40edo: Difference between revisions

{{EDO intro|40}}

Up to this point, all the multiples of 5 have had the 720 cent blackwood 5th as their best approximation of 3/2. 35edo combined the small circles of blackwood and whitewood 5ths, almost equally far from just, requiring the use of both to reach all keys. 40edo adds a diatonic 5th that's closer to just. However, it is still the second flattest diatonic 5th, only exceeded by 47edo in error, which results in it being inconsistent in the 5-limit - combining the best major and minor third will result in the blackwood 5th instead. So some may not consider it a valid perfect fifth.

Despite all keys being reachable by stacking this 5th, it does not qualify as meantone either, tempering out [[177147/163840]] and [[1053/1024]] in the patent val instead of [[81/80]], which means 4 5ths make a near perfect [[16/13|tridecimal neutral 3rd]] and it takes a full 11 to reach the 5th harmonic.  

81/80 is only tempered out in the 40c alternative val where the aforementioned high neutral 3rd is equated with 5/4 instead of the much more accurate 390-cent interval. The resulting [[5L 2s]] scale has large steps of 6 intervals and small ones of 5, putting sharps and flats right next to letters without any ups or downs in between and requiring a lot of them to notate more distant keys. It tempers out [[648/625]] in the 5-limit; [[225/224]] and [[16807/16384]] in the 7-limit; [[99/98]], [[121/120]] and [[176/175]] in the 11-limit; and [[66/65]] in the 13-limit.

40edo can be treated as a [[dual-fifth system]] in the 2.3+.3-.5.7.11 subgroup, or the 2.3+.3-.5.7.11.13.19.23 subgroup for those who aren’t intimidated by lots of [[basis element]]s. As a dual-fifth system, it really shines, as both of its fifths have low enough [[harmonic entropy]] to sound [[consonant]] to many listeners, giving two consonant intervals for the price of one.

! style="text-align:center;" |Cents

! colspan="3" |Notation

! colspan="2" |Approximate ratios

!Difference

!0

!0¢

|perfect unison

|P1

|D

|1:1

|<small>''0''</small>

|0

!1

!30

|augmented 1sn

|A1

|D#

|59:58

|<small>''29.5944''</small>

|0.40553

!2

!60

|double-aug 1sn

|AA1

|Dx

|29:28

|<small>''60.7512''</small>

| -0.75128

!3

!90

|double-dim 2nd

|dd2

|D#x, Ebbb

|20:19

|<small>''88.8006''</small>

|1.19930

!4

!120

|diminished 2nd

|d2

|Ebb

|15:14

|<small>''119.4428''</small>

|0.55719

!5

!150

|minor 2nd

|m2

|Eb

|12:11

|<small>''150.6370''</small>

| -0.63705

!6

!180

|major 2nd

|M2

|E

|10:9

|<small>''182.4037''</small>

| -2.40371

!7

!210

|augmented 2nd

|A2

|E#

|9:8

|<small>''203.9100''</small>

|6.08999

!8

!240

|double-aug 2nd

|AA2

|Ex

|8:7

|<small>''231.1741''</small>

|8.82590

!9

!270

|double-dim 3rd

|dd3

|Fbb

|7:6

|<small>''266.8709''</small>

|3.12909

!10

!300

|diminished 3rd

|d3

|Fb

|19:16

|<small>''297.5130''</small>

|2.48698

!11

!330

|minor 3rd

|m3

|F

|6:5

|<small>''315.6412''</small>

|14.3587

!12

!360

|major 3rd

|M3

|F#

|16:13

|<small>''359.4723''</small>

|0.52766

!13

!390

|augmented 3rd

|A3

|Fx

|5:4

|<small>''386.3137''</small>

|3.68628

!14

!420

|double-aug 3rd

|AA3

|F#x, Gbbb

|14:11

|<small>''417.5079''</small>

|2.49203

!15

!450

|double-dim 4th

|dd4

|Gbb

|22:17

|<small>''446.3625''</small>

|3.63746

!16

!480

|diminished 4th

|d4

|Gb

|21:16

|<small>''470.781''</small>

|9.219

!17

!510

|perfect 4th

|P4

|G

|4:3

|<small>''498.0449''</small>

|11.9550

!18

!540

|augmented 4th

|A4

|G#

|11:8

|<small>''551.3179''</small>

| -11.3179

!19

!570

|double-aug 4th

|AA4

|G##

|25:18

|<small>''568.7174''</small>

|1.2825

!20

!600

|triple-aug 4th,

|AAA4,

|Gx#, Abbb

|7:5

|<small>''582.5121''</small>

|17.4878

!21

!630

|double-dim 5th

|dd5

|Abb

|23:16

|<small>''628.2743''</small>

|1.72565

!22

!660

|diminished 5th

|d5

|Ab

|16:11

|<small>''648.6820''</small>

|11.3179

!23

!690

|perfect 5th

|P5

|A

|3:2

|<small>''701.9550''</small>

| -11.9550

!24

!720

|augmented 5th

|A5

|A#

|32:21

|<small>''729.2191''</small>

| -9.219

!25

!750

|double-aug 5th

|AA5

|Ax

|17:11

|<small>''753.6374''</small>

| -3.63746

!26

!780

|double-dim 6th

|dd6

|A#x, Bbbb

|11:7

|<small>''782.4920''</small>

| -2.49203

!27

!810

|diminished 6th

|d6

|Bbb

| style="text-align:center;" |8:5

|<small>''813.6862''</small>

| -3.68628

!28

!840

|minor 6th

|m6

|Bb

|13:8

|<small>''840.5276''</small>

| -0.52766

!29

!870

|major 6th

|M6

|B

| style="text-align:center;" |5:3

|<small>''884.3587''</small>

| -14.3587

!30

!900

|augmented 6th

|A6

|B#

| style="text-align:center;" |32:19

|<small>''902.4869''</small>

| -2.48698

!31

!930

|double-aug 6th

|AA6

|Bx

| style="text-align:center;" |12:7

|<small>''933.1291''</small>

| -3.12909

!32

!960

|double-dim 7th

|dd7

|Cbb

| style="text-align:center;" |7:4

|<small>''968.8259''</small>

| -8.82590

!33

!990

|diminished 7th

|d7

|Cb

| style="text-align:center;" |16:9

|<small>''996.0899''</small>

| -6.08999

!34

!1020

|minor 7th

|m7

|C

| style="text-align:center;" |9:5

|<small>''1017.5962''</small>

|2.40371

!35

!1050

|major 7th

|M7

|C#

| style="text-align:center;" |11:6

|<small>''1049.3629''</small>

|0.63705

!36

!1080

|augmented 7th

|A7

|Cx

| style="text-align:center;" |28:15

|<small>''1080.5571''</small>

| -0.55719

!37

!1110

|double-aug 7th

|AA7

|C#x, Dbbb

| style="text-align:center;" |19:10

|<small>''1111.1993''</small>

| -1.19930

!38

!1140

|double-dim 8ve

|dd8

|Dbb

| style="text-align:center;" |56:29

|<small>''1139.2487''</small>

|0.75128

!39

!1170

|diminished 8ve

|d8

|Db

| style="text-align:center;" |116:59

|<small>''1170.4055''</small>

| -0.40553

!40

!1200

|perfect octave

|P8

|D

| style="text-align:center;" |2:1

|<small>''1200''</small>

|0

===Sagittal notation===

{{todo|inline=1|complete section|add examples}}

== Instruments ==

* [[Lumatone mapping for 40edo]]