
|-
| | 2
| 2
| | 2\46
| 2\46
| | 52.174
| 52.174
| | [[Shrutar|Shrutar]]
| [[Shrutar]]
| | 2L 2s (4-tone)
| 2L 2s (4-tone)


[[2L_4s|2L 4s]] (6-tone)
[[2L 4s]] (6-tone)


[[2L_6s|2L 6s]] (8-tone)
[[2L 6s]] (8-tone)


[[2L_8s|2L 8s]] (10-tone)
[[2L 8s]] (10-tone)


[[2L_10s|2L 10s]] (12-tone)
[[2L 10s]] (12-tone)


2L 12s (14-tone)
2L 12s (14-tone)
Line 964: Line 975:


22L 2s (24-tone)
22L 2s (24-tone)
| | 21:2
| 21:2


19:2
19:2
Line 986: Line 997:
2:1 ~ QE
2:1 ~ QE
|-
|-
| | 2
| 2
| | 3\46
| 3\46
| | 78.261
| 78.261
| | [[Semivalentine|Semivalentine]]
| [[Semivalentine]]
| | 2L 2s (4-tone)
| 2L 2s (4-tone)


[[2L_4s|2L 4s]] (6-tone)
[[2L 4s]] (6-tone)


[[2L_6s|2L 6s]] (8-tone)
[[2L 6s]] (8-tone)


[[2L_8s|2L 8s]] (10-tone)
[[2L 8s]] (10-tone)


[[2L_10s|2L 10s]] (12-tone)
[[2L 10s]] (12-tone)


2L 12s (14-tone)
2L 12s (14-tone)
Line 1,005: Line 1,016:


16L 14s (30-tone)
16L 14s (30-tone)
| | 20:3
| 20:3


17:3
17:3
Line 1,021: Line 1,032:
2:1 ~ QE
2:1 ~ QE
|-
|-
| | 2
| 2
| | 4\46
| 4\46
| | 104.348
| 104.348
| | [[Srutal|Srutal]]/[[Diaschismic|diaschismic]]
| [[Srutal]]/[[Diaschismic|diaschismic]]
| | 2L 2s (4-tone)
| 2L 2s (4-tone)


[[2L_4s|2L 4s]] (6-tone)
[[2L 4s]] (6-tone)


[[2L_6s|2L 6s]] (8-tone)
[[2L 6s]] (8-tone)


[[2L_8s|2L 8s]] (10-tone)
[[2L 8s]] (10-tone)


[[10L_2s|10L 2s]] (12-tone)
[[10L 2s]] (12-tone)


12L 10s (22-tone)
12L 10s (22-tone)


12L 22s (34-tone)
12L 22s (34-tone)
| | 19:4
| 19:4


15:4
15:4
Line 1,052: Line 1,063:
2:1 ~ QE
2:1 ~ QE
|-
|-
| | 2
| 2
| | 5\46
| 5\46
| | 130.435
| 130.435
| |  
|  
| | 2L 2s (4-tone)
| 2L 2s (4-tone)


[[2L_4s|2L 4s]] (6-tone)
[[2L 4s]] (6-tone)


[[2L_6s|2L 6s]] (8-tone)
[[2L 6s]] (8-tone)


[[8L_2s|8L 2s]] (10-tone)
[[8L 2s]] (10-tone)


8L 10s (18-tone)
8L 10s (18-tone)


18L 10s (28-tone)
18L 10s (28-tone)
| | 18:5
| 18:5


13:5
13:5
Line 1,079: Line 1,090:
2:1 ~ QE
2:1 ~ QE
|-
|-
| | 2
| 2
| | 6\46
| 6\46
| | 156.522
| 156.522
| | [[Bison|Bison]]
| [[Bison]]
| | 2L 2s (4-tone)
| 2L 2s (4-tone)


[[2L_4s|2L 4s]] (6-tone)
[[2L 4s]] (6-tone)


[[6L_2s|6L 2s]] (8-tone)
[[6L 2s]] (8-tone)


8L 6s (14-tone)
8L 6s (14-tone)
Line 1,096: Line 1,107:


8L 30s (38-tone
8L 30s (38-tone
| | 17:6
| 17:6


11:6
11:6
Line 1,110: Line 1,121:
2:1 ~ QE
2:1 ~ QE
|-
|-
| | 2
| 2
| | 7\46
| 7\46
| | 182.609
| 182.609
| | [[Unidec|Unidec]]/[[Hendec|hendec]]
| [[Unidec]]/[[Hendec|hendec]]
| | 2L 2s (4-tone)
| 2L 2s (4-tone)


[[2L_4s|2L 4s]] (6-tone)
[[2L 4s]] (6-tone)


[[6L_2s|6L 2s]] (8-tone)
[[6L 2s]] (8-tone)


6L 8s (14-tone)
6L 8s (14-tone)
Line 1,125: Line 1,136:


20L 6s (26-tone)
20L 6s (26-tone)
| | 16:7
| 16:7


9:7
9:7
Line 1,137: Line 1,148:
2:1 ~ QE
2:1 ~ QE
|-
|-
| | 2
| 2
| | 8\46
| 8\46
| | 208.696
| 208.696
| | [[Abigail|Abigail]]
| [[Abigail]]
| | 2L 2s (4-tone)
| 2L 2s (4-tone)


[[4L_2s|4L 2s]] (6-tone)
[[4L 2s]] (6-tone)


[[6L_2s|6L 2s]] (8-tone)
[[6L 2s]] (8-tone)


6L 8s (14-tone)
6L 8s (14-tone)
Line 1,158: Line 1,169:


6L 38s (44-tone)
6L 38s (44-tone)
| | 15:8
| 15:8


8:7 ~ QE
8:7 ~ QE
Line 1,176: Line 1,187:
2:1 ~ QE
2:1 ~ QE
|-
|-
| | 2
| 2
| | 9\46
| 9\46
| | 234.783
| 234.783
| | [[Echidnic|Echidnic]]
| [[Echidnic]]
| | 2L 2s (4-tone)
| 2L 2s (4-tone)


[[4L_2s|4L 2s]] (6-tone)
[[4L 2s]] (6-tone)


[[6L_4s|6L 4s]] (10-tone)
[[6L 4s]] (10-tone)


10L 6s (16-tone)
10L 6s (16-tone)
Line 1,191: Line 1,202:


10L 26s (36-tone)
10L 26s (36-tone)
| | 14:9
| 14:9


9:5
9:5
Line 1,203: Line 1,214:
2:1 ~ QE
2:1 ~ QE
|-
|-
| | 2
| 2
| | 10\46
| 10\46
| | 260.87
| 260.87
| | [[Bamity|Bamity]]
| [[Bamity]]
| | 2L 2s (4-tone)
| 2L 2s (4-tone)


[[4L_2s|4L 2s]] (6-tone)
[[4L 2s]] (6-tone)


[[4L_6s|4L 6s]] (10-tone)
[[4L 6s]] (10-tone)


4L 10s (14-tone)
4L 10s (14-tone)
Line 1,218: Line 1,229:


14L 18s (32-tone)
14L 18s (32-tone)
| | 13:10
| 13:10


10:3
10:3
Line 1,230: Line 1,241:
2:1 ~ QE
2:1 ~ QE
|-
|-
| | 2
| 2
| | 11\46
| 11\46
| | 286.957
| 286.957
| | [[Vines|Vines]]
| [[Vines]]
| | 2L 2s (4-tone)
| 2L 2s (4-tone)


[[4L_2s|4L 2s]] (6-tone)
[[4L 2s]] (6-tone)


[[4L_6s|4L 6s]] (10-tone)
[[4L 6s]] (10-tone)


4L 10s (14-tone)
4L 10s (14-tone)
Line 1,255: Line 1,266:


4L 38s (42-tone)
4L 38s (42-tone)
| | 12:11 ~ QE
| 12:11 ~ QE


11:1
11:1
Line 1,277: Line 1,288:
2:1 ~ QE
2:1 ~ QE
|-
|-
| | 23
| 23
| | 1\46
| 1\46
| | 26.087
| 26.087
| |  
|  
| |  
|  
| |  
|  
|}
|}


=Scales=
== Scales ==
*[[plum|plum]]
*[[plum]]
*[[sensi5|sensi5]]
*[[sensi5]]
*[[sensi8|sensi8]]
*[[sensi8]]
*[[sensi11|sensi11]]
*[[sensi11]]
*[[sensi19|sensi19]]
*[[sensi19]]


==Approximation to Mode 8 of the Harmonic Series==
=== Approximation to Mode 8 of the Harmonic Series ===


46edo represents [[overtone|overtone]]s 8 through 16 (written as [[JI|JI]] ratios 8:9:10:11:12:13:14:15:16) with degrees 0, 8, 15, 21, 27, 32, 37, 42, 46. In steps-in-between, that's 8, 7, 6, 6, 5, 5, 5, 4.
46edo represents [[Overtone|overtones]] 8 through 16 (written as [[JI]] ratios 8:9:10:11:12:13:14:15:16) with degrees 0, 8, 15, 21, 27, 32, 37, 42, 46. In steps-in-between, that's 8, 7, 6, 6, 5, 5, 5, 4.


8\46edo (208.696¢) stands in for frequency ratio [[9/8|9:8]] (203.910¢).
8\46edo (208.696¢) stands in for frequency ratio [[9/8|9:8]] (203.910¢).
Line 1,306: Line 1,317:
4\46edo (104.348¢) stands in for [[16/15|16:15]] (111.731¢).
4\46edo (104.348¢) stands in for [[16/15|16:15]] (111.731¢).


=Music=
== Music ==
[http://aaronkristerjohnson.bandcamp.com/track/satiesque Satiesque] by [[Aaron_Krister_Johnson|Aaron Krister Johnson.]]
[http://aaronkristerjohnson.bandcamp.com/track/satiesque Satiesque] by [[Aaron Krister Johnson.]]


[http://www.archive.org/details/Chromosounds Chromosounds] [http://clones.soonlabel.com/public/micro/gene_ward_smith/chromosounds/GWS-GPO-Jazz-chromosounds.mp3 play] by [[Gene_Ward_Smith|Gene Ward Smith.]]
[http://www.archive.org/details/Chromosounds Chromosounds] [http://clones.soonlabel.com/public/micro/gene_ward_smith/chromosounds/GWS-GPO-Jazz-chromosounds.mp3 play] by [[Gene Ward Smith.]]


[http://www.archive.org/details/MusicForYourEars Music For Your Ears] [http://www.archive.org/download/MusicForYourEars/musicfor.mp3 play] by [[Gene_Ward_Smith|Gene Ward Smith.]] The central portion is in [[27edo|27edo]], the rest is in 46edo.
[http://www.archive.org/details/MusicForYourEars Music For Your Ears] [http://www.archive.org/download/MusicForYourEars/musicfor.mp3 play] by [[Gene Ward Smith.]] The central portion is in [[27edo]], the rest is in 46edo.


[http://andrewheathwaite.bandcamp.com/track/rats Rats] [http://micro.soonlabel.com/gene_ward_smith/Others/Heathwaite/Newbeams/Andrew%20Heathwaite%20-%20Newbeams%20-%2001%20Rats.mp3 play] by [[Andrew_Heathwaite|Andrew Heathwaite]].
[http://andrewheathwaite.bandcamp.com/track/rats Rats] [http://micro.soonlabel.com/gene_ward_smith/Others/Heathwaite/Newbeams/Andrew%20Heathwaite%20-%20Newbeams%20-%2001%20Rats.mp3 play] by [[Andrew Heathwaite]].


[http://andrewheathwaite.bandcamp.com/track/tumbledown-stew Tumbledown Stew]  
[http://andrewheathwaite.bandcamp.com/track/tumbledown-stew Tumbledown Stew]  
[http://micro.soonlabel.com/gene_ward_smith/Others/Heathwaite/Newbeams/Andrew%20Heathwaite%20-%20Newbeams%20-%2012%20Tumbledown%20Stew.mp3 play] by [[Andrew_Heathwaite|Andrew Heathwaite]].
[http://micro.soonlabel.com/gene_ward_smith/Others/Heathwaite/Newbeams/Andrew%20Heathwaite%20-%20Newbeams%20-%2012%20Tumbledown%20Stew.mp3 play] by [[Andrew Heathwaite]].


[http://andrewheathwaite.bandcamp.com/track/hypnocloudsmack-1 Hypnocloudsmack 1] [http://micro.soonlabel.com/gene_ward_smith/Others/Heathwaite/Newbeams/Andrew%20Heathwaite%20-%20Newbeams%20-%2004%20Hypnocloudsmack%201.mp3 play] by [[Andrew_Heathwaite|Andrew Heathwaite]].
[http://andrewheathwaite.bandcamp.com/track/hypnocloudsmack-1 Hypnocloudsmack 1] [http://micro.soonlabel.com/gene_ward_smith/Others/Heathwaite/Newbeams/Andrew%20Heathwaite%20-%20Newbeams%20-%2004%20Hypnocloudsmack%201.mp3 play] by [[Andrew Heathwaite]].


[http://andrewheathwaite.bandcamp.com/track/hypnocloudsmack-2 Hypnocloudsmack 2]  
[http://andrewheathwaite.bandcamp.com/track/hypnocloudsmack-2 Hypnocloudsmack 2]  
[http://micro.soonlabel.com/gene_ward_smith/Others/Heathwaite/Newbeams/Andrew%20Heathwaite%20-%20Newbeams%20-%2009%20Hypnocloudsmack%202.mp3 play] by [[Andrew_Heathwaite|Andrew Heathwaite]].  
[http://micro.soonlabel.com/gene_ward_smith/Others/Heathwaite/Newbeams/Andrew%20Heathwaite%20-%20Newbeams%20-%2009%20Hypnocloudsmack%202.mp3 play] by [[Andrew Heathwaite]].  


[http://andrewheathwaite.bandcamp.com/track/hypnocloudsmack-3 Hypnocloudsmack 3]  
[http://andrewheathwaite.bandcamp.com/track/hypnocloudsmack-3 Hypnocloudsmack 3]  
[http://micro.soonlabel.com/gene_ward_smith/Others/Heathwaite/Newbeams/Andrew%20Heathwaite%20-%20Newbeams%20-%2013%20Hypnocloudsmack%203.mp3 play] by [[Andrew_Heathwaite|Andrew Heathwaite]].
[http://micro.soonlabel.com/gene_ward_smith/Others/Heathwaite/Newbeams/Andrew%20Heathwaite%20-%20Newbeams%20-%2013%20Hypnocloudsmack%203.mp3 play] by [[Andrew Heathwaite]].


[http://soonlabel.com/xenharmonic/wp-content/uploads/2013/10/Bach_BWV_1029_E46-Alto-Sax-+-Harpsichord.mp3 Bach BWV 1029 in 46 equal] Claudi Meneghin version
[http://soonlabel.com/xenharmonic/wp-content/uploads/2013/10/Bach_BWV_1029_E46-Alto-Sax-+-Harpsichord.mp3 Bach BWV 1029 in 46 equal] Claudi Meneghin version
Line 1,330: Line 1,341:
[http://soonlabel.com/xenharmonic/wp-content/uploads/2014/02/Bach_Contrapunctus_4-Jeux14-E46.mp3 Bach Contrapunctus 4] Claudi Meneghin version
[http://soonlabel.com/xenharmonic/wp-content/uploads/2014/02/Bach_Contrapunctus_4-Jeux14-E46.mp3 Bach Contrapunctus 4] Claudi Meneghin version


[http://micro.soonlabel.com/gene_ward_smith/Others/Freivald/a_seed_planted-starling-46-EDO.mp3 A Seed Planted - (Yet another version: 46 EDO)] by [https://soundcloud.com/jdfreivald/a-seed-planted-yet-another Jake Freivald]    
[http://micro.soonlabel.com/gene_ward_smith/Others/Freivald/a_seed_planted-starling-46-EDO.mp3 A Seed Planted - (Yet another version: 46 EDO)] by [https://soundcloud.com/jdfreivald/a-seed-planted-yet-another Jake Freivald]
 
[[Category:46edo]]
[[Category:46edo]]
[[Category:chromosounds]]
[[Category:chromosounds]]
Retrieved from "https://en.xen.wiki/w/46edo"