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__FORCETOC__
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=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]]

Revision as of 02:26, 21 June 2020


Theory

The 46 equal temperament, often abbreviated 46-tET, 46-EDO, or 46-ET, is the scale derived by dividing the octave into 46 equally-sized steps. Each step represents a frequency ratio of 26.087 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 it supports include sensi, valentine, shrutar, rodan, leapday and unidec. The 11-limit minimax tuning for 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, though others award that distinction to 41edo. In fact, while 41 is a zeta integral edo but not a 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, Margo Schulter) prefer, sometimes strongly, over both the 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's separated by an interval of 26.087 cents.

Shrutar22 as srutis describes a possible use of 46edo for Indian music.

Intervals

Cents Approximate Ratios* Ups and Downs Notation Solfege
0 0.000 1/1 perfect unison P1 D do
1 26.087 81/80, 64/63 up unison ^1 ^D di
2 52.174 28/27, 33/32 downminor 2nd vm2 vEb ro
3 78.261 25/24, 21/20, 22/21, 24/23, 23/22 minor 2nd m2 Eb rih
4 104.348 16/15, 17/16, 18/17 upminor 2nd ^m2 ^Eb ra
5 130.435 13/12, 14/13, 15/14 downmid 2nd v~2 ^^Eb ru (as in supraminor)
6 156.522 12/11, 11/10, 23/21 upmid 2nd ^~2 vvE ruh (as in submajor)
7 182.609 10/9 downmajor 2nd vM2 vE reh
8 208.696 9/8 major 2nd M2 E re
9 234.783 8/7, 23/20 upmajor 2nd ^M2 ^E ri
10 260.870 7/6 downminor 3rd vm3 vF ma
11 286.957 13/11, 20/17 minor 3rd m3 F meh
12 313.043 6/5 upminor 3rd ^m3 ^F me
13 339.130 11/9, 17/14 downmid 3rd v~3 ^^F mu
14 365.217 16/13, 26/21, 21/17 upmid 3rd ^~3 vvF# muh
15 391.304 5/4 downmajor 3rd vM3 vF# mi
16 417.391 14/11, 23/18 major 3rd M3 F# maa
17 443.478 9/7, 13/10, 22/17 upmajor 3rd ^M3 ^F# mo
18 469.565 21/16, 17/13 down 4th v4 vG fe
19 495.652 4/3 perfect 4th P4 G fa
20 521.739 23/17 up 4th ^4 ^G fih
21 547.826 11/8 downmid 4th v~4 ^^G fu
22 573.913 7/5, 18/13, 32/23 upmid 4th, dim 5th ^~4, d5 vvG#, Ab fi
23 600.000 17/12, 24/17 downaug 4th, updim 5th vA4, ^d5 vG#, ^Ab seh
24 626.087 10/7, 13/9, 23/16 aug 4th, downmid 5th A4, v~5 G#, ^^Ab se
25 652.174 16/11 double-down 5th ^~5 vvA su
26 678.261 34/23 down 5th v5 vA sih
27 704.348 3/2 perfect 5th P5 A sol
28 730.435 32/21, 26/17 up 5th ^5 ^A si
29 756.522 14/9, 20/13, 17/11 downminor 6th vm6 vBb lo
30 782.609 11/7 minor 6th m6 Bb leh
31 808.696 8/5 upminor 6th ^m6 ^Bb le
32 834.783 13/8, 21/13, 34/21 downmid 6th v~6 ^^Bb lu
33 860.870 18/11, 28/17, 23/14 upmid 6th ^~6 vvB luh
34 886.957 5/3 downmajor 6th vM6 vB la
35 913.043 22/13, 17/10 major 6th M6 B laa
36 939.130 12/7 upmajor 6th ^M6 ^B li
37 965.217 7/4, 40/23 downminor 7th vm7 vC ta
38 991.304 16/9, 23/13 minor 7th m7 C teh
39 1017.391 9/5 upminor 7th ^m7 ^C te
40 1043.478 11/6, 20/11, 42/23 downmid 7th v~7 ^^C tu
41 1069.565 24/13, 13/7, 28/15 upmid 7th ^~7 vvC# tuh
42 1095.652 15/8, 32/17, 17/9 downmajor 7th vM7 vC# ti
43 1121.739 48/25, 40/21, 21/11, 23/12, 44/23 major 7th M7 C# taa
44 1147.826 27/14, 64/33 upmajor 7th ^M7 ^C# to
45 1173.913 160/81, 63/32 down 8ve v8 vD da
46 1200.000 2/1 perfect 8ve P8 D do

* 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.

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

quality color monzo format examples
downminor zo {a, b, 0, 1} 7/6, 7/4
minor fourthward wa {a, b}, b < -1 32/27, 16/9
upminor gu {a, b, -1} 6/5, 9/5
downmid ilo {a, b, 0, 0, 1} 11/9, 11/6
upmid 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 > 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. 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:

color of the 3rd JI chord notes as edosteps notes of C chord written name spoken name
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 Cv~ C downmid
lu 22:27:33 0-14-27 C vvE G C^~ C upmid
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.

Selected just intervals by error

prime 2 prime 3 prime 5 prime 7 prime 11 prime 13 prime 17 prime 19 prime 23
Error absolute (¢) 0.0 +2.4 +5.0 -3.6 -3.5 -5.7 -0.61 +10.6 -2.1
relative (%) 0.0 +9.2 +19.1 -13.8 -13.4 -22.0 -2.3 -40.5 -8.4
Fifthspan 0 +1 +21 +15 +11 +8 -22 -3 +6

The following table shows how some prominent just intervals are represented in 46edo (ordered by absolute error).

Interval, complement Error (abs., in cents)
14/11, 11/7 0.117
10/9, 9/5 0.205
14/13, 13/7 2.137
13/11, 22/13 2.253
4/3, 3/2 2.393
6/5, 5/3 2.598
11/8, 16/11 3.492
8/7, 7/4 3.609
9/8, 16/9 4.786
5/4, 8/5 4.991
16/13, 13/8 5.745
12/11, 11/6 5.885
7/6, 12/7 6.001
16/15, 15/8 7.383
13/12, 24/13 8.138
11/9, 18/11 8.278
9/7, 14/9 8.394
11/10, 20/11 8.482
7/5, 10/7 8.599
18/13, 13/9 10.531
13/10, 20/13 10.736
15/11, 22/15 10.875
15/14, 28/15 10.992
15/13, 26/15 12.958

Linear temperaments

Periods
per octave 		Generator Cents Temperaments MOS/DE Scales available L:s
1 1\46 26.087
1 3\46 78.261 Valentine 1L 14s (15-tone)

15L 1s (16-tone)

16L 15s (31-tone)

4:3 ~ quasi-equal

3:1

2:1 ~ QE

1 5\46 130.435 Twothirdtonic 1L 8s (9-tone)

9L_1s (10-tone)

9L 10s (19-tone)

9L 19s (28-tone)

9L 28s (37-tone)

6:5 ~ QE

5:1

4:1

3:1

2:1 ~ QE

1 7\46 182.609 Minortone 1L 5s (6-tone)

6L 1s (7-tone)

7L 6s (13-tone)

13L 7s (20-tone)

13L 20s (33-tone)

11:7

7:4

4:3 ~ QE

3:1

2:1 ~ QE

1 9\46 234.783 Rodan 1L 4s (5-tone)

1L 5s (6-tone)

5L 6s (11-tone)

5L 11s (16-tone)

5L 16s (21-tone)

5L 21s (26-tone)

5L 26s (31-tone)

5L 31s (36-tone)

5L 36s (41-tone)

10:9 ~QE

9:1

8:1

7:1

6:1

5:1

4:1

3:1

2:1 ~ QE

1 11\46 286.957 4L 1s (5-tone)

4L 5s (9-tone)

4L 9s (13-tone)

4L 13s (17-tone)

4L 17s (21-tone)

21L 4s (25-tone)

11:2

9:2

7:2

5:2

3:2 ~ QE, Golden

2:1 ~ QE

1 13\46 339.130 Amity/hitchcock 4L 3s (7-tone)

7L 4s (11-tone)

7L 11s (18-tone)

7L 18s (25-tone)

7L 25s (32-tone)

7L 32s (39-tone)

7:6 ~ QE

6:1

5:1

4:1

3:1

2:1 ~ QE

1 15\46 391.304 Amigo 1L 2s (3-tone)

3L 1s (4-tone)

3L 4s (7-tone)

3L 7s (10-tone)

3L 10s (13-tone)

3L 13s (16-tone)

3L 16s (19-tone)

3L 19s (21-tone)

3L 21s (24-tone)

3L 24s (27-tone)

3L 27s (30-tone)

3L 30s (33-tone)

3L 33s (36-tone)

3L 36s (39-tone)

3L 39s (42-tone)

16:15 ~ QE

15:1

14:1

13:1

12:1

11:1

10:1

9:1

8:1

7:1

6:1

5:1

4:1

3:1

2:1 ~ QE

1 17\46 443.478 Sensi 3L 2s (5-tone)

3L 5s (8-tone)

8L 3s (11-tone)

8L 11s (19-tone)

19L 8s (27-tone)

12:5

7:5

5:2

3:2 ~ QE, Golden

2:1

1 19\46 495.652 Leapday 2L 3s (5-tone)

5L 2s (7-tone)

5L 7s (12-tone)

12L 5s (17-tone)

17L 12s (29-tone)

11:8

8:3

5:3 ~ Golden

3:2 ~ QE, Golden

2:1 ~ QE

1 21\46 547.826 Heinz 2L 3s (5-tone)

2L 5s (7-tone)

2L 7s (9-tone)

2L 9s (11-tone)

11L 2s (13-tone)

11L 13s (24-tone)

11L 24s (35-tone)

17:4

13:4

9:4

5:4 ~ QE

4:1

3:1

2:1 ~ QE

2 1\46 26.087 Ketchup
2 2\46 52.174 Shrutar 2L 2s (4-tone)

2L 4s (6-tone)

2L 6s (8-tone)

2L 8s (10-tone)

2L 10s (12-tone)

2L 12s (14-tone)

2L 14s (16-tone)

2L 16s (18-tone)

2L 18s (20-tone)

2L 20s (22-tone)

22L 2s (24-tone)

21:2

19:2

17:2

15:2

13:2

11:2

9:2

7:2

5:2

3:2 ~ QE, Golden

2:1 ~ QE

2 3\46 78.261 Semivalentine 2L 2s (4-tone)

2L 4s (6-tone)

2L 6s (8-tone)

2L 8s (10-tone)

2L 10s (12-tone)

2L 12s (14-tone)

14L 2s (16-tone)

16L 14s (30-tone)

20:3

17:3

14:3

11:3

8:3

5:3 ~ Golden

3:2 ~ QE, Golden

2:1 ~ QE

2 4\46 104.348 Srutal/diaschismic 2L 2s (4-tone)

2L 4s (6-tone)

2L 6s (8-tone)

2L 8s (10-tone)

10L 2s (12-tone)

12L 10s (22-tone)

12L 22s (34-tone)

19:4

15:4

11:4

7:4

4:3 ~ QE

3:1

2:1 ~ QE

2 5\46 130.435 2L 2s (4-tone)

2L 4s (6-tone)

2L 6s (8-tone)

8L 2s (10-tone)

8L 10s (18-tone)

18L 10s (28-tone)

18:5

13:5

8:5 ~ Golden

5:3 ~ Golden

3:2 ~ QE, Golden

2:1 ~ QE

2 6\46 156.522 Bison 2L 2s (4-tone)

2L 4s (6-tone)

6L 2s (8-tone)

8L 6s (14-tone)

8L 14s (22-tone)

8L 22s (30-tone)

8L 30s (38-tone

17:6

11:6

6:5 ~ QE

5:1

4:1

3:1

2:1 ~ QE

2 7\46 182.609 Unidec/hendec 2L 2s (4-tone)

2L 4s (6-tone)

6L 2s (8-tone)

6L 8s (14-tone)

6L 14s (20-tone)

20L 6s (26-tone)

16:7

9:7

7:2

5:2

3:2 ~ QE, Golden

2:1 ~ QE

2 8\46 208.696 Abigail 2L 2s (4-tone)

4L 2s (6-tone)

6L 2s (8-tone)

6L 8s (14-tone)

6L 14s (20-tone)

6L 20s (26-tone)

6L 26s (32-tone)

6L 32s (38-tone)

6L 38s (44-tone)

15:8

8:7 ~ QE

8:1

7:1

6:1

5:1

4:1

3:1

2:1 ~ QE

2 9\46 234.783 Echidnic 2L 2s (4-tone)

4L 2s (6-tone)

6L 4s (10-tone)

10L 6s (16-tone)

10L 16s (26-tone)

10L 26s (36-tone)

14:9

9:5

5:4 ~ QE

4:1

3:1

2:1 ~ QE

2 10\46 260.87 Bamity 2L 2s (4-tone)

4L 2s (6-tone)

4L 6s (10-tone)

4L 10s (14-tone)

14L 4s (18-tone)

14L 18s (32-tone)

13:10

10:3

7:3

4:3 ~ QE

3:1

2:1 ~ QE

2 11\46 286.957 Vines 2L 2s (4-tone)

4L 2s (6-tone)

4L 6s (10-tone)

4L 10s (14-tone)

4L 14s (18-tone)

4L 18s (22-tone)

4L 22s (26-tone)

4L 26s (30-tone)

4L 30s (34-tone)

4L 34s (38-tone)

4L 38s (42-tone)

12:11 ~ QE

11:1

10:1

9:1

8:1

7:1

6:1

5:1

4:1

3:1

2:1 ~ QE

23 1\46 26.087

Scales

Approximation to Mode 8 of the Harmonic Series

46edo represents 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 (203.910¢).

7\46edo (182.609¢) stands in for 10:9 (182.404¢).

6\46edo (156.522¢) stands in for 11:10 (165.004¢) and 12:11 (150.637¢).

5\46edo (130.435¢) stands in for 13:12 (138.573¢), 14:13 (128.298¢) and 15:14 (119.443¢).

4\46edo (104.348¢) stands in for 16:15 (111.731¢).

Music

Satiesque by Aaron Krister Johnson.

Chromosounds play by Gene Ward Smith.

Music For Your Ears play by Gene Ward Smith. The central portion is in 27edo, the rest is in 46edo.

Rats play by Andrew Heathwaite.

Tumbledown Stew play by Andrew Heathwaite.

Hypnocloudsmack 1 play by Andrew Heathwaite.

Hypnocloudsmack 2 play by Andrew Heathwaite.

Hypnocloudsmack 3 play by Andrew Heathwaite.

Bach BWV 1029 in 46 equal Claudi Meneghin version

Bach Contrapunctus 4 Claudi Meneghin version

A Seed Planted - (Yet another version: 46 EDO) by Jake Freivald

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