40edo

User:IlL/Template:RTT restriction 40edo is the equal division of the octave into 40 parts of exactly 30 cents each. 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 you to use 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. As such, calling it a perfect 5th seems very much a misnomer. Despite all keys being reachable by stacking this 5th, it does not qualify as meantone either, as stacking 4 of them results in a near perfect tridecimal neutral third rather than a major one. 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 in the 7-limit; 99/98, 121/120 and 176/175 in the 11-limit; and 66/65 in the 13-limit.

40edo is more accurate on the 2.9.5.21.33.13.51.19 2*40 subgroup, where it offers the same tuning as 80edo, and tempers out the same commas. It is also the first equal temperament to approximate both the 23rd and 19th harmonic, by tempering out the 9 cent comma to 4-edo, with 10 divisions therein.

Step #	ET	Just		Difference (ET minus Just)	Notation
Step #	Cents	Interval	Cents
0	0¢	1:1	0	0	Unison	1	D
1	30	59:58	29.5944	0.40553	Up Unison		D#
2	60	29:28	60.7512	-0.75128	Downminor 2nd		D##
3	90	20:19	88.8006	1.19930	Minor 2nd		D###/Ebbb
4	120	15:14	119.4428	0.55719	Upminor 2nd		Ebb
5	150	12:11	150.6370	-0.63705	Downmajor 2nd		Eb
6	180	10:9	182.4037	-2.40371	Major 2nd		E
7	210	9:8	203.9100	6.08999	Upmajor 2nd		E#
8	240	8:7	231.1741	8.82590	Augmented 2nd		E##
9	270	7:6	266.8709	3.12909	Diminished 3rd		Fbb
10	300	19:16	297.5130	2.48698	Downminor 3rd		Fb
11	330	6:5	315.6412	14.3587	Upminor 3rd		F
12	360	16:13	359.4723	0.52766	Neutral 3rd		F#
13	390	5:4	386.3137	3.68628	Major 3rd		F##
14	420	14:11	417.5079	2.49203	Augmented 3rd		F###/Gbbb
15	450	22:17	446.3625	3.63746	Diminished 4th		Gbb
16	480	21:16	470.781	9.219	Blackwood 4th		Gb
17	510	4:3	498.0449	11.9550	Diatonic 4th		G
18	540	11:8	551.3179	-11.3179	Augmented 4th		G#
19	570	25:18	568.7174	1.2825	Minor Tritone		G##
20	600	7:5	582.5121	17.4878	Perfect Tritone		G###/Abbb
21	630	23:16	628.2743	1.72565	Major Tritone		Abb
22	660	16:11	648.6820	11.3179	Diminished 5th		Ab
23	690	3:2	701.9550	-11.9550	Diatonic 5th		A
24	720	32:21	729.2191	-9.219	Blackwood 5th		A#
25	750	17:11	753.6374	-3.63746	Augmented 5th		A##
26	780	11:7	782.4920	-2.49203	Diminished 6th		A###/Bbbb
27	810	8:5	813.6862	-3.68628	Minor 6th		Bbb
28	840	13:8	840.5276	-0.52766	Neutral 6th		Bb
29	870	5:3	884.3587	-14.3587	Downmajor 6th		B
30	900	32:19	902.4869	-2.48698	Upmajor 6th		B#
31	930	12:7	933.1291	-3.12909	Augmented 6th		B##
32	960	7:4	968.8259	-8.82590	Harmonic 7th		Cbb
33	990	16:9	996.0899	-6.08999	Downminor 7th		Cb
34	1020	9:5	1017.5962	2.40371	Minor 7th		C
35	1050	11:6	1049.3629	0.63705	Upminor 7th		C#
36	1080	28:15	1080.5571	-0.55719	Downmajor 7th		C##
37	1110	19:10	1111.1993	-1.19930	Major 7th		C###/Dbbb
38	1140	56:29	1139.2487	0.75128	Upmajor 7th		Dbb
39	1170	116:59	1170.4055	-0.40553	Down Octave		Db
40	1200	2:1	1200	0	Octave		D