User:Lucius Chiaraviglio/Musical Mad Science

For now, my Musical Mad Science Musings and Ramblings will go on this page.

The Accuracy Fever Dream

This morning I had this weird fever dream about an increasingly futile effort to achieve perfect 5-limit and 7-limit accuracy by way of representation of pitches using DNA fragments (which I sometimes have to manipulate in my day job, although not for the purpose of representing pitches — that's a new one on me). DNA fragments representing the syntonic comma, the schisma, the septimal comma, and the garischisma were in little tubes of DNA, but as the commas got divided up in the ever more frantic quest for accuracy, the tubes of increasingly chopped-up DNA fragments representing increasingly smaller comma fractions got all mixed up into little pieces parts that nobody could any longer keep track of, and eventually I had to give up and pick some no-longer-identified DNA fragment representing (maybe) a random syntonic comma fragment and make a post with it concluding that we had reached saturation to the point of just-noticeable difference in the quest for accuracy, and that parts is parts.

Added: Lucius Chiaraviglio (talk) 20:18, 17 November 2024 (UTC)

Various Lumatone mappings

Moved to Keyboard Layout lab: Lucius Chiaraviglio (talk) 06:55, 28 March 2025 (UTC)

Musical Mad Science Musings on Diatonicized Chromaticism

Thoughts on Diatonicized Chromaticism (11L 2s) moved here from Talk:11L 2s Lucius Chiaraviglio (talk) 08:59, 18 March 2025 (UTC)

These ramblings are copied from my comments (minus the aggravation of YouTube eating posts, and plus Xenharmonic Wikification of links, including sadly decapitalizing EDO to get said links to work) on Claudi Meneghin's YouTube instance of his arrangement of John Dowland's Pavana «Lachrimae» (from «Seven Tears») in 50edo, the hearing of which led me to realize that 50edo can actually support Ivan Wyschnegradsky's Diatonicized Chromaticism (11L 2s) scale.

My YouTube comments start here

I just had a crazy idea for your next musical mad science experiment (and it potentially includes 50edo): See if it is possible to retune some of the quarter-tone (24edo, "diatonicized chromatic")11L 2s (L/s = 2) scale works of Ivan Wyschnegradsky into other tuning systems that support 11L 2s and have a good approximation and single circle of 11/8 (or 16/11). Plausible candidate tuning systems on the soft side are 37edo (L/s = 3/2, and has a super-good 11/8), 61edo (L/s = 5/3, but 61edo is big enough to be pushing the limits of plausibility), and 50edo (L/s = 4/3 -- might be too soft). Plausible candidate tuning systems on the hard side are 35edo (L/s = 3), 59edo (L/s = 7/3, but 59edo is big enough to be pushing the limits of plausibility), and 46edo (L/s = 4/1 -- might be too hard).

Most of Ivan Wyschnegradsky's quarter-tone pieces are for 2 pianos tuned a quarter tone apart (in a few cases with other instruments); he did have a couple of quarter-tone pianos and even a quarter-tone harmonium built, but was not very satisfied with them (based on quarter-tone piano photos and video footage, I am going to hazard a guess that this was for ergonomic reasons); I think that with the way he wrote this music, it really does need the resonance and timbre of pianos.

The easiest pieces to deal with in this way would probably be a couple of his 24 Preludes (my favorites are III, VII, and VIII, but that is not an exclusive list of good choices).

If you look up a certain music organization named after Greek mythological figures responsible for inspiring artists that also has sheet music but that YouTube's censoring algorithm seems to think is a terrorist organization or something MuseScore, and there you look up Ivan Wyschnegradsky, they have several of his compositions. I looked at the 24 Preludes, and they offer several formats, including MIDI as well as some formats displayable as sheet music, although I haven't tested their output myself.

It is also worth going to Wikipedia and looking him up to get a list of compositions, and then searching for them and listening to them on YouTube. These compositions (other than the early ones in 12edo or 12WT) are not just somewhat xenharmonic like something that was originally written in quarter-comma meantone or some well-temperament -- they are seriously xenharmonic out of the box.

For the 24 Preludes, I would recommend Ivan Wyschnegradsky - 24 Quarter-Tone Preludes for two pianos Op. 22 (audio + sheet music) for an introduction that has the English translation of his own writing on diatonicized chromaticism followed by a complete set of the 24 Preludes well-performed with sheet music so that you can see the dynamic and tempo markings (not sure how much of that makes it through in the MIDI files or even in the MuseScore sheet music). The only downside of that one is that reading the music as 2 12edo piano parts for instruments tuned a quarter-tone apart might be problematic. In that case, a subset of the 24 Preludes are available (after you scroll down some ways) as individual videos on the YouTube channel of musicaignotus which have sheet music written for an actual quarter-tone piano. Ivan Wyschnegradsky used a non-standard semiflat symbol -- instead of looking like a (possibly narrower) backwards flat, it looks like a normal flat that has the bottom peeled open so that it looks like a cross between a normal flat and an 'h'.

Got a chance to look into this a bit more. For the fifths in the basic 11L 2s (Wyschnegradsky) diatonicized chromatic scale (the version that 24edo yields), the 11/8-span of a patent fifth is a stack of 10 intervals of 11/8, octave-reduced.

On the soft side, this still works for 37edo and 61edo, but if you go further afield, with 50edo you instead get the 5edo fifth, so that is too far on the soft end of the scale tree (although might still be good to include it for instructional purposes).

On the hard side, this still works for 35edo (the flat fifth ends up being the patent fifth by a hair), and for 59edo you get the not the patent fifth but the alternate flat fifth, which is barely further away from just and is still in the range of flattone, so we can call that still sort of working; but with 46edo you instead get the 23edo flat fifth, so that is too far on the hard end of the scale tree (although might still be good to include it for instructional purposes).

I'll do this separately for the major third, although the 24edo major third is at best mediocre in terms of relative error, being as sharp as that of 12edo, but in the context of increments of half the size.

Doing this for the 5/4 major third: The 16/11-span (goes the other way around the circle of 11/8) of this is in 24edo is 8.

On the soft side, this works very well for 37edo, for which the 5/4 major third is nearly just, as well as for 50edo, for which the 5/4 major third is just slightly flat, and for 61edo, for which the 5/4 major third is mildly sharp (although in terms of relative error it ends up being even worse than for 24edo).

On the hard side, this already quits working for 35edo, for which the 5/4 major third is fairly flat (you instead get a very sharp alternative major third, a bit sharp of a Pythagorean major third, and almost up to 33/26). For 46edo, it gets even worse, giving an alternate-alternate sharp major third (the patent 5/4 (sub-)major third is already slightly sharp), actually more like 14/11. For 59edo, the situation is similar, even though the 59edo 5/4 (sub-)maor third is just barely sharp of just, instead giving 19/15. So even the mildly hard side of the 11L 2s tuning spectrum doesn't work for the 5th harmonic, despite working for the third harmonic.

I didn't do the 7th harmonic, because Ivan Wyschnegradsky himself wrote in the text at the beginning of the 24 Quarter-Tone Preludes for two pianos linked above that you really need something more than 24edo to get the 7th harmonic (his choice for this in almost all of his compositions that could have reasonably dealt with it was 36edo or 72edo, although he also wrote a single 31edo composition for the Fokker organ; but none of those support his diatonicized chromatic scale, other than 72edo in redundant form (being 3 * 24edo). It would be interesting to do as a future back-extension to the 24 & 37 temperament, but it is understandable why he didn't use it, since 24edo has a bad 7th harmonic, and you have to go to either the superhard (46edo) region or the soft region (37edo through 50edo) to get a good 7th harmonic within the 11L 2s tuning spectrum.

Added: Lucius Chiaraviglio (talk) 10:18, 25 January 2025 (UTC)
Last modified: Lucius Chiaraviglio (talk) 08:25, 26 April 2025 (UTC)

Comma for getting the fifth on the circle of 11/8 or 16/11 in the middle of the 11L 2s tuning spectrum

The comma |-33 -1 0 0 10⟩ (11.224¢) equates a stack of ten 11/8 (octave-reduced) to 3/2. However, this only gives the patent fifth in more or less the range 35EDO to 37EDO. For 50EDO (as noted above) it gives the Blackwood (pentatonic) fifth; while for 46EDO it gives the 23EDO flat fifth.

Still need comma for back-extension to 5th harmonic and maybe back extension to 7th harmonic.

Added: Lucius Chiaraviglio (talk) 11:04, 16 February 2025 (UTC)
Last modified: Lucius Chiaraviglio (talk) 07:00, 9 April 2025 (UTC)

Rationale for leaving some of this on the Talk:11L 2s page

Even after the above gets its own space on a temperament page (if that ever happens), something I would like to see (but have no way to generate myself) on the 11L 2s page would be a musical samples akin to what the 5L 3s page has for musical samples. Ivan Wyschnegradsky wrote a considerable volume of diatonicized chromatic music, including Préludes dans tous les tons de l'échelle chromatique diatonisée à 13 sons (24), for 2 pianos in quarter tones, Op. 22 (1934, rev. 1960). If a short one of these works (such as one or two of the Preludes) could be gotten in MIDI form and processed through a synthesizer with a high-quality piano emulation in various tuning systems (as noted in the temperament discussion above), this would provide an excellent Musical Samples section for this page.

Added: Lucius Chiaraviglio (talk) 09:29, 18 March 2025 (UTC)

Table of odd harmonics for various EDO values supporting 11L 2s

This table (actually a collection of tables for now) is for tracking trends in odd harmonics along the tuning spectrum of 11L 2s; it is intended to match the organization of the corresponding scale tree:

13edo (L=1, s=1, ~16/11 = 7\13) — Equalized 11L 2s

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	+36.51	-17.08	-45.75	-19.29	+2.53	-9.76	+19.42	-12.65	-20.59	-9.24	+17.88	-34.17	+17.21	-14.19	-37.34	+39.03	+29.48	+25.58	+26.75	+32.48	+42.33	-36.38	-19.35	+0.81	+23.86	-42.74	-14.55	+15.92
	Relative (%)	+39.5	-18.5	-49.6	-20.9	+2.7	-10.6	+21.0	-13.7	-22.3	-10.0	+19.4	-37.0	+18.6	-15.4	-40.5	+42.3	+31.9	+27.7	+29.0	+35.2	+45.9	-39.4	-21.0	+0.9	+25.8	-46.3	-15.8	+17.2
Steps (reduced)		21 (8)	30 (4)	36 (10)	41 (2)	45 (6)	48 (9)	51 (12)	53 (1)	55 (3)	57 (5)	59 (7)	60 (8)	62 (10)	63 (11)	64 (12)	66 (1)	67 (2)	68 (3)	69 (4)	70 (5)	71 (6)	71 (6)	72 (7)	73 (8)	74 (9)	74 (9)	75 (10)	76 (11)

76edo (L=6, s=5, ~16/11 = 41\76)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	-7.22	-7.37	-5.67	+1.35	+1.31	-3.69	+1.20	+5.57	+2.49	+2.90	+3.30	+1.06	-5.87	-3.26	+7.60	-5.90	+2.76	+1.29	+4.89	-2.75	-6.25	-6.01	-2.35	+4.45	-1.65	-5.08	-6.05	-4.73
	Relative (%)	-45.7	-46.7	-35.9	+8.6	+8.3	-23.3	+7.6	+35.3	+15.8	+18.4	+20.9	+6.7	-37.1	-20.7	+48.1	-37.4	+17.4	+8.2	+30.9	-17.4	-39.6	-38.1	-14.9	+28.2	-10.4	-32.2	-38.3	-30.0
Steps (reduced)		120 (44)	176 (24)	213 (61)	241 (13)	263 (35)	281 (53)	297 (69)	311 (7)	323 (19)	334 (30)	344 (40)	353 (49)	361 (57)	369 (65)	377 (73)	383 (3)	390 (10)	396 (16)	402 (22)	407 (27)	412 (32)	417 (37)	422 (42)	427 (47)	431 (51)	435 (55)	439 (59)	443 (63)

63edo (L=5, s=4, ~16/11 = 34\63)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	+2.81	-5.36	+2.60	+5.61	+1.06	-2.43	-2.55	+9.33	+7.25	+5.41	+0.30	+8.32	+8.42	-1.01	-2.18	+3.87	-2.76	-3.72	+0.37	+9.03	+2.77	+0.25	+1.16	+5.21	-6.91	+2.69	-4.30	-8.99
	Relative (%)	+14.7	-28.1	+13.7	+29.5	+5.6	-12.8	-13.4	+49.0	+38.1	+28.4	+1.6	+43.7	+44.2	-5.3	-11.4	+20.3	-14.5	-19.6	+2.0	+47.4	+14.5	+1.3	+6.1	+27.3	-36.3	+14.1	-22.6	-47.2
Steps (reduced)		100 (37)	146 (20)	177 (51)	200 (11)	218 (29)	233 (44)	246 (57)	258 (6)	268 (16)	277 (25)	285 (33)	293 (41)	300 (48)	306 (54)	312 (60)	318 (3)	323 (8)	328 (13)	333 (18)	338 (23)	342 (27)	346 (31)	350 (35)	354 (39)	357 (42)	361 (46)	364 (49)	367 (52)

113edo (L=9, s=7, ~16/11 = 61\113)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	-1.07	-4.01	-2.45	-2.14	+0.89	-1.59	-5.08	+1.24	-0.17	-3.52	-1.73	+2.59	-3.21	+0.51	+1.87	-0.18	+4.15	+3.52	-2.66	-4.28	-1.78	+4.47	+3.52	-4.91	+0.17	-2.71	-3.12	-1.24
	Relative (%)	-10.1	-37.8	-23.1	-20.2	+8.4	-15.0	-47.9	+11.7	-1.6	-33.2	-16.3	+24.4	-30.2	+4.8	+17.6	-1.7	+39.1	+33.2	-25.0	-40.3	-16.8	+42.1	+33.1	-46.2	+1.6	-25.5	-29.4	-11.7
Steps (reduced)		179 (66)	262 (36)	317 (91)	358 (19)	391 (52)	418 (79)	441 (102)	462 (10)	480 (28)	496 (44)	511 (59)	525 (73)	537 (85)	549 (97)	560 (108)	570 (5)	580 (15)	589 (24)	597 (32)	605 (40)	613 (48)	621 (56)	628 (63)	634 (69)	641 (76)	647 (82)	653 (88)	659 (94)

50edo (L=4, s=3, ~16/11 = 27\50) — Supersoft 11L 2s

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	-5.96	-2.31	-8.83	-11.91	+0.68	-0.53	-8.27	-8.96	-9.51	+9.22	-4.27	-4.63	+6.13	+2.42	+6.96	-5.27	-11.14	-11.34	-6.48	+2.94	-7.52	+9.78	+6.49	+6.35	+9.09	-9.50	-1.63	+8.53
	Relative (%)	-24.8	-9.6	-36.8	-49.6	+2.8	-2.2	-34.5	-37.3	-39.6	+38.4	-17.8	-19.3	+25.6	+10.1	+29.0	-22.0	-46.4	-47.3	-27.0	+12.2	-31.3	+40.7	+27.1	+26.5	+37.9	-39.6	-6.8	+35.5
Steps (reduced)		79 (29)	116 (16)	140 (40)	158 (8)	173 (23)	185 (35)	195 (45)	204 (4)	212 (12)	220 (20)	226 (26)	232 (32)	238 (38)	243 (43)	248 (48)	252 (2)	256 (6)	260 (10)	264 (14)	268 (18)	271 (21)	275 (25)	278 (28)	281 (31)	284 (34)	286 (36)	289 (39)	292 (42)

137edo (L=11, s=8, ~16/11 = 74\137)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	-1.23	-0.91	+3.44	-2.45	+0.51	+0.35	-2.14	+0.15	+0.30	+2.21	+2.38	-1.82	-3.68	+4.00	+2.41	-0.72	+2.52	+2.67	-0.88	+0.13	-3.49	-3.36	+0.19	-1.89	-1.07	+2.41	-0.41	-0.93
	Relative (%)	-14.0	-10.4	+39.2	-28.0	+5.8	+4.0	-24.4	+1.8	+3.4	+25.3	+27.2	-20.8	-42.0	+45.7	+27.5	-8.2	+28.8	+30.5	-10.0	+1.5	-39.8	-38.4	+2.1	-21.5	-12.2	+27.5	-4.6	-10.6
Steps (reduced)		217 (80)	318 (44)	385 (111)	434 (23)	474 (63)	507 (96)	535 (124)	560 (12)	582 (34)	602 (54)	620 (72)	636 (88)	651 (103)	666 (118)	679 (131)	691 (6)	703 (18)	714 (29)	724 (39)	734 (49)	743 (58)	752 (67)	761 (76)	769 (84)	777 (92)	785 (100)	792 (107)	799 (114)

87edo (L=7, s=5, ~16/11 = 47\87)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	+1.49	-0.11	-3.31	+2.99	+0.41	+0.85	+1.39	+5.39	+5.94	-1.82	+6.21	-0.21	+4.48	+4.91	-0.21	+1.90	-3.42	-3.07	+2.34	-1.48	-1.17	+2.88	-3.44	-6.62	+6.88	-4.54	+0.30	-6.36
	Relative (%)	+10.8	-0.8	-24.0	+21.7	+2.9	+6.2	+10.1	+39.1	+43.0	-13.2	+45.0	-1.5	+32.5	+35.6	-1.5	+13.8	-24.8	-22.2	+17.0	-10.7	-8.5	+20.9	-24.9	-48.0	+49.9	-32.9	+2.2	-46.1
Steps (reduced)		138 (51)	202 (28)	244 (70)	276 (15)	301 (40)	322 (61)	340 (79)	356 (8)	370 (22)	382 (34)	394 (46)	404 (56)	414 (66)	423 (75)	431 (83)	439 (4)	446 (11)	453 (18)	460 (25)	466 (31)	472 (37)	478 (43)	483 (48)	488 (53)	494 (59)	498 (63)	503 (68)	507 (72)

124edo (L=10, s=7, ~16/11 = 67\124)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	+4.50	+0.78	-1.08	-0.68	+0.29	+1.41	-4.40	+1.50	+2.49	+3.41	+0.76	+1.57	+3.81	-3.77	-3.10	+4.79	-0.30	+0.27	-3.77	-3.26	+1.39	+0.10	+2.24	-2.17	-3.68	-2.54	+1.08	-2.69
	Relative (%)	+46.5	+8.1	-11.2	-7.1	+3.0	+14.5	-45.4	+15.5	+25.7	+35.3	+7.8	+16.2	+39.4	-39.0	-32.0	+49.5	-3.1	+2.8	-39.0	-33.6	+14.3	+1.0	+23.1	-22.4	-38.1	-26.2	+11.1	-27.8
Steps (reduced)		197 (73)	288 (40)	348 (100)	393 (21)	429 (57)	459 (87)	484 (112)	507 (11)	527 (31)	545 (49)	561 (65)	576 (80)	590 (94)	602 (106)	614 (118)	626 (6)	636 (16)	646 (26)	655 (35)	664 (44)	673 (53)	681 (61)	689 (69)	696 (76)	703 (83)	710 (90)	717 (97)	723 (103)

37edo (L=3, s=2, ~16/11 is 20\37) — Soft 11L 2s

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	+11.56	+2.88	+4.15	-9.32	+0.03	+2.72	+14.43	-7.66	-5.62	+15.71	-12.06	+5.75	+2.24	+8.26	-9.90	+11.59	+7.02	+8.12	+14.27	-7.44	+7.40	-6.44	+15.57	+8.29	+3.90	+2.17	+2.91	+5.94
	Relative (%)	+35.6	+8.9	+12.8	-28.7	+0.1	+8.4	+44.5	-23.6	-17.3	+48.4	-37.2	+17.7	+6.9	+25.5	-30.5	+35.7	+21.7	+25.0	+44.0	-22.9	+22.8	-19.9	+48.0	+25.6	+12.0	+6.7	+9.0	+18.3
Steps (reduced)		59 (22)	86 (12)	104 (30)	117 (6)	128 (17)	137 (26)	145 (34)	151 (3)	157 (9)	163 (15)	167 (19)	172 (24)	176 (28)	180 (32)	183 (35)	187 (2)	190 (5)	193 (8)	196 (11)	198 (13)	201 (16)	203 (18)	206 (21)	208 (23)	210 (25)	212 (27)	214 (29)	216 (31)

135edo (L=11, s=7, ~16/11 = 73\135)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	+0.27	-4.09	+0.06	+0.53	-0.21	+3.92	-3.82	+1.71	-4.18	+0.33	+2.84	+0.71	+0.80	+1.53	+1.63	+0.06	-4.03	-2.46	+4.18	-2.40	+4.04	-3.56	+1.16	+0.13	+1.98	-2.39	-4.30	-3.91
	Relative (%)	+3.0	-46.0	+0.7	+6.0	-2.3	+44.1	-43.0	+19.3	-47.0	+3.7	+31.9	+7.9	+9.0	+17.3	+18.3	+0.7	-45.3	-27.6	+47.1	-27.0	+45.4	-40.0	+13.1	+1.4	+22.3	-26.9	-48.4	-44.0
Steps (reduced)		214 (79)	313 (43)	379 (109)	428 (23)	467 (62)	500 (95)	527 (122)	552 (12)	573 (33)	593 (53)	611 (71)	627 (87)	642 (102)	656 (116)	669 (129)	681 (6)	692 (17)	703 (28)	714 (39)	723 (48)	733 (58)	741 (66)	750 (75)	758 (83)	766 (91)	773 (98)	780 (105)	787 (112)

98edo (L=8, s=5, ~16/11 = 53\98)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	-4.00	+5.52	-1.48	+4.25	-0.30	+4.37	+1.53	+5.25	-3.64	-5.47	-3.78	-1.20	+0.26	-1.01	+5.98	-4.29	+4.04	+5.80	+0.37	-0.49	+2.77	-2.47	-4.28	-2.96	+1.25	-4.12	+5.23	+4.61
	Relative (%)	-32.6	+45.1	-12.1	+34.7	-2.4	+35.7	+12.5	+42.9	-29.7	-44.7	-30.9	-9.8	+2.1	-8.2	+48.9	-35.1	+33.0	+47.4	+3.1	-4.0	+22.6	-20.2	-35.0	-24.2	+10.2	-33.6	+42.7	+37.7
Steps (reduced)		155 (57)	228 (32)	275 (79)	311 (17)	339 (45)	363 (69)	383 (89)	401 (9)	416 (24)	430 (38)	443 (51)	455 (63)	466 (74)	476 (84)	486 (94)	494 (4)	503 (13)	511 (21)	518 (28)	525 (35)	532 (42)	538 (48)	544 (54)	550 (60)	556 (66)	561 (71)	567 (77)	572 (82)

159edo (L=13, s=8, ~16/11 = 86\159)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	-0.07	-1.41	-2.79	-0.14	-0.37	-2.79	-1.48	+0.70	-3.17	-2.86	-1.86	-2.82	-0.20	-3.16	+2.13	-0.44	+3.35	-2.29	-2.86	+1.13	+1.69	-1.54	-1.36	+1.97	+0.64	+1.97	-1.78	-3.24
	Relative (%)	-0.9	-18.7	-36.9	-1.8	-5.0	-37.0	-19.6	+9.3	-42.0	-37.8	-24.6	-37.3	-2.7	-41.9	+28.3	-5.9	+44.4	-30.3	-37.9	+14.9	+22.4	-20.5	-18.0	+26.1	+8.4	+26.1	-23.6	-43.0
Steps (reduced)		252 (93)	369 (51)	446 (128)	504 (27)	550 (73)	588 (111)	621 (144)	650 (14)	675 (39)	698 (62)	719 (83)	738 (102)	756 (120)	772 (136)	788 (152)	802 (7)	816 (21)	828 (33)	840 (45)	852 (57)	863 (68)	873 (78)	883 (88)	893 (98)	902 (107)	911 (116)	919 (124)	927 (132)

61edo (L=5, s=3, ~16/11 = 33\61) — Semisoft 11L 2s

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	+6.24	+7.13	-4.89	-7.19	-0.50	+5.37	-6.30	-6.59	-2.43	+1.35	+1.23	-5.41	-0.95	-6.63	-4.05	+5.74	+2.24	+4.39	-8.06	+3.72	-0.04	-0.06	+3.35	-9.78	-0.35	-7.93	+6.63	+3.81
	Relative (%)	+31.7	+36.2	-24.9	-36.5	-2.5	+27.3	-32.0	-33.5	-12.4	+6.9	+6.3	-27.5	-4.8	-33.7	-20.6	+29.2	+11.4	+22.3	-41.0	+18.9	-0.2	-0.3	+17.0	-49.7	-1.8	-40.3	+33.7	+19.4
Steps (reduced)		97 (36)	142 (20)	171 (49)	193 (10)	211 (28)	226 (43)	238 (55)	249 (5)	259 (15)	268 (24)	276 (32)	283 (39)	290 (46)	296 (52)	302 (58)	308 (3)	313 (8)	318 (13)	322 (17)	327 (22)	331 (26)	335 (30)	339 (34)	342 (37)	346 (41)	349 (44)	353 (48)	356 (51)

146edo (L=12, s=7, ~16/11 = 79\146)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	-3.32	-0.01	+1.04	+1.57	-0.63	-2.17	-3.34	+1.89	-1.62	-2.29	-3.62	-0.02	-1.76	-2.18	-2.57	-3.96	+1.02	+3.45	+2.72	-1.67	-1.93	+1.56	+0.25	+2.07	-1.43	-2.27	-0.65	+3.27
	Relative (%)	-40.5	-0.2	+12.6	+19.1	-7.7	-26.4	-40.6	+23.0	-19.7	-27.8	-44.0	-0.3	-21.4	-26.5	-31.3	-48.2	+12.5	+42.0	+33.1	-20.3	-23.5	+18.9	+3.0	+25.2	-17.4	-27.6	-7.9	+39.8
Steps (reduced)		231 (85)	339 (47)	410 (118)	463 (25)	505 (67)	540 (102)	570 (132)	597 (13)	620 (36)	641 (57)	660 (76)	678 (94)	694 (110)	709 (125)	723 (139)	736 (6)	749 (19)	761 (31)	772 (42)	782 (52)	792 (62)	802 (72)	811 (81)	820 (90)	828 (98)	836 (106)	844 (114)	852 (122)

85edo (L=7, s=4, ~16/11 = 46\85)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	+3.93	-5.14	+5.29	-6.26	-0.73	+6.53	-1.21	-6.13	-1.04	-4.90	+7.02	+3.84	-2.34	+1.01	-1.51	+3.20	+0.15	+2.77	-3.66	-5.53	-3.28	+2.72	-1.98	-3.53	-2.20	+1.79	-5.87	+2.88
	Relative (%)	+27.8	-36.4	+37.5	-44.4	-5.2	+46.3	-8.6	-43.4	-7.4	-34.7	+49.7	+27.2	-16.5	+7.2	-10.7	+22.6	+1.1	+19.6	-25.9	-39.2	-23.3	+19.2	-14.0	-25.0	-15.6	+12.7	-41.6	+20.4
Steps (reduced)		135 (50)	197 (27)	239 (69)	269 (14)	294 (39)	315 (60)	332 (77)	347 (7)	361 (21)	373 (33)	385 (45)	395 (55)	404 (64)	413 (73)	421 (81)	429 (4)	436 (11)	443 (18)	449 (24)	455 (30)	461 (36)	467 (42)	472 (47)	477 (52)	482 (57)	487 (62)	491 (66)	496 (71)

109edo (L=9, s=5, ~16/11 = 59\109)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	+2.63	-0.99	-0.02	+5.26	-0.86	-3.83	+1.64	+5.14	-0.27	+2.61	-0.75	-1.99	-3.11	+5.29	-0.08	+1.77	-1.01	+1.87	-1.20	+0.30	-5.10	+4.27	-4.96	-0.04	-3.24	-3.78	-1.85	+2.37
	Relative (%)	+23.9	-9.0	-0.2	+47.8	-7.8	-34.8	+14.9	+46.7	-2.4	+23.7	-6.8	-18.0	-28.3	+48.0	-0.7	+16.1	-9.2	+17.0	-10.9	+2.7	-46.3	+38.8	-45.0	-0.3	-29.4	-34.3	-16.8	+21.5
Steps (reduced)		173 (64)	253 (35)	306 (88)	346 (19)	377 (50)	403 (76)	426 (99)	446 (10)	463 (27)	479 (43)	493 (57)	506 (70)	518 (82)	530 (94)	540 (104)	550 (5)	559 (14)	568 (23)	576 (31)	584 (39)	591 (46)	599 (54)	605 (60)	612 (67)	618 (73)	624 (79)	630 (85)	636 (91)

24edo (L=2, s=1, ~16/11 = 13\24) — Basic 11L 2s

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	-1.96	+13.69	-18.83	-3.91	-1.32	+9.47	+11.73	-4.96	+2.49	-20.78	+21.73	-22.63	-5.87	+20.42	+4.96	-3.27	-5.14	-1.34	+7.52	+20.94	-11.52	+9.78	-15.51	+12.35	-6.91	-23.50	+12.37	+0.53
	Relative (%)	-3.9	+27.4	-37.7	-7.8	-2.6	+18.9	+23.5	-9.9	+5.0	-41.6	+43.5	-45.3	-11.7	+40.8	+9.9	-6.5	-10.3	-2.7	+15.0	+41.9	-23.0	+19.6	-31.0	+24.7	-13.8	-47.0	+24.7	+1.1
Steps (reduced)		38 (14)	56 (8)	67 (19)	76 (4)	83 (11)	89 (17)	94 (22)	98 (2)	102 (6)	105 (9)	109 (13)	111 (15)	114 (18)	117 (21)	119 (23)	121 (1)	123 (3)	125 (5)	127 (7)	129 (9)	130 (10)	132 (12)	133 (13)	135 (15)	136 (16)	137 (17)	139 (19)	140 (20)

107edo (L=9, s=4, ~16/11 = 58\107)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	+4.59	-5.01	-4.34	-2.04	-1.79	+0.59	-0.42	-4.02	+5.29	+0.25	-0.24	+1.20	+2.55	+2.20	-1.11	+2.80	+1.87	-4.62	+5.18	-2.89	+4.37	+4.17	-3.82	+2.54	+0.57	+1.26	+4.42	-1.34
	Relative (%)	+40.9	-44.6	-38.7	-18.2	-15.9	+5.3	-3.7	-35.9	+47.2	+2.2	-2.1	+10.7	+22.7	+19.6	-9.9	+25.0	+16.7	-41.2	+46.2	-25.8	+39.0	+37.2	-34.1	+22.6	+5.0	+11.3	+39.5	-11.9
Steps (reduced)		170 (63)	248 (34)	300 (86)	339 (18)	370 (49)	396 (75)	418 (97)	437 (9)	455 (27)	470 (42)	484 (56)	497 (69)	509 (81)	520 (92)	530 (102)	540 (5)	549 (14)	557 (22)	566 (31)	573 (38)	581 (46)	588 (53)	594 (59)	601 (66)	607 (72)	613 (78)	619 (84)	624 (89)

83edo (L=7, s=3, ~16/11 = 45\83)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	+6.48	+4.05	-0.15	-1.50	-1.92	-1.97	-3.93	-3.75	+6.10	+6.33	-6.59	-6.36	+4.98	-3.07	-2.87	+4.56	+3.90	-5.56	+4.51	+4.67	-5.49	+2.55	-0.45	-0.30	+2.73	-6.03	+2.13	-1.88
	Relative (%)	+44.8	+28.0	-1.0	-10.4	-13.3	-13.6	-27.2	-25.9	+42.2	+43.8	-45.6	-44.0	+34.4	-21.2	-19.8	+31.5	+27.0	-38.5	+31.2	+32.3	-38.0	+17.6	-3.1	-2.1	+18.9	-41.7	+14.7	-13.0
Steps (reduced)		132 (49)	193 (27)	233 (67)	263 (14)	287 (38)	307 (58)	324 (75)	339 (7)	353 (21)	365 (33)	375 (43)	385 (53)	395 (63)	403 (71)	411 (79)	419 (4)	426 (11)	432 (17)	439 (24)	445 (30)	450 (35)	456 (41)	461 (46)	466 (51)	471 (56)	475 (60)	480 (65)	484 (69)

142edo (L=12, s=5, ~16/11 = 77\142)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	-0.55	+2.42	+3.01	-1.09	-2.02	-3.91	+1.87	-3.55	-1.74	+2.46	-2.92	-3.61	-1.64	+1.41	-4.19	-2.57	-3.03	+2.18	+4.00	+1.92	+3.98	+1.33	+2.10	-2.44	-4.09	-3.08	+0.40	-2.28
	Relative (%)	-6.5	+28.6	+35.6	-12.9	-23.9	-46.2	+22.2	-42.0	-20.6	+29.1	-34.6	-42.8	-19.4	+16.7	-49.6	-30.4	-35.8	+25.8	+47.3	+22.8	+47.0	+15.7	+24.8	-28.9	-48.4	-36.5	+4.7	-27.0
Steps (reduced)		225 (83)	330 (46)	399 (115)	450 (24)	491 (65)	525 (99)	555 (129)	580 (12)	603 (35)	624 (56)	642 (74)	659 (91)	675 (107)	690 (122)	703 (135)	716 (6)	728 (18)	740 (30)	751 (41)	761 (51)	771 (61)	780 (70)	789 (79)	797 (87)	805 (95)	813 (103)	821 (111)	828 (118)

59edo (L=5, s=2, ~16/11 = 32\59) — Semihard 11L 2s

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	+9.91	+0.13	+7.45	-0.52	-2.17	-6.63	+10.04	-3.26	+7.57	-2.98	+2.23	+0.25	+9.39	+7.71	-6.05	+7.74	+7.57	-7.28	+3.28	-1.94	-3.04	-0.39	+5.68	-5.45	+6.65	+1.07	-2.04	-2.86
	Relative (%)	+48.7	+0.6	+36.6	-2.6	-10.6	-32.6	+49.3	-16.0	+37.2	-14.7	+11.0	+1.2	+46.2	+37.9	-29.8	+38.1	+37.2	-35.8	+16.1	-9.6	-15.0	-1.9	+27.9	-26.8	+32.7	+5.3	-10.0	-14.1
Steps (reduced)		94 (35)	137 (19)	166 (48)	187 (10)	204 (27)	218 (41)	231 (54)	241 (5)	251 (15)	259 (23)	267 (31)	274 (38)	281 (45)	287 (51)	292 (56)	298 (3)	303 (8)	307 (12)	312 (17)	316 (21)	320 (25)	324 (29)	328 (33)	331 (36)	335 (40)	338 (43)	341 (46)	344 (49)

153edo (L=13, s=5, ~16/11 = 83\153)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	-3.92	-2.00	+3.72	+0.01	-2.30	-1.31	+1.93	-2.99	+0.53	-0.19	-0.82	+3.84	-3.90	-2.13	+0.06	+1.63	+1.72	-0.36	+2.62	+2.31	-1.71	-1.99	+1.16	-0.40	+0.93	-2.92	+3.54	-3.39
	Relative (%)	-49.9	-25.5	+47.5	+0.1	-29.3	-16.7	+24.6	-38.2	+6.7	-2.5	-10.5	+49.0	-49.8	-27.1	+0.8	+20.8	+22.0	-4.6	+33.3	+29.5	-21.9	-25.4	+14.8	-5.1	+11.9	-37.2	+45.2	-43.2
Steps (reduced)		242 (89)	355 (49)	430 (124)	485 (26)	529 (70)	566 (107)	598 (139)	625 (13)	650 (38)	672 (60)	692 (80)	711 (99)	727 (115)	743 (131)	758 (146)	772 (7)	785 (20)	797 (32)	809 (44)	820 (55)	830 (65)	840 (75)	850 (85)	859 (94)	868 (103)	876 (111)	885 (120)	892 (127)

94edo (L=8, s=3, ~16/11 = 51\94)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	+0.17	-3.33	+1.39	+0.35	-2.38	+2.03	-3.16	-2.83	-3.90	+1.56	-2.74	+6.10	+0.52	+4.47	+3.90	-2.21	-1.95	+3.98	+2.20	+4.98	-0.88	-2.99	-1.68	+2.77	-2.66	-5.42	-5.72	-3.72
	Relative (%)	+1.4	-26.1	+10.9	+2.7	-18.7	+15.9	-24.8	-22.2	-30.5	+12.2	-21.5	+47.8	+4.1	+35.0	+30.6	-17.3	-15.3	+31.1	+17.2	+39.0	-6.9	-23.4	-13.1	+21.7	-20.8	-42.5	-44.8	-29.2
Steps (reduced)		149 (55)	218 (30)	264 (76)	298 (16)	325 (43)	348 (66)	367 (85)	384 (8)	399 (23)	413 (37)	425 (49)	437 (61)	447 (71)	457 (81)	466 (90)	474 (4)	482 (12)	490 (20)	497 (27)	504 (34)	510 (40)	516 (46)	522 (52)	528 (58)	533 (63)	538 (68)	543 (73)	548 (78)

129edo (L=11, s=4, ~16/11 = 70\129)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	-4.28	+4.38	-1.38	+0.74	-2.48	-3.32	+0.10	-2.63	+0.16	+3.64	+4.28	-0.53	-3.54	+2.98	-0.85	+2.54	+3.00	-0.18	+1.70	-1.16	+0.11	-4.18	+4.26	-2.77	+2.39	+0.91	+1.90	-4.12
	Relative (%)	-46.0	+47.1	-14.9	+8.0	-26.7	-35.7	+1.1	-28.3	+1.7	+39.1	+46.1	-5.7	-38.0	+32.0	-9.1	+27.3	+32.2	-1.9	+18.3	-12.4	+1.2	-44.9	+45.8	-29.8	+25.7	+9.8	+20.5	-44.3
Steps (reduced)		204 (75)	300 (42)	362 (104)	409 (22)	446 (59)	477 (90)	504 (117)	527 (11)	548 (32)	567 (51)	584 (68)	599 (83)	613 (97)	627 (111)	639 (123)	651 (6)	662 (17)	672 (27)	682 (37)	691 (46)	700 (55)	708 (63)	717 (72)	724 (79)	732 (87)	739 (94)	746 (101)	752 (107)

35edo (L=3, s=1, ~16/11 = 19\35) — Hard 11L 2s

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	-16.24	-9.17	-8.83	+1.80	-2.75	+16.62	+8.87	-2.10	+11.06	+9.22	-11.13	+15.94	-14.44	-1.01	-13.61	+15.30	+16.29	-11.34	+0.37	+16.65	+2.77	-7.37	-14.08	+16.63	+15.95	-16.36	-11.92	-5.18
	Relative (%)	-47.4	-26.7	-25.7	+5.3	-8.0	+48.5	+25.9	-6.1	+32.3	+26.9	-32.5	+46.5	-42.1	-2.9	-39.7	+44.6	+47.5	-33.1	+1.1	+48.6	+8.1	-21.5	-41.1	+48.5	+46.5	-47.7	-34.8	-15.1
Steps (reduced)		55 (20)	81 (11)	98 (28)	111 (6)	121 (16)	130 (25)	137 (32)	143 (3)	149 (9)	154 (14)	158 (18)	163 (23)	166 (26)	170 (30)	173 (33)	177 (2)	180 (5)	182 (7)	185 (10)	188 (13)	190 (15)	192 (17)	194 (19)	197 (22)	199 (24)	200 (25)	202 (27)	204 (29)

116edo (L=10, s=3, ~16/11 = 63\116)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	+1.49	-3.56	+3.59	+2.99	-3.04	-2.60	-2.06	-1.51	+2.49	+5.08	+2.76	+3.23	+4.48	+4.91	+3.24	-1.55	+0.03	-3.07	-1.10	-4.92	-4.62	-0.57	-3.44	-3.17	-0.01	-4.54	+3.75	+3.98
	Relative (%)	+14.4	-34.4	+34.7	+28.9	-29.4	-25.1	-19.9	-14.6	+24.0	+49.1	+26.7	+31.3	+43.3	+47.4	+31.3	-15.0	+0.3	-29.7	-10.7	-47.6	-44.7	-5.5	-33.2	-30.6	-0.1	-43.9	+36.2	+38.5
Steps (reduced)		184 (68)	269 (37)	326 (94)	368 (20)	401 (53)	429 (81)	453 (105)	474 (10)	493 (29)	510 (46)	525 (61)	539 (75)	552 (88)	564 (100)	575 (111)	585 (5)	595 (15)	604 (24)	613 (33)	621 (41)	629 (49)	637 (57)	644 (64)	651 (71)	658 (78)	664 (84)	671 (91)	677 (97)

81edo (L=7, s=2, ~16/11 = 44\81)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	-5.66	-1.13	-5.86	+3.50	-3.17	+3.92	-6.79	-1.25	-1.22	+3.29	-6.05	-2.26	-2.16	-7.35	-4.29	+5.99	-6.99	+0.51	-1.74	+0.57	+7.00	+2.37	+1.16	+3.09	-6.91	+0.57	-4.30	-6.88
	Relative (%)	-38.2	-7.6	-39.6	+23.6	-21.4	+26.4	-45.8	-8.4	-8.2	+22.2	-40.9	-15.2	-14.6	-49.6	-29.0	+40.4	-47.2	+3.4	-11.8	+3.8	+47.3	+16.0	+7.8	+20.9	-46.6	+3.8	-29.0	-46.4
Steps (reduced)		128 (47)	188 (26)	227 (65)	257 (14)	280 (37)	300 (57)	316 (73)	331 (7)	344 (20)	356 (32)	366 (42)	376 (52)	385 (61)	393 (69)	401 (77)	409 (4)	415 (10)	422 (17)	428 (23)	434 (29)	440 (35)	445 (40)	450 (45)	455 (50)	459 (54)	464 (59)	468 (63)	472 (67)

127edo (L=11, s=3, ~16/11 = 69\127)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	-2.74	+1.09	+4.40	+3.96	-3.29	+0.42	-1.65	-1.02	-4.60	+1.66	-4.65	+2.18	+1.22	+0.34	-1.73	+3.42	-3.96	+3.77	-2.33	-3.87	-1.28	-4.40	-4.09	-0.64	-3.76	-4.21	-2.20	+2.11
	Relative (%)	-29.0	+11.5	+46.6	+42.0	-34.8	+4.4	-17.5	-10.8	-48.7	+17.6	-49.2	+23.0	+12.9	+3.6	-18.3	+36.2	-41.9	+39.9	-24.6	-40.9	-13.6	-46.5	-43.3	-6.8	-39.8	-44.6	-23.3	+22.3
Steps (reduced)		201 (74)	295 (41)	357 (103)	403 (22)	439 (58)	470 (89)	496 (115)	519 (11)	539 (31)	558 (50)	574 (66)	590 (82)	604 (96)	617 (109)	629 (121)	641 (6)	651 (16)	662 (27)	671 (36)	680 (45)	689 (54)	697 (62)	705 (70)	713 (78)	720 (85)	727 (92)	734 (99)	741 (106)

46edo (L=4, s=1, ~16/11 = 25\46) — Superhard 11L 2s

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	+2.39	+4.99	-3.61	+4.79	-3.49	-5.75	+7.38	-0.61	-10.56	-1.22	-2.19	+9.98	+7.18	-12.19	+2.79	-1.10	+1.38	+9.53	-3.35	-11.67	+10.22	+9.78	+12.75	-7.22	+1.79	-12.63	+1.50	-8.16
	Relative (%)	+9.2	+19.1	-13.8	+18.3	-13.4	-22.0	+28.3	-2.3	-40.5	-4.7	-8.4	+38.3	+27.5	-46.7	+10.7	-4.2	+5.3	+36.5	-12.9	-44.7	+39.2	+37.5	+48.9	-27.7	+6.8	-48.4	+5.7	-31.3
Steps (reduced)		73 (27)	107 (15)	129 (37)	146 (8)	159 (21)	170 (32)	180 (42)	188 (4)	195 (11)	202 (18)	208 (24)	214 (30)	219 (35)	223 (39)	228 (44)	232 (2)	236 (6)	240 (10)	243 (13)	246 (16)	250 (20)	253 (23)	256 (26)	258 (28)	261 (31)	263 (33)	266 (36)	268 (38)

103edo (L=9, s=2, ~16/11 = 56\103)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	-2.93	-1.85	-1.84	+5.80	-3.75	-1.69	-4.77	-0.10	+5.40	-4.76	+0.85	-3.70	+2.87	-4.33	-3.29	+4.98	-3.68	+4.97	-4.62	+2.01	+1.10	+3.95	-1.43	-3.67	-3.03	+0.28	-5.59	+2.47
	Relative (%)	-25.1	-15.9	-15.8	+49.8	-32.1	-14.5	-41.0	-0.9	+46.3	-40.9	+7.3	-31.7	+24.7	-37.2	-28.2	+42.7	-31.6	+42.6	-39.6	+17.2	+9.5	+33.9	-12.3	-31.5	-26.0	+2.4	-48.0	+21.2
Steps (reduced)		163 (60)	239 (33)	289 (83)	327 (18)	356 (47)	381 (72)	402 (93)	421 (9)	438 (26)	452 (40)	466 (54)	478 (66)	490 (78)	500 (88)	510 (98)	520 (5)	528 (13)	537 (22)	544 (29)	552 (37)	559 (44)	566 (51)	572 (57)	578 (63)	584 (69)	590 (75)	595 (80)	601 (86)

57edo (L=5, s=1, ~16/11 = 31\57)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	-7.22	-7.37	-0.40	+6.62	-3.95	+1.58	+6.47	+0.31	-2.78	-7.62	+3.30	+6.32	-0.60	+2.00	-8.19	+9.88	-7.77	+1.29	-5.64	-8.01	-6.25	-0.75	+8.18	-0.81	-6.91	-10.35	+9.74	-9.99
	Relative (%)	-34.3	-35.0	-1.9	+31.4	-18.8	+7.5	+30.7	+1.5	-13.2	-36.2	+15.7	+30.0	-2.9	+9.5	-38.9	+47.0	-36.9	+6.1	-26.8	-38.0	-29.7	-3.6	+38.8	-3.8	-32.8	-49.1	+46.2	-47.5
Steps (reduced)		90 (33)	132 (18)	160 (46)	181 (10)	197 (26)	211 (40)	223 (52)	233 (5)	242 (14)	250 (22)	258 (30)	265 (37)	271 (43)	277 (49)	282 (54)	288 (3)	292 (7)	297 (12)	301 (16)	305 (20)	309 (24)	313 (28)	317 (32)	320 (35)	323 (38)	326 (41)	330 (45)	332 (47)

68edo (L=6, s=1, ~16/11 = 37\68)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	+3.93	+1.92	+1.76	+7.85	-4.26	+6.53	+5.85	+0.93	+2.49	+5.69	+7.02	+3.84	-5.87	-6.05	+2.02	-0.33	+3.68	-4.29	-7.19	-5.53	+0.25	-7.87	+5.08	+3.52	+4.85	-8.80	-2.34	+6.41
	Relative (%)	+22.3	+10.9	+10.0	+44.5	-24.1	+37.0	+33.1	+5.3	+14.1	+32.2	+39.8	+21.8	-33.2	-34.3	+11.5	-1.9	+20.9	-24.3	-40.7	-31.4	+1.4	-44.6	+28.8	+20.0	+27.5	-49.9	-13.2	+36.3
Steps (reduced)		108 (40)	158 (22)	191 (55)	216 (12)	235 (31)	252 (48)	266 (62)	278 (6)	289 (17)	299 (27)	308 (36)	316 (44)	323 (51)	330 (58)	337 (65)	343 (3)	349 (9)	354 (14)	359 (19)	364 (24)	369 (29)	373 (33)	378 (38)	382 (42)	386 (46)	389 (49)	393 (53)	397 (57)

11edo (L=1, s=0, ~16/11 = 6\11) — Collapsed 11L 2s

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	-47.41	+50.05	+12.99	+14.27	-5.86	+32.20	+2.64	+4.14	+29.76	-34.42	+26.27	-8.99	-33.14	-47.76	-54.13	-53.27	-46.05	-33.16	-15.21	+7.30	+33.94	-44.77	-10.96	+25.98	-43.27	-0.78	+44.19	-17.65
	Relative (%)	-43.5	+45.9	+11.9	+13.1	-5.4	+29.5	+2.4	+3.8	+27.3	-31.5	+24.1	-8.2	-30.4	-43.8	-49.6	-48.8	-42.2	-30.4	-13.9	+6.7	+31.1	-41.0	-10.0	+23.8	-39.7	-0.7	+40.5	-16.2
Steps (reduced)		17 (6)	26 (4)	31 (9)	35 (2)	38 (5)	41 (8)	43 (10)	45 (1)	47 (3)	48 (4)	50 (6)	51 (7)	52 (8)	53 (9)	54 (10)	55 (0)	56 (1)	57 (2)	58 (3)	59 (4)	60 (5)	60 (5)	61 (6)	62 (7)	62 (7)	63 (8)	64 (9)	64 (9)

Note that 11/8 (the dark generator, and thereby the bright generator 16/11) remains stable throughout the entire currently posted 11L 2s table &emdash; the worst relative error is -34.8%, at 127edo.

(Need a way to combine the collection of tables into a single table for better readability.)

In preliminary observations of scrolling through the above table group, I started noticing interesting things, like how even though the 11th harmonic is the only one with stable mapping all the way through 11L 2s, some of the others have stable mapping in sections, like the 3rd harmonic has stable mapping in the middle section but is all over the place in both the hard and soft ends, but the 9th harmonic actually does okay in the hard end, as does the 17th harmonic (both of these get to be all over the place in the soft end), and the 5th and 13th harmonics have stable mapping in the soft end as long as the EDO values are not too large.

Added: Lucius Chiaraviglio (talk) 07:00, 9 April 2025 (UTC)
Last modified: Lucius Chiaraviglio (talk) 09:25, 21 June 2025 (UTC)

Musical Mad Science Musings on Diatonicized Third-Tone Sub-Chromaticism

The 36edo equivalent of Diatonicized Chromaticism is 17L 2s. (Originally I had this down as Diatonicized Sixth-Tone Sub-Chromaticism, following after the example of Ivan Wyschnegradsky, but Diatonicized Chromaticism is really named after the large step in its 11L 2s scale, so this should liewise be named after the large step in its 17L 2s scale, which approximates a third-tone.) So I've been giving a bit of thought to how to start constructing a temperament (or set thereof) that uses this scale. (And it has turned out to be a real rabbit hole, which suggests the name Wonderland for the temperament if I ever get to that point before somebody else takes that name for some other purpose.) This is (unfortunately still) in a very rudimentary stage, but so far I have observed:

1. As the number of L intervals in a nL 2s scale grows, the range of qualifying generator sizes shrinks, and so the scale becomes more brittle to tempering of the generator, and it becomes hard to find good ratios for specifying the generator. Considering the wider of each pair of generators, the range of 5L 2s (as in Meantone, Superpyth, and their relatives) is very wide range — you have to have a bad fifth to land outside of its range. The range of 7L 2s is still fairly wide, going from barely over 52/35 down to somewhat under 81/55; 9L 2s is narrower, going from barely over 25/17 down to somewhat under 19/13; 11L 2s (Ivan Wyschnegradsky's original Diatonicized Chromatic scale) brackets 16/11; and the ranges get progressively narrower and the ratios more complicated until by the time we get to 19L 2s, the range falls between two ratios, the second of which is not even all that simple: 13/9 and 36/25. The first is too sharp by somewhat over 1 ¢, and the second is barely too flat; although since it is near-just as 10 steps of 19edo, which is equalized 19L 2s, we can count it as snapping to the lower end. It is possible to come up with more complicated ratios by mediation between these slightly out-of-bounds endpoints, such as 75/52 and 49/34, or even 62/43 in the middle, but the latter uses such large primes as to give difficulty (otherwise it would be very good), while the previous ratios and even 36/25 itself fail to map properly in the patent vals of some of the equal temperaments within the range of 17L 2s (this flaw of 36/25 making it tempting to use the slightly flatter 23/16, so before considering the next point, it seems better to specify the generator as a tempered 36/25 ~ 13/9, or perhaps even 23/16 ~ 13/9, either way with the proviso that the generator can never reach the just value of either endpoint without going out of range. Also worthy of consideration is the generator ~59049/40960 = ~|-13 10 -1⟩ of the established Alphatricot family, although this only works for a narrow band in the hard to super-hard region of the 17L 2s scale tree, plus 17c. (The Alphatricot family also has recognition of ~13/9 and ~75/52 for its 13-limit extensions.) But the choice of generator tempering comma will need to depend upon which subgroup(s) counts as the core of this temperament, so let's not throw out any of the above intervals just yet. (More painstaking inspection has revealed that ~23/16 or ~13/9 are usable after all in significant parts of the scale tree with only minimal use of warts, while ~62/43 is usable throughout again with only minimal use of warts — therefore this needs to be rewritten to prefer ~23/16 or ~13/9, followed by ~49/34 or ~75/52, with ~62/43 as a fallback, and the more complex sliding generator with the 53rd harmonic component only used as a last resort.)
2. In 36edo, the original inspiration for this attempt at a temperament, 19L 2s lends itself to making good use of 36edo as a 2.3.7... subgroup temperament, with the generator 19\36. With this scale, it is possible to choose a mode of this scale (UDP 11|7, cyclic order 14, LLLLLsLLLLLLLLLsLLL, no mode name assigned yet) that includes the following key 2.3.7 intervals: root (0\36), 9/8 (6\36), 7/6 (8\36), both flavors of split neutral third (10\36 and 11\36), 9/7 (13\36), 4/3 (15\36), 3/2 (21\36), 7/4 (29\36), 16/9 (30\36), and on to the root, all the while filling in the scale with 2\36 stacked to various extents. It also includes the generator interval 19\36, but let's not assign the generator a (tempered) ratio just yet. The choice of other modes enables use of other intervals relative to the root, while a decent subset of them still support both the 3-limit fourth and fifth. (But see later parts of this analysis, in which it is actually necessary to assign the 2.3.5... subgroup mapping first, at least for the hard half of the scale tree   an early warning sign of this quest turning into a rabbit hole.)
3. It is noteworthy that harmonics 3 and 23 are very stable over the 17L 2s scale tree scale tree (at least for EDO values up into the mid double digits, except for needing a wart at 112b), although the 23rd harmonic is guaranteed to be sharp, meaning that at larger EDO values, increasingly fine divisions of the octave will cause the mapping to disagree with 10\19 and 9\17 (and thereby with 19\36), thus requiring an 'i' wart. The 7th harmonic is also reasonably stable, although it changes enough over the scale tree to get rather bad at the extremes; the 5th harmonic is definitely not stable in the hard half of the spectrum, but is fairly stable in the soft half (although warts are needed for a few of the larger EDOs).
4. Tried assigning the generator as 23/16 ~ 13/9, tempering out 208/207 (the vicetone comma). But the problem is that — as can be seen in the table of harmonics below — the 13th harmonic is not stable enough for the entire 17L 2s scale tree, although it gets close in the hard half of the scale tree (closer to just 13/9, including having the best 3rd harmonic within the scale tree). Need to split the scale tree of 17L 2s into 2 or more temperaments. For the soft half, the 5th and 53rd harmonics are stable enough to team up with the 3rd and 23rd harmonics to get a usable generator, except not in the right-most column of the scale tree. Nobody in their right mind is going to want to actually use the 53rd harmonic for constructing intervals, but for tuning the generator, it will have to do (however, see below for the meantone subset of the soft half, for which the 23rd harmonic is actually stable enough). The bright generator (basic 19\36, spectrum from 10\19 soft to 9\17 hard) is therefore constituted as 23/16 (|-4 0 0 0 0 0 0 0 1⟩, 628.274347 ¢) ~ 384/265 (|7 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1⟩, 642.1367415 ¢), tempered together using the unnamed comma 6144/6095 (|11 1 -1 0 0 0 0 0 -1 0 0 0 0 0 0 -1⟩, 13.8623942563 ¢); this comma is in fact tempered out in most of the EDOs on the soft half of the scale tree, plus 17edo constituted as the often-used (and barely further from just) 17c val. (In a rigorous test of the mapping of ~384/265, it did not perform as well as expected, missing the boat for the entire right-most column of the scale tree and in some other parts of the table, with no opportunities anywhere in the table to fix the mapping by adding a wart other than 17c. Need to rewrite this section to stop depending upon 384/265, since it only works for the very complicated tuned generator.) It follows that the dark generator is constituted as 32/23 (|5 0 0 0 0 0 0 0 -1⟩, 571.725653 ¢) ~ 265/192 (|-6 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 1⟩, 557.863258 ¢), tempered together using the same comma. A plausible alternative to the generator spectrum is ~62/43 ; on the one hand, this would have the advantage of simplifying the description (if not the mathematics) of extensions; on the other hand, it has the disadvantage of replacing one moderately high harmonic having fairly stable mapping and one extremely high harmonic having rock-solid stable mapping with 2 very high harmonics having very unstable mappings, so the subgroup would have to map this ratio by direct approximation, as in 2.3...43/31 (direct-approximated 43/31 being the dark generator), or it would be necessary to add 'k' and 'n' warts as needed to achieve the same effect (amazingly, these harmonics covary sofficiently well that a wart is needed for only one EDO in the whole set: 142k — need to rework this section to take advantage of this near-rock-solid mapping stability of the ratio despite instability of the mappings of the individual harmonics, and should do the corresponding due diligence for ~13/9). (Coming in the future: More work on the hard half of the scale tree. Maybe the 7th and 17th harmonics are stable enough for the middle of the 17L 2s scale tree?)
5. To get ~3/1, we stack 3 bright generators (and then octave reduction gets us ~3/2), putting this temperament on the alpha-tricot part of the ploidacot system (not to be confused with the Alphatricot family temperaments, and not to be confused with Mothra, which is tricot without the alpha, which only works for the subset of the following EDOs that also have octaves divisible by 3). The comma for this is 12288/12167 (|12 1 0 0 0 0 0 0 -3⟩, 17.13195906 ¢) in the 23-limit, and 18874368/18609625 (|21 2 -3 0 0 0 0 0 0 0 0 0 0 0 0 -3⟩, 24.45522370876 ¢) in the 53-limit. Except as noted with warts, this works for the patent vals of all of the smaller and mid-size EDOs in the 17L 2s scale tree (leaving out the far right column of the scale tree apart from its top and bottom ends, since the EDO sizes in between the top and bottom ends get very large). As noted above, things are different between the soft half (up to and including 36edo) and the hard half (36edo onwards) later on, so even though they work the same way for the 3rd harmonic, I split the EDO list in half, although oddly enough 17c works as if it was on the soft half despite being at the hard end (collapsed 17L 2s). EDO list for soft half of scale tree: 19, 36, 55, 74, 91c, 93, 112b, 127cci, 129, 146i (and note that 112b just barely misses being the patent val of the dual-fifth 112edo). EDO list for hard half of scale tree: 17, 36, 53, 70, 87, 89, 104, 123, 125, 142 (this list is manually generated — Graham Breed's x31eq temperament finder has trouble with the high prime limit of the subgroup, and only finds part of the spectrum, and shows some other EDOs in addition). Note that some of the commas in each comma spectrum listed below (corresponding to the generator spectrum above) have negative just intonation values, because each spectrum crosses through 0.
6. Dividing the EDO list into hard and soft halves is helpful for looking at these EDOs as 5-limit temperaments. For the soft half, except as noted with warts, these are all Meantone temperaments, or conntortions under Meantone temperaments: 19, 36 (contorted under 12), 55, 74, 91c, 93 (contorted under 31), 112b, 127cc, 129 (contorted under 43), 146c. Without the warts, 91edo and 127edo fall on Python (currently still named Lalagu by Graham Breed's x31eq temperament finder), for which 16 fifths (octave-reduced) are needed to reach 5/4; while 146edo falls on the currently unnamed diploid temperament that flattens the fifth by (optimally) close to 1/10 of |-27 20 -2⟩ to get ~5/4. Once the temperaments other than 112b that have warts have been removed, the remaining temperaments have a 23rd harmonic with stable enough mapping to use as the bright generator, even though it always maps sharp and is often inconsistent with nearby (particularly lower) harmonics. For the hard half (to be dealt with later), ~13/9 appears to be usable as the generator, only needing a wart in one instance: 125f (which just barely misses being the patent val, while the wart improves consistency with nearby harmonics).
7. For the Meantone set, since 4 fifths get the ~5/4, and 3 (unnamed-53-limit-comma-tempered) bright generators get the fifth, this means that 12 of these bright generators get the ~5/4. (Originally this was going to be a 2.3.5.13.23 subgroup extension of Meantone, with the Vicetone comma already having a fitting name, leading "Vicetone" as the name for this extension; but the 13th harmonic mapping just wasn't stable enough. Too bad. For now, going to have to go with something weird like Fitho-vicesimotertial for the comma and temperament name.) Dropping EDOs for which warts greatly degrade accuracy (91c, 127cci, and 142ci), the optimal ET sequence is: 17c, 19, 36, 55, 74, 93, 112b, 129 (this list is manually generated — see above about x31eq having trouble with the high prime limit of the subgroup.) The comma for this is made from the syntonic comma by substituting each instance of ~3/2 with an octave-reduced stack of 3 of our bright generator, which produces a spectrum of commas from |52 0 1 0 0 0 0 0 -12⟩ ~ |-80 -12 13 0 0 0 0 0 0 0 0 0 0 0 0 12⟩, of which |19 -3 4 0 0 0 0 0 -9 0 0 0 0 0 0 3⟩ (made by substituting 9 instances of (3/2)^(1/3) by 23/16 and the other 3 instances of (3/2)^(1/3) by 384/265) has the closest 53-limit just intonation value to 0 (5.4343638749 ¢). Naturally, 81/80 itself is also tempered out.
8. It is natural to ask next for the 7-limit infill extension (after all, the original idea was to get a 2.3.7... temperament).
   1. For some of these EDOs the normal Septimal Meantone extension gives the proper ~7/4, only having the generator number multiplied by 3 since 3 bright generators are needed to get 1 fifth — optimal ET sequence: 19, 55d, 74, 93, 112b; of these, only 93 qualifies for Mothra. The comma for this is Harrison's comma with instance of 3/2 substituted by on octave-reduced stack of 3 of our bright generator, which produces a spectrum of enormously complicated commas from |-133 0 0 -1 0 0 0 0 30⟩ ~ |197 30 -30 -1 0 0 0 0 0 0 0 0 0 0 0 -30⟩, of which |-34 9 -9 -1 0 0 0 0 21 0 0 0 0 0 0 -9⟩ (made by substituting 21 of 30 instances of (3/2)^(1/3) by 23/16 and the other 9 of 30 instances of (3/2)^(1/3) by 384/265) has the closest 53-limit just intonation value to 0 (4.16605989 ¢), while still clocking in at 66 digits.
   2. For those EDOs having a less flat fifth (but also including 19), the extension is actually much simpler, needing only 11 bright generators to get ~7/4 — optimal ET sequence: 17c, 19, 36, 55 of these, only 36 qualifies for Mothra. The comma for this is a spectrum of less complicated (but still very complicated) commas from |47 0 0 1 0 0 0 0 -11⟩ ~ |-74 -11 11 1 0 0 0 0 0 0 0 0 0 0 0 11⟩, of which |3 -4 4 1 0 0 0 0 -7 0 0 0 0 0 0 4⟩ has the closest 53-limit just intonation value to 0 (2.35850949135 ¢, made using 7 instances of 23/16 and 4 instances of 384/265).
   3. This leaves out 129edo, which we don't want to miss because it has a very accurate 7th harmonic; for 129edo, if we want a strong extension, we need -44 bright generators (which is +44 dark generators) — optimal ET sequence: 55, 74, 129; however, this is overly complex for all 3 members, since 55 and 74 also belong to much simpler strong extensions, while 129 qualifies for Mothra. For 129edo, this means that we can proceed by -3 bright generators (+3 dark generators), octave-reduce, and divide by 3, which simplifies to +1 dark generator and +1/3 octave; furthermore, this also works for the other EDO sizes divisible by 3, which suggests the name Alpha-Mothra (since these are both tricot and alpha-tricot, which simplifies to triploid alpha-dark_generator); optimal ET sequence: 36, 93, 129. The comma for the simplified form is a spectrum of merely highly complicated commas from 4173281/4194304 (|3 -4 4 1 0 0 0 0 -7 0 0 0 0 0 0 4⟩) ~ 18966528/18609625 (|11 3 -3 3 0 0 0 0 0 0 0 0 0 0 0 -3⟩), of which 544341/542720 (|-11 1 -1 3 0 0 0 0 2 0 0 0 0 0 0 -1⟩) has the closest 53-limit just intonation value to 0 (5.1631554689 ¢, smaller than the Gamelisma for which it substitutes).

Added: Lucius Chiaraviglio (talk) 08:20, 4 April 2025 (UTC)
Last modified: Lucius Chiaraviglio (talk) 08:07, 18 June 2025 (UTC)

Table of odd harmonics for various EDO values supporting 17L 2s

The following table (actually a collection of tables for now) is for tracking trends in odd harmonics along the scale tree of 17L 2s; it is intended to match the organization of the corresponding scale tree (but this needs to be updated for addition of another column on 2025-05-12 — this is in a very early stage). For each EDO, it includes a list of plausible candidates for the 17L 2s bright generator (BrightGen), with candidates failing to map to the bright generator in italics, along with plausible wart fixes (if any).

19edo (L=1, s=1, BrightGen is 10\19; patent ~13/9 = 10\19; patent ~23/16 = 10\19; patent ~49/34 = 9\19; patent ~62/43 = 10\19; patent ~75/52 = 10\19; patent ~384/265 = 10\19; patent ~59049/40960 = 9\19) — Equalized 17L 2s

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	-7.22	-7.37	-21.46	-14.44	+17.10	-19.48	-14.58	+21.36	+18.28	-28.68	+3.30	-14.73	-21.65	-19.05	-8.19	+9.88	-28.82	+1.29	-26.69	+13.04	-6.25	-21.80	+29.23	+20.24	+14.14	+10.71	+9.74	+11.06
	Relative (%)	-11.4	-11.7	-34.0	-22.9	+27.1	-30.8	-23.1	+33.8	+28.9	-45.4	+5.2	-23.3	-34.3	-30.2	-13.0	+15.7	-45.6	+2.0	-42.3	+20.7	-9.9	-34.5	+46.3	+32.1	+22.4	+17.0	+15.4	+17.5
Steps (reduced)		30 (11)	44 (6)	53 (15)	60 (3)	66 (9)	70 (13)	74 (17)	78 (2)	81 (5)	83 (7)	86 (10)	88 (12)	90 (14)	92 (16)	94 (18)	96 (1)	97 (2)	99 (4)	100 (5)	102 (7)	103 (8)	104 (9)	106 (11)	107 (12)	108 (13)	109 (14)	110 (15)	111 (16)

131edo (L=7, s=6, BrightGen is 69\131; patent ~13/9 = _\131; patent ~23/16 = _\131; patent ~49/34 = _\131; patent ~62/43 = _\131; patent ~75/52 = _\131; patent ~384/265 = _\131; patent ~59049/40960 = _\131)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	+3.39	-1.58	+2.17	-2.38	-1.70	+2.22	+1.81	-4.19	-4.38	-3.61	+3.79	-3.16	+1.01	-3.62	+0.00	+1.69	+0.59	-4.02	-3.55	+1.47	+1.46	-3.96	+3.20	+4.33	-0.80	-3.28	-3.28	-0.99
	Relative (%)	+37.0	-17.3	+23.7	-26.0	-18.6	+24.2	+19.7	-45.8	-47.9	-39.4	+41.3	-34.5	+11.0	-39.6	+0.0	+18.4	+6.4	-43.8	-38.8	+16.1	+15.9	-43.3	+34.9	+47.3	-8.8	-35.8	-35.8	-10.9
Steps (reduced)		208 (77)	304 (42)	368 (106)	415 (22)	453 (60)	485 (92)	512 (119)	535 (11)	556 (32)	575 (51)	593 (69)	608 (84)	623 (99)	636 (112)	649 (125)	661 (6)	672 (17)	682 (27)	692 (37)	702 (47)	711 (56)	719 (64)	728 (73)	736 (81)	743 (88)	750 (95)	757 (102)	764 (109)

112edo (L=6, s=5, BrightGen is 59\112; patent ~13/9 = 58\112; b val ~13/9 = 60\112; patent ~23/16 = 59\112; patent ~49/34 = 58\112; patent ~62/43 = 59\112; patent ~75/52 = 60\112, 'b' or 'f' wart usable; patent ~384/265 = 60\112; b val ~384/255 = 61\112; patent ~59049/40960 = 64\112; b val ~59049/40960 = 54\112)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	+5.19	-0.60	-4.54	-0.34	-4.89	-4.81	+4.59	+2.19	+2.49	+0.65	+3.87	-1.20	+4.85	-1.01	+1.39	+0.30	-5.14	-4.92	+0.37	-0.49	+2.77	-0.94	-1.22	+1.63	-3.34	+5.07	+5.23	-3.04
	Relative (%)	+48.4	-5.6	-42.4	-3.2	-45.6	-44.9	+42.8	+20.4	+23.2	+6.0	+36.1	-11.2	+45.3	-9.4	+13.0	+2.8	-48.0	-45.9	+3.5	-4.6	+25.8	-8.8	-11.4	+15.2	-31.2	+47.3	+48.8	-28.4
Steps (reduced)		178 (66)	260 (36)	314 (90)	355 (19)	387 (51)	414 (78)	438 (102)	458 (10)	476 (28)	492 (44)	507 (59)	520 (72)	533 (85)	544 (96)	555 (107)	565 (5)	574 (14)	583 (23)	592 (32)	600 (40)	608 (48)	615 (55)	622 (62)	629 (69)	635 (75)	642 (82)	648 (88)	653 (93)

205edo (L=11, s=9, BrightGen is 108\205; patent ~13/9 = _\205; patent ~23/16 = _\205; patent ~49/34 = _\205; patent ~62/43 = _\205; patent ~75/52 = _\205; patent ~384/265 = _\205\205; patent ~59049/40960 = _\205)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	+0.48	+0.03	+2.88	+0.97	-1.07	+2.40	+0.51	+0.41	+1.02	-2.49	-1.93	+0.06	+1.45	+0.67	+2.28	-0.59	+2.91	+0.36	+2.88	-1.75	-2.25	+1.00	+1.81	-0.09	+0.89	-1.31	-1.05	+1.51
	Relative (%)	+8.3	+0.5	+49.2	+16.5	-18.3	+41.0	+8.7	+7.0	+17.5	-42.5	-33.0	+0.9	+24.8	+11.4	+39.0	-10.1	+49.7	+6.2	+49.3	-29.8	-38.4	+17.0	+30.9	-1.6	+15.3	-22.4	-17.9	+25.8
Steps (reduced)		325 (120)	476 (66)	576 (166)	650 (35)	709 (94)	759 (144)	801 (186)	838 (18)	871 (51)	900 (80)	927 (107)	952 (132)	975 (155)	996 (176)	1016 (196)	1034 (9)	1052 (27)	1068 (43)	1084 (59)	1098 (73)	1112 (87)	1126 (101)	1139 (114)	1151 (126)	1163 (138)	1174 (149)	1185 (160)	1196 (171)

93edo (L=5, s=4, BrightGen is 49\93; patent ~13/9 = 50\93; patent ~23/16 = 49\93; patent ~49/34 = 49\93; patent ~62/43 = 49\93; patent ~75/52 = 49\93; patent ~384/265 = 49\93; patent ~59049/40960 = 45\93)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	-5.18	+0.78	-1.08	+2.54	+3.52	-1.82	-4.40	-1.73	-0.74	-6.26	+3.98	+1.57	-2.64	+2.68	+3.35	-1.66	-0.30	-6.18	+5.90	-3.26	+4.61	+3.32	+5.46	-2.17	+5.99	+3.91	+4.30	-5.92
	Relative (%)	-40.2	+6.1	-8.4	+19.7	+27.3	-14.1	-34.1	-13.4	-5.7	-48.6	+30.9	+12.1	-20.5	+20.8	+26.0	-12.9	-2.3	-47.9	+45.8	-25.2	+35.7	+25.8	+42.3	-16.8	+46.4	+30.3	+33.4	-45.9
Steps (reduced)		147 (54)	216 (30)	261 (75)	295 (16)	322 (43)	344 (65)	363 (84)	380 (8)	395 (23)	408 (36)	421 (49)	432 (60)	442 (70)	452 (80)	461 (89)	469 (4)	477 (12)	484 (19)	492 (27)	498 (33)	505 (40)	511 (46)	517 (52)	522 (57)	528 (63)	533 (68)	538 (73)	542 (77)

260edo (L=14, s=11, BrightGen is 137\260; patent ~13/9 = _\260; patent ~23/16 = _\260; patent ~49/34 = _\260; patent ~62/43 = _\260; patent ~75/52 = _\260; patent ~384/265 = _\260; patent ~59049/40960 = _\260)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	-0.42	+1.38	+0.40	-0.83	-2.09	-0.53	+0.96	+1.20	-2.13	-0.01	-0.58	-1.86	-1.25	-0.35	-0.42	+2.11	+1.78	-2.11	-0.94	+0.17	+0.79	+0.55	-0.89	+0.81	+0.78	-1.20	-0.71	+2.07
	Relative (%)	-9.0	+29.9	+8.8	-18.1	-45.2	-11.4	+20.8	+26.0	-46.1	-0.3	-12.6	-40.3	-27.1	-7.5	-9.1	+45.8	+38.6	-45.8	-20.5	+3.6	+17.1	+11.8	-19.3	+17.5	+16.9	-25.9	-15.4	+44.9
Steps (reduced)		412 (152)	604 (84)	730 (210)	824 (44)	899 (119)	962 (182)	1016 (236)	1063 (23)	1104 (64)	1142 (102)	1176 (136)	1207 (167)	1236 (196)	1263 (223)	1288 (248)	1312 (12)	1334 (34)	1354 (54)	1374 (74)	1393 (93)	1411 (111)	1428 (128)	1444 (144)	1460 (160)	1475 (175)	1489 (189)	1503 (203)	1517 (217)

167edo (L=9, s=7, BrightGen is 88\167; patent ~13/9 = 88\167; patent ~23/16 = 87\167; patent ~49/34 = 88\167; patent ~62/43 = 88\167; patent ~75/52 = 89\167, 'b' wart usable; patent ~384/265 = 89\167; patent ~59049/40960 = 91\167)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	+2.24	+1.71	+1.23	-2.71	+1.98	+0.19	-3.24	+2.83	-2.90	+3.47	-3.12	+3.42	-0.48	-2.03	-2.52	-2.97	+2.94	+0.15	+2.43	+2.08	-1.34	-1.00	+2.76	+2.47	-2.12	+3.14	-3.50	-0.67
	Relative (%)	+31.1	+23.8	+17.2	-37.7	+27.5	+2.7	-45.1	+39.4	-40.4	+48.3	-43.5	+47.6	-6.6	-28.3	-35.1	-41.4	+41.0	+2.1	+33.8	+28.9	-18.6	-13.9	+38.4	+34.3	-29.5	+43.7	-48.7	-9.3
Steps (reduced)		265 (98)	388 (54)	469 (135)	529 (28)	578 (77)	618 (117)	652 (151)	683 (15)	709 (41)	734 (66)	755 (87)	776 (108)	794 (126)	811 (143)	827 (159)	842 (7)	857 (22)	870 (35)	883 (48)	895 (60)	906 (71)	917 (82)	928 (93)	938 (103)	947 (112)	957 (122)	965 (130)	974 (139)

241edo (L=13, s=10, BrightGen is 127\241; patent ~13/9 = _\241; patent ~23/16 = _\241; patent ~49/34 = _\241; patent ~62/43 = _\241; patent ~75/52 = _\241; patent ~384/265 = _\241; patent ~59049/40960 = _\241)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	+0.12	+2.07	+2.13	+0.24	+1.38	+0.97	+2.19	-0.39	+1.24	+2.25	-0.89	-0.84	+0.36	+1.13	+0.19	+1.50	-0.78	-2.38	+1.09	-0.85	+1.35	+2.31	+1.71	-0.72	-0.27	-2.14	-1.53	+1.36
	Relative (%)	+2.4	+41.5	+42.7	+4.8	+27.7	+19.4	+43.9	-7.9	+24.9	+45.2	-17.8	-16.9	+7.2	+22.7	+3.9	+30.1	-15.7	-47.8	+21.8	-17.0	+27.0	+46.3	+34.4	-14.5	-5.5	-42.9	-30.8	+27.4
Steps (reduced)		382 (141)	560 (78)	677 (195)	764 (41)	834 (111)	892 (169)	942 (219)	985 (21)	1024 (60)	1059 (95)	1090 (126)	1119 (155)	1146 (182)	1171 (207)	1194 (230)	1216 (11)	1236 (31)	1255 (50)	1274 (69)	1291 (86)	1308 (103)	1324 (119)	1339 (134)	1353 (148)	1367 (162)	1380 (175)	1393 (188)	1406 (201)

74edo (L=4, s=3, BrightGen is 39\74; patent ~13/9 = 40\74; patent ~23/16 = 39\74; patent ~49/34 = 40\74, 'g' wart usable; patent ~62/43 = 39\74; patent ~75/52 = 39\74; patent ~384/265 = 39\74; patent ~59049/40960 = 36\74) — Supersoft 17L 2s

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	-4.66	+2.88	+4.15	+6.90	+0.03	+2.72	-1.78	-7.66	-5.62	-0.51	+4.16	+5.75	+2.24	-7.96	+6.32	-4.62	+7.02	-8.10	-1.94	-7.44	+7.40	-6.44	-0.64	-7.92	+3.90	+2.17	+2.91	+5.94
	Relative (%)	-28.7	+17.7	+25.6	+42.6	+0.2	+16.7	-11.0	-47.2	-34.7	-3.1	+25.6	+35.5	+13.8	-49.1	+38.9	-28.5	+43.3	-50.0	-12.0	-45.9	+45.6	-39.7	-4.0	-48.9	+24.1	+13.4	+17.9	+36.6
Steps (reduced)		117 (43)	172 (24)	208 (60)	235 (13)	256 (34)	274 (52)	289 (67)	302 (6)	314 (18)	325 (29)	335 (39)	344 (48)	352 (56)	359 (63)	367 (71)	373 (3)	380 (10)	385 (15)	391 (21)	396 (26)	402 (32)	406 (36)	411 (41)	415 (45)	420 (50)	424 (54)	428 (58)	432 (62)

277edo (L=15, s=11, BrightGen is 146\277; patent ~13/9 = _\277; patent ~23/16 = _\277; patent ~49/34 = _\277; patent ~62/43 = _\277; patent ~75/52 = _\277; patent ~384/265 = _\277; patent ~59049/40960 = _\277)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	-0.15	-0.75	+1.57	-0.30	-1.14	-0.09	-0.90	-0.98	+1.40	+1.42	-0.12	-1.51	-0.45	+1.47	-1.35	-1.29	+0.82	-0.08	-0.24	-0.18	-0.33	-1.05	+1.64	-1.19	-1.13	+1.59	-1.89	+1.25
	Relative (%)	-3.5	-17.4	+36.3	-6.9	-26.3	-2.2	-20.9	-22.7	+32.4	+32.8	-2.7	-34.8	-10.4	+33.9	-31.2	-29.7	+18.9	-1.9	-5.6	-4.2	-7.5	-24.3	+37.9	-27.5	-26.2	+36.6	-43.7	+28.9
Steps (reduced)		439 (162)	643 (89)	778 (224)	878 (47)	958 (127)	1025 (194)	1082 (251)	1132 (24)	1177 (69)	1217 (109)	1253 (145)	1286 (178)	1317 (209)	1346 (238)	1372 (264)	1397 (12)	1421 (36)	1443 (58)	1464 (79)	1484 (99)	1503 (118)	1521 (136)	1539 (154)	1555 (170)	1571 (186)	1587 (202)	1601 (216)	1616 (231)

203edo (L=11, s=8, BrightGen is 107\203; patent ~13/9 = 107\203; patent ~23/16 = 106\203; patent ~49/34 = 107\203; patent ~62/43 = 107\203; patent ~75/52 = 107\203; patent ~384/265 = 109\203; patent ~59049/40960 = 110\203)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	+1.49	-2.08	+0.63	-2.92	-1.56	-1.12	-0.58	+1.45	-1.95	+2.13	-1.67	+1.76	-1.43	-1.01	+1.76	-0.07	-1.45	+2.84	+0.37	+2.46	+2.77	+0.91	+2.47	+1.26	+2.94	+1.37	+2.27	-0.45
	Relative (%)	+25.3	-35.1	+10.7	-49.5	-26.5	-18.9	-9.9	+24.5	-32.9	+36.0	-28.3	+29.7	-24.2	-17.0	+29.8	-1.2	-24.4	+48.1	+6.3	+41.7	+46.8	+15.4	+41.8	+21.4	+49.8	+23.2	+38.4	-7.7
Steps (reduced)		322 (119)	471 (65)	570 (164)	643 (34)	702 (93)	751 (142)	793 (184)	830 (18)	862 (50)	892 (80)	918 (106)	943 (131)	965 (153)	986 (174)	1006 (194)	1024 (9)	1041 (26)	1058 (43)	1073 (58)	1088 (73)	1102 (87)	1115 (100)	1128 (113)	1140 (125)	1152 (137)	1163 (148)	1174 (159)	1184 (169)

332edo (L=18, s=13, BrightGen is 175\332; patent ~13/9 = _\332; patent ~23/16 = _\332\332; patent ~49/34 = _\332; patent ~62/43 = _\332; patent ~75/52 = _\332; patent ~384/265 = _\332; patent ~59049/40960 = _\332)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	-0.75	+0.43	-0.15	-1.50	+1.69	+1.64	-0.32	-0.14	-1.13	-0.90	+0.64	+0.87	+1.36	+0.54	+0.75	+0.94	+0.28	+1.67	+0.89	+1.06	+1.74	-1.07	-0.45	-0.30	-0.89	+1.19	-1.49	+1.74
	Relative (%)	-20.8	+12.0	-4.2	-41.5	+46.9	+45.4	-8.8	-3.8	-31.2	-24.9	+17.7	+24.0	+37.7	+15.0	+20.7	+26.1	+7.8	+46.1	+24.6	+29.3	+48.0	-29.5	-12.3	-8.4	-24.5	+33.0	-41.1	+48.1
Steps (reduced)		526 (194)	771 (107)	932 (268)	1052 (56)	1149 (153)	1229 (233)	1297 (301)	1357 (29)	1410 (82)	1458 (130)	1502 (174)	1542 (214)	1579 (251)	1613 (285)	1645 (317)	1675 (15)	1703 (43)	1730 (70)	1755 (95)	1779 (119)	1802 (142)	1823 (163)	1844 (184)	1864 (204)	1883 (223)	1902 (242)	1919 (259)	1937 (277)

129edo (L=7, s=5, BrightGen is 68\129; patent ~13/9 = 69\129; patent ~23/16 = 68\129; patent ~49/34 = 68\129; patent ~62/43 = 68\129; patent ~75/52 = 69\129; patent ~384/265 = 68\129; patent ~59049/40960 = 63\129)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	-4.28	+4.38	-1.38	+0.74	-2.48	-3.32	+0.10	-2.63	+0.16	+3.64	+4.28	-0.53	-3.54	+2.98	-0.85	+2.54	+3.00	-0.18	+1.70	-1.16	+0.11	-4.18	+4.26	-2.77	+2.39	+0.91	+1.90	-4.12
	Relative (%)	-46.0	+47.1	-14.9	+8.0	-26.7	-35.7	+1.1	-28.3	+1.7	+39.1	+46.1	-5.7	-38.0	+32.0	-9.1	+27.3	+32.2	-1.9	+18.3	-12.4	+1.2	-44.9	+45.8	-29.8	+25.7	+9.8	+20.5	-44.3
Steps (reduced)		204 (75)	300 (42)	362 (104)	409 (22)	446 (59)	477 (90)	504 (117)	527 (11)	548 (32)	567 (51)	584 (68)	599 (83)	613 (97)	627 (111)	639 (123)	651 (6)	662 (17)	672 (27)	682 (37)	691 (46)	700 (55)	708 (63)	717 (72)	724 (79)	732 (87)	739 (94)	746 (101)	752 (107)

313edo (L=17, s=12, BrightGen is 165\313; patent ~13/9 = _\313; patent ~23/16 = _\313; patent ~49/34 = _\313; patent ~62/43 = _\313; patent ~75/52 = _\313; patent ~384/265 = _\313; patent ~59049/40960 = _\313)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	-0.36	+0.91	+1.14	-0.72	+0.76	-0.91	+0.55	-1.44	+1.53	+0.78	+0.48	+1.81	-1.07	+1.73	+1.29	+0.40	-1.78	+1.69	-1.27	+0.33	-1.61	+0.19	+1.59	-1.55	-1.80	+0.62	+1.67	+1.17
	Relative (%)	-9.3	+23.7	+29.8	-18.7	+19.8	-23.8	+14.3	-37.6	+39.9	+20.5	+12.5	+47.3	-28.0	+45.2	+33.7	+10.5	-46.6	+44.1	-33.1	+8.6	-42.1	+5.0	+41.4	-40.4	-46.9	+16.1	+43.4	+30.5
Steps (reduced)		496 (183)	727 (101)	879 (253)	992 (53)	1083 (144)	1158 (219)	1223 (284)	1279 (27)	1330 (78)	1375 (123)	1416 (164)	1454 (202)	1488 (236)	1521 (269)	1551 (299)	1579 (14)	1605 (40)	1631 (66)	1654 (89)	1677 (112)	1698 (133)	1719 (154)	1739 (174)	1757 (192)	1775 (210)	1793 (228)	1810 (245)	1826 (261)

184edo (L=10, s=7, BrightGen is 97\184; patent ~13/9 = 97\184; patent ~23/16 = 96\184, 'i' wart usable; patent ~49/34 = 98\184; patent ~62/43 = 97\184; patent ~75/52 = 97\184; patent ~384/265 = 99\184; patent ~59049/40960 = 101\184)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	+2.39	-1.53	+2.91	-1.74	+3.03	+0.78	+0.86	-0.61	+2.49	-1.22	-2.19	-3.06	+0.66	+0.86	+2.79	-1.10	+1.38	+3.00	+3.17	+1.37	-2.82	+3.25	-0.29	-0.70	+1.79	+0.41	+1.50	-1.64
	Relative (%)	+36.7	-23.5	+44.7	-26.6	+46.5	+11.9	+13.2	-9.3	+38.1	-18.6	-33.5	-47.0	+10.1	+13.1	+42.8	-16.9	+21.2	+46.1	+48.6	+21.0	-43.3	+49.9	-4.4	-10.7	+27.4	+6.3	+23.0	-25.2
Steps (reduced)		292 (108)	427 (59)	517 (149)	583 (31)	637 (85)	681 (129)	719 (167)	752 (16)	782 (46)	808 (72)	832 (96)	854 (118)	875 (139)	894 (158)	912 (176)	928 (8)	944 (24)	959 (39)	973 (53)	986 (66)	998 (78)	1011 (91)	1022 (102)	1033 (113)	1044 (124)	1054 (134)	1064 (144)	1073 (153)

239edo (L=13, s=9, BrightGen is 126\239; patent ~13/9 = _\239; patent ~23/16 = _\239; patent ~49/34 = _\239; patent ~62/43 = _\239; patent ~75/52 = _\239; patent ~384/265 = _\239; patent ~59049/40960 = _\239)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	+0.97	+0.30	+0.21	+1.95	+0.98	-2.03	+1.27	+0.48	-1.28	+1.19	-0.66	+0.59	-2.10	-0.29	-0.27	+1.96	+0.51	-0.30	-1.06	-2.28	+0.62	+2.24	+2.28	+0.42	+1.46	+0.14	+1.28	-0.30
	Relative (%)	+19.4	+5.9	+4.2	+38.8	+19.6	-40.5	+25.3	+9.6	-25.5	+23.6	-13.1	+11.8	-41.8	-5.7	-5.3	+39.0	+10.1	-5.9	-21.1	-45.5	+12.3	+44.7	+45.3	+8.4	+29.0	+2.7	+25.5	-6.1
Steps (reduced)		379 (140)	555 (77)	671 (193)	758 (41)	827 (110)	884 (167)	934 (217)	977 (21)	1015 (59)	1050 (94)	1081 (125)	1110 (154)	1136 (180)	1161 (205)	1184 (228)	1206 (11)	1226 (31)	1245 (50)	1263 (68)	1280 (85)	1297 (102)	1313 (118)	1328 (133)	1342 (147)	1356 (161)	1369 (174)	1382 (187)	1394 (199)

55edo (L=3, s=2, BrightGen is 29\55; patent ~13/9 = 30\55; patent ~23/16 = 29\55; patent ~49/34 = 28\55; patent ~62/43 = 29\55; patent ~75/52 = 29\55; patent ~384/265 = 29\55; patent ~59049/40960 = 27\55) — Soft 17L 2s

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	-3.77	+6.41	-8.83	-7.55	-5.86	+10.38	+2.64	+4.14	+7.94	+9.22	+4.45	-8.99	+10.50	-4.12	-10.49	-9.64	-2.41	+10.47	+6.61	+7.30	-9.70	-1.13	+10.86	+4.17	+0.36	-0.78	+0.55	+4.17
	Relative (%)	-17.3	+29.4	-40.5	-34.6	-26.9	+47.6	+12.1	+19.0	+36.4	+42.3	+20.4	-41.2	+48.1	-18.9	-48.1	-44.2	-11.1	+48.0	+30.3	+33.5	-44.5	-5.2	+49.8	+19.1	+1.7	-3.6	+2.5	+19.1
Steps (reduced)		87 (32)	128 (18)	154 (44)	174 (9)	190 (25)	204 (39)	215 (50)	225 (5)	234 (14)	242 (22)	249 (29)	255 (35)	262 (42)	267 (47)	272 (52)	277 (2)	282 (7)	287 (12)	291 (16)	295 (20)	298 (23)	302 (27)	306 (31)	309 (34)	312 (37)	315 (40)	318 (43)	321 (46)

256edo (L=14, s=9, BrightGen is 135\256; patent ~13/9 = _\256; patent ~23/16 = _\256; patent ~49/34 = _\256; patent ~62/43 = _\256; patent ~75/52 = _\256; patent ~384/265 = _\256; patent ~59049/40960 = _\256)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	+1.17	-1.94	+1.49	+2.34	+1.81	-1.47	-0.77	-1.83	-2.20	-2.03	-0.15	+0.81	-1.18	+1.67	-1.29	-1.71	-0.45	+1.78	-0.30	+2.19	-0.58	+0.40	+0.12	-1.71	-0.66	-1.63	-0.13	-1.03
	Relative (%)	+25.0	-41.4	+31.7	+49.9	+38.6	-31.3	-16.4	-39.0	-46.9	-43.3	-3.2	+17.3	-25.1	+35.7	-27.4	-36.5	-9.6	+38.0	-6.3	+46.7	-12.4	+8.6	+2.5	-36.6	-14.1	-34.8	-2.8	-22.0
Steps (reduced)		406 (150)	594 (82)	719 (207)	812 (44)	886 (118)	947 (179)	1000 (232)	1046 (22)	1087 (63)	1124 (100)	1158 (134)	1189 (165)	1217 (193)	1244 (220)	1268 (244)	1291 (11)	1313 (33)	1334 (54)	1353 (73)	1372 (92)	1389 (109)	1406 (126)	1422 (142)	1437 (157)	1452 (172)	1466 (186)	1480 (200)	1493 (213)

201edo (L=11, s=7, BrightGen is 106\201; patent ~13/9 = 106\201; patent ~23/16 = 105\201; patent ~49/34 = 105\201, 'g' wart usable; patent ~62/43 = 106\201; patent ~75/52 = 107\201, 'b' wart usable; patent ~384/265 = 108\201; patent ~59049/40960 = 110\201)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	+2.52	+1.75	-1.66	-0.92	-2.06	+1.26	-1.70	+2.51	+0.99	+0.86	-1.41	-2.48	+1.60	-2.71	+1.23	+0.46	+0.08	-0.60	-2.18	+0.79	+1.92	+0.82	-2.82	+2.65	-0.94	-1.86	-0.32	-2.45
	Relative (%)	+42.3	+29.2	-27.8	-15.5	-34.6	+21.2	-28.5	+42.0	+16.7	+14.4	-23.6	-41.5	+26.8	-45.4	+20.7	+7.7	+1.4	-10.0	-36.6	+13.2	+32.1	+13.8	-47.2	+44.3	-15.7	-31.2	-5.3	-41.1
Steps (reduced)		319 (118)	467 (65)	564 (162)	637 (34)	695 (92)	744 (141)	785 (182)	822 (18)	854 (50)	883 (79)	909 (105)	933 (129)	956 (152)	976 (172)	996 (192)	1014 (9)	1031 (26)	1047 (42)	1062 (57)	1077 (72)	1091 (86)	1104 (99)	1116 (111)	1129 (124)	1140 (135)	1151 (146)	1162 (157)	1172 (167)

347edo (L=19, s=12, BrightGen is 183\347; patent ~13/9 = _\347; patent ~23/16 = _\347; patent ~49/34 = _\347; patent ~62/43 = _\347; patent ~75/52 = _\347; patent ~384/265 = _\347; patent ~59049/40960 = _\347)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	+0.06	+1.01	-0.53	+0.12	-1.46	-0.18	+1.07	-1.21	-0.11	-0.46	+1.12	-1.45	+0.19	+0.97	-0.37	-1.40	+0.48	+1.11	-0.12	-0.24	+0.30	+1.13	-1.53	-1.05	-1.15	+1.42	-0.46	-0.04
	Relative (%)	+1.8	+29.1	-15.2	+3.6	-42.3	-5.3	+30.9	-35.0	-3.1	-13.4	+32.4	-41.8	+5.4	+28.1	-10.6	-40.5	+13.9	+32.0	-3.5	-7.1	+8.6	+32.7	-44.2	-30.4	-33.2	+41.2	-13.2	-1.3
Steps (reduced)		550 (203)	806 (112)	974 (280)	1100 (59)	1200 (159)	1284 (243)	1356 (315)	1418 (30)	1474 (86)	1524 (136)	1570 (182)	1611 (223)	1650 (262)	1686 (298)	1719 (331)	1750 (15)	1780 (45)	1808 (73)	1834 (99)	1859 (124)	1883 (148)	1906 (171)	1927 (192)	1948 (213)	1968 (233)	1988 (253)	2006 (271)	2024 (289)

146edo (L=8, s=5, BrightGen is 77\146; patent ~13/9 = 78\146; patent ~23/16 = 76\146\146; patent ~49/34 = 77\146; patent ~62/43 = 77\146; patent ~75/52 = 77\146; patent ~384/265 = 78\146; patent ~59049/40960 = 73\146)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	-3.32	-0.01	+1.04	+1.57	-0.63	-2.17	-3.34	+1.89	-1.62	-2.29	-3.62	-0.02	-1.76	-2.18	-2.57	-3.96	+1.02	+3.45	+2.72	-1.67	-1.93	+1.56	+0.25	+2.07	-1.43	-2.27	-0.65	+3.27
	Relative (%)	-40.5	-0.2	+12.6	+19.1	-7.7	-26.4	-40.6	+23.0	-19.7	-27.8	-44.0	-0.3	-21.4	-26.5	-31.3	-48.2	+12.5	+42.0	+33.1	-20.3	-23.5	+18.9	+3.0	+25.2	-17.4	-27.6	-7.9	+39.8
Steps (reduced)		231 (85)	339 (47)	410 (118)	463 (25)	505 (67)	540 (102)	570 (132)	597 (13)	620 (36)	641 (57)	660 (76)	678 (94)	694 (110)	709 (125)	723 (139)	736 (6)	749 (19)	761 (31)	772 (42)	782 (52)	792 (62)	802 (72)	811 (81)	820 (90)	828 (98)	836 (106)	844 (114)	852 (122)

383edo (L=21, s=13, BrightGen is 202\383; patent ~13/9 = _\383; patent ~23/16 = _\383; patent ~49/34 = _\383; patent ~62/43 = _\383; patent ~75/52 = _\383; patent ~384/265 = _\383; patent ~59049/40960 = _\383)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	-0.13	-0.94	-0.68	-0.25	+0.12	-0.84	-1.06	-1.56	+0.14	-0.81	+1.49	+1.26	-0.38	+1.23	-1.43	-0.01	+1.52	-0.69	-0.97	+0.18	-0.81	-1.19	-1.28	-1.36	+1.44	+0.65	-0.82	+0.01
	Relative (%)	-4.1	-29.8	-21.7	-8.1	+3.8	-26.8	-33.9	-49.8	+4.4	-25.8	+47.6	+40.3	-12.2	+39.3	-45.7	-0.3	+48.5	-22.1	-30.9	+5.8	-25.9	-38.0	-40.8	-43.4	+46.1	+20.6	-26.1	+0.3
Steps (reduced)		607 (224)	889 (123)	1075 (309)	1214 (65)	1325 (176)	1417 (268)	1496 (347)	1565 (33)	1627 (95)	1682 (150)	1733 (201)	1779 (247)	1821 (289)	1861 (329)	1897 (365)	1932 (17)	1965 (50)	1995 (80)	2024 (109)	2052 (137)	2078 (163)	2103 (188)	2127 (212)	2150 (235)	2173 (258)	2194 (279)	2214 (299)	2234 (319)

237edo (L=13, s=8, BrightGen is 125\237; patent ~13/9 = 125\237; patent ~23/16 = 124\237; patent ~49/34 = 124\237; patent ~62/43 = 125\237; patent ~75/52 = 125\237; patent ~384/265 = 127\237; patent ~59049/40960 = 129\237)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	+1.84	-1.50	-1.74	-1.38	+0.58	-0.02	+0.34	+1.37	+1.22	+0.11	-0.43	+2.06	+0.46	-1.73	-0.73	+2.42	+1.82	+1.82	+1.82	+1.32	-0.13	+2.18	-2.22	+1.59	-1.85	+2.44	-0.92	-2.00
	Relative (%)	+36.4	-29.7	-34.3	-27.2	+11.5	-0.4	+6.7	+27.1	+24.1	+2.1	-8.4	+40.6	+9.2	-34.1	-14.5	+47.9	+36.0	+36.0	+36.0	+26.0	-2.5	+43.1	-43.8	+31.4	-36.5	+48.3	-18.2	-39.5
Steps (reduced)		376 (139)	550 (76)	665 (191)	751 (40)	820 (109)	877 (166)	926 (215)	969 (21)	1007 (59)	1041 (93)	1072 (124)	1101 (153)	1127 (179)	1151 (203)	1174 (226)	1196 (11)	1216 (31)	1235 (50)	1253 (68)	1270 (85)	1286 (101)	1302 (117)	1316 (131)	1331 (146)	1344 (159)	1358 (173)	1370 (185)	1382 (197)

328edo (L=18, s=11, BrightGen is 173\328; patent ~13/9 = _\328; patent ~23/16 = _\328; patent ~49/34 = _\328; patent ~62/43 = _\328; patent ~75/52 = _\328; patent ~384/265 = _\328; patent ~59049/40960 = _\328)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	+0.48	+1.49	+0.69	+0.97	+1.12	+0.94	-1.68	+1.14	-1.17	+1.17	+0.99	-0.68	+1.45	-1.53	+0.09	+1.61	-1.48	+1.09	+1.42	-1.01	+0.68	-1.20	+0.35	+1.37	+1.63	+0.89	-1.05	-0.69
	Relative (%)	+13.2	+40.8	+18.8	+26.5	+30.6	+25.6	-46.0	+31.2	-32.0	+32.0	+27.2	-18.5	+39.7	-41.8	+2.4	+43.9	-40.5	+29.9	+38.8	-27.7	+18.5	-32.8	+9.5	+37.5	+44.4	+24.2	-28.6	-18.8
Steps (reduced)		520 (192)	762 (106)	921 (265)	1040 (56)	1135 (151)	1214 (230)	1281 (297)	1341 (29)	1393 (81)	1441 (129)	1484 (172)	1523 (211)	1560 (248)	1593 (281)	1625 (313)	1655 (15)	1682 (42)	1709 (69)	1734 (94)	1757 (117)	1780 (140)	1801 (161)	1822 (182)	1842 (202)	1861 (221)	1879 (239)	1896 (256)	1913 (273)

91edo (L=5, s=3, BrightGen is 48\91; patent ~13/9 = 49\91; patent ~23/16 = 48\91; patent ~49/34 = 47\91; patent ~62/43 = 48\91; patent ~75/52 = 47\91; patent ~384/265 = 49\91; patent ~59049/40960 = 46\91) — Semisoft 17L 2s

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	-3.05	-3.90	-6.19	-6.11	+2.53	+3.43	+6.24	+0.54	+5.78	+3.94	+4.69	+5.39	+4.03	-1.01	+2.22	-0.53	+3.10	-0.79	+0.37	+6.10	+2.77	+3.18	-6.17	+0.81	-2.51	-3.17	-1.37	+2.73
	Relative (%)	-23.2	-29.5	-46.9	-46.3	+19.2	+26.0	+47.3	+4.1	+43.9	+29.9	+35.6	+40.9	+30.5	-7.6	+16.8	-4.0	+23.5	-6.0	+2.8	+46.3	+21.0	+24.1	-46.8	+6.1	-19.1	-24.1	-10.4	+20.7
Steps (reduced)		144 (53)	211 (29)	255 (73)	288 (15)	315 (42)	337 (64)	356 (83)	372 (8)	387 (23)	400 (36)	412 (48)	423 (59)	433 (69)	442 (78)	451 (87)	459 (4)	467 (12)	474 (19)	481 (26)	488 (33)	494 (39)	500 (45)	505 (50)	511 (56)	516 (61)	521 (66)	526 (71)	531 (76)

309edo (L=17, s=10, BrightGen is 163\309; patent ~13/9 = _\309; patent ~23/16 = _\309; patent ~49/34 = _\309; patent ~62/43 = _\309; patent ~75/52 = _\309; patent ~384/265 = _\309; patent ~59049/40960 = _\309)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	+0.96	-1.85	-1.84	+1.92	+0.14	-1.69	-0.89	-0.10	+1.52	-0.88	+0.85	+0.19	-1.01	-0.45	+0.60	+1.10	+0.20	+1.08	-0.74	-1.88	+1.10	+0.07	-1.43	+0.21	+0.86	+0.28	-1.71	-1.41
	Relative (%)	+24.7	-47.6	-47.3	+49.3	+3.6	-43.6	-22.9	-2.6	+39.0	-22.6	+21.9	+4.8	-26.0	-11.6	+15.3	+28.2	+5.2	+27.9	-18.9	-48.4	+28.4	+1.7	-36.8	+5.5	+22.1	+7.3	-44.0	-36.3
Steps (reduced)		490 (181)	717 (99)	867 (249)	980 (53)	1069 (142)	1143 (216)	1207 (280)	1263 (27)	1313 (77)	1357 (121)	1398 (162)	1435 (199)	1469 (233)	1501 (265)	1531 (295)	1559 (14)	1585 (40)	1610 (65)	1633 (88)	1655 (110)	1677 (132)	1697 (152)	1716 (171)	1735 (190)	1753 (208)	1770 (225)	1786 (241)	1802 (257)

218edo (L=12, s=7, BrightGen is 115\218; patent ~13/9 = 115\218; patent ~23/16 = 114\218; patent ~49/34 = 115\218; patent ~62/43 = 115\218; patent ~75/52 = 115\218; patent ~384/265 = 117\218; patent ~59049/40960 = 120\218)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	+2.63	-0.99	-0.02	-0.24	-0.86	+1.67	+1.64	-0.37	-0.27	+2.61	-0.75	-1.99	+2.39	-0.22	-0.08	+1.77	-1.01	+1.87	-1.20	+0.30	+0.41	-1.23	+0.55	-0.04	+2.26	+1.72	-1.85	+2.37
	Relative (%)	+47.8	-18.0	-0.3	-4.4	-15.6	+30.4	+29.8	-6.7	-4.8	+47.5	-13.7	-36.1	+43.5	-4.0	-1.5	+32.2	-18.4	+33.9	-21.8	+5.4	+7.4	-22.4	+10.0	-0.7	+41.1	+31.3	-33.6	+43.0
Steps (reduced)		346 (128)	506 (70)	612 (176)	691 (37)	754 (100)	807 (153)	852 (198)	891 (19)	926 (54)	958 (86)	986 (114)	1012 (140)	1037 (165)	1059 (187)	1080 (208)	1100 (10)	1118 (28)	1136 (46)	1152 (62)	1168 (78)	1183 (93)	1197 (107)	1211 (121)	1224 (134)	1237 (147)	1249 (159)	1260 (170)	1272 (182)

345edo (L=19, s=11, BrightGen is 182\345; patent ~13/9 = _\345; patent ~23/16 = _\345; patent ~49/34 = _\345; patent ~62/43 = _\345; patent ~75/52 = _\345; patent ~384/265 = _\345; patent ~59049/40960 = _\345)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	+0.65	-0.23	+1.61	+1.31	+1.73	+1.21	+0.43	-0.61	+1.62	-1.22	+1.29	-0.45	-1.52	-0.01	-0.69	-1.10	+1.38	-0.91	-1.61	-1.24	-0.21	+1.08	-1.16	-0.26	+0.05	-0.46	+1.50	-1.21
	Relative (%)	+18.8	-6.5	+46.3	+37.6	+49.6	+34.8	+12.3	-17.5	+46.5	-35.0	+37.1	-13.0	-43.6	-0.3	-19.8	-31.6	+39.7	-26.1	-46.4	-35.5	-6.1	+31.1	-33.3	-7.5	+1.3	-13.3	+43.1	-34.7
Steps (reduced)		547 (202)	801 (111)	969 (279)	1094 (59)	1194 (159)	1277 (242)	1348 (313)	1410 (30)	1466 (86)	1515 (135)	1561 (181)	1602 (222)	1640 (260)	1676 (296)	1709 (329)	1740 (15)	1770 (45)	1797 (72)	1823 (98)	1848 (123)	1872 (147)	1895 (170)	1916 (191)	1937 (212)	1957 (232)	1976 (251)	1995 (270)	2012 (287)

127edo (L=7, s=4, BrightGen is 67\127; patent ~13/9 = 68\127; patent ~23/16 = 66\127, 'i' wart usable; patent ~49/34 = 68\127; patent ~62/43 = 67\127; patent ~75/52 = 67\127; patent ~384/265 = 68\127; patent ~59049/40960 = 64\127)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	-2.74	+1.09	+4.40	+3.96	-3.29	+0.42	-1.65	-1.02	-4.60	+1.66	-4.65	+2.18	+1.22	+0.34	-1.73	+3.42	-3.96	+3.77	-2.33	-3.87	-1.28	-4.40	-4.09	-0.64	-3.76	-4.21	-2.20	+2.11
	Relative (%)	-29.0	+11.5	+46.6	+42.0	-34.8	+4.4	-17.5	-10.8	-48.7	+17.6	-49.2	+23.0	+12.9	+3.6	-18.3	+36.2	-41.9	+39.9	-24.6	-40.9	-13.6	-46.5	-43.3	-6.8	-39.8	-44.6	-23.3	+22.3
Steps (reduced)		201 (74)	295 (41)	357 (103)	403 (22)	439 (58)	470 (89)	496 (115)	519 (11)	539 (31)	558 (50)	574 (66)	590 (82)	604 (96)	617 (109)	629 (121)	641 (6)	651 (16)	662 (27)	671 (36)	680 (45)	689 (54)	697 (62)	705 (70)	713 (78)	720 (85)	727 (92)	734 (99)	741 (106)

290edo (L=16, s=9, BrightGen is 153\290; patent ~13/9 = _\290; patent ~23/16 = _\290; patent ~49/34 = _\290; patent ~62/43 = _\290; patent ~75/52 = _\290; patent ~384/265 = _\290; patent ~59049/40960 = _\290)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	+1.49	-1.49	-0.55	-1.15	-0.97	-0.53	+0.01	-1.51	+0.42	+0.94	+0.69	+1.17	+0.34	+0.77	+1.17	+0.52	-2.04	+1.07	+0.97	+1.28	+1.59	+1.50	+0.70	-1.10	-0.01	-0.40	+1.68	+1.91
	Relative (%)	+36.1	-35.9	-13.3	-27.8	-23.5	-12.8	+0.2	-36.4	+10.1	+22.8	+16.7	+28.2	+8.3	+18.6	+28.3	+12.6	-49.2	+25.9	+23.3	+31.0	+38.3	+36.3	+16.9	-26.6	-0.3	-9.7	+40.6	+46.2
Steps (reduced)		460 (170)	673 (93)	814 (234)	919 (49)	1003 (133)	1073 (203)	1133 (263)	1185 (25)	1232 (72)	1274 (114)	1312 (152)	1347 (187)	1379 (219)	1409 (249)	1437 (277)	1463 (13)	1487 (37)	1511 (61)	1533 (83)	1554 (104)	1574 (124)	1593 (143)	1611 (161)	1628 (178)	1645 (195)	1661 (211)	1677 (227)	1692 (242)

163edo (L=9, s=5, BrightGen is 86\163; patent ~13/9 = 87\163; patent ~23/16 = 85\163; patent ~49/34 = 87\163;; patent ~62/43 = 87\163, 'k' wart usable; patent ~75/52 = 85\163; patent ~384/265 = 87\163; patent ~59049/40960 = 83\163)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	-2.57	-3.49	+2.95	+2.22	+0.83	-1.26	+1.30	-1.89	-3.03	+0.38	-2.51	+0.38	-0.34	+1.10	+3.43	-1.74	-0.54	-1.04	+3.53	-2.07	-3.54	-1.27	-2.93	-1.46	+2.91	+2.57	-2.66	+1.76
	Relative (%)	-34.9	-47.4	+40.1	+30.2	+11.3	-17.2	+17.7	-25.6	-41.2	+5.2	-34.1	+5.1	-4.7	+14.9	+46.6	-23.6	-7.3	-14.1	+47.9	-28.1	-48.1	-17.2	-39.8	-19.8	+39.5	+34.9	-36.2	+23.9
Steps (reduced)		258 (95)	378 (52)	458 (132)	517 (28)	564 (75)	603 (114)	637 (148)	666 (14)	692 (40)	716 (64)	737 (85)	757 (105)	775 (123)	792 (140)	808 (156)	822 (7)	836 (21)	849 (34)	862 (47)	873 (58)	884 (69)	895 (80)	905 (90)	915 (100)	925 (110)	934 (119)	942 (127)	951 (136)

199edo (L=11, s=6, BrightGen is 105\199; patent ~13/9 = _\199; patent ~23/16 = _\199; patent ~49/34 = _\199; patent ~62/43 = _\199; patent ~75/52 = _\199; patent ~384/265 = _\199; patent ~59049/40960 = _\199)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	-2.46	-0.38	+2.03	+1.12	-2.57	-2.34	-2.84	-2.44	-2.04	-0.43	-1.14	-0.77	-1.34	+1.58	+0.69	+1.00	+1.64	+1.92	+1.24	-0.92	+1.05	+0.73	-2.19	-1.97	+1.13	+0.87	-2.96	+1.54
	Relative (%)	-40.8	-6.4	+33.6	+18.5	-42.7	-38.8	-47.1	-40.5	-33.8	-7.1	-18.9	-12.7	-22.3	+26.2	+11.5	+16.6	+27.3	+31.9	+20.5	-15.3	+17.3	+12.1	-36.3	-32.7	+18.7	+14.4	-49.1	+25.5
Steps (reduced)		315 (116)	462 (64)	559 (161)	631 (34)	688 (91)	736 (139)	777 (180)	813 (17)	845 (49)	874 (78)	900 (104)	924 (128)	946 (150)	967 (171)	986 (190)	1004 (9)	1021 (26)	1037 (42)	1052 (57)	1066 (71)	1080 (85)	1093 (98)	1105 (110)	1117 (122)	1129 (134)	1140 (145)	1150 (155)	1161 (166)

36edo (L=2, s=1, BrightGen is 19\36; patent ~13/9 = 19\36; patent ~23/16 = 19\36; patent ~49/34 = 19\36; patent ~62/43 = 19\36; patent ~75/52 = 20\36; patent ~384/265 = 19\36; patent ~59049/40960 = 18\36) — Basic 17L 2s

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	-1.96	+13.69	-2.16	-3.91	+15.35	-7.19	+11.73	-4.96	+2.49	-4.11	+5.06	-5.96	-5.87	+3.76	-11.70	+13.39	+11.53	+15.32	-9.15	+4.27	-11.52	+9.78	+1.16	-4.32	-6.91	-6.84	-4.30	+0.53
	Relative (%)	-5.9	+41.1	-6.5	-11.7	+46.0	-21.6	+35.2	-14.9	+7.5	-12.3	+15.2	-17.9	-17.6	+11.3	-35.1	+40.2	+34.6	+46.0	-27.4	+12.8	-34.6	+29.3	+3.5	-13.0	-20.7	-20.5	-12.9	+1.6
Steps (reduced)		57 (21)	84 (12)	101 (29)	114 (6)	125 (17)	133 (25)	141 (33)	147 (3)	153 (9)	158 (14)	163 (19)	167 (23)	171 (27)	175 (31)	178 (34)	182 (2)	185 (5)	188 (8)	190 (10)	193 (13)	195 (15)	198 (18)	200 (20)	202 (22)	204 (24)	206 (26)	208 (28)	210 (30)

197edo (L=11, s=5, BrightGen is 104\197; patent ~13/9 = _\197; patent ~23/16 = _\197; patent ~49/34 = _\197; patent ~62/43 = _\197; patent ~75/52 = _\197; patent ~384/265 = _\197; patent ~59049/40960 = _\197;)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	-1.45	-2.56	-0.30	-2.89	+3.00	+0.08	+2.09	-1.40	+0.96	-1.75	-0.86	+0.98	+1.75	-0.14	+0.14	+1.55	-2.86	-1.60	-1.37	-2.67	+0.16	+0.64	-1.55	-0.60	-2.85	-2.44	+0.44	-0.48
	Relative (%)	-23.8	-42.0	-4.9	-47.5	+49.2	+1.3	+34.3	-23.0	+15.8	-28.7	-14.2	+16.0	+28.7	-2.2	+2.3	+25.4	-46.9	-26.2	-22.4	-43.8	+2.6	+10.5	-25.4	-9.8	-46.8	-40.0	+7.2	-7.9
Steps (reduced)		312 (115)	457 (63)	553 (159)	624 (33)	682 (91)	729 (138)	770 (179)	805 (17)	837 (49)	865 (77)	891 (103)	915 (127)	937 (149)	957 (169)	976 (188)	994 (9)	1010 (25)	1026 (41)	1041 (56)	1055 (70)	1069 (84)	1082 (97)	1094 (109)	1106 (121)	1117 (132)	1128 (143)	1139 (154)	1149 (164)

161edo (L=9, s=4, BrightGen is 85\161; patent ~13/9 = 86\161; patent ~23/16 = 84\161;; patent ~49/34 = 85\161; patent ~62/43 = 85\161; patent ~75/52 = 85\161; patent ~384/265 = 86\161; patent ~59049/40960 = 83\161)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	-1.33	+1.26	+0.12	-2.67	+0.23	+1.71	-0.07	-0.61	+0.62	-1.22	-2.19	+2.53	+3.45	-1.01	+2.79	-1.10	+1.38	+2.07	+0.37	+3.24	+2.77	-1.40	-2.15	+0.24	-1.94	-1.45	+1.50	-0.71
	Relative (%)	-17.9	+17.0	+1.6	-35.8	+3.2	+22.9	-0.9	-8.2	+8.4	-16.3	-29.3	+33.9	+46.3	-13.5	+37.4	-14.7	+18.5	+27.8	+5.0	+43.4	+37.1	-18.8	-28.9	+3.2	-26.0	-19.5	+20.1	-9.5
Steps (reduced)		255 (94)	374 (52)	452 (130)	510 (27)	557 (74)	596 (113)	629 (146)	658 (14)	684 (40)	707 (63)	728 (84)	748 (104)	766 (122)	782 (138)	798 (154)	812 (7)	826 (21)	839 (34)	851 (46)	863 (58)	874 (69)	884 (79)	894 (89)	904 (99)	913 (108)	922 (117)	931 (126)	939 (134)

286edo (L=16, s=7, BrightGen is 151\286; patent ~13/9 = _\286; patent ~23/16 = _\286; patent ~49/34 = _\286; patent ~62/43 = _\286; patent ~75/52 = _\286; patent ~384/265 = _\286; patent ~59049/40960 = _\286)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	-1.26	-0.30	+0.40	+1.68	-1.67	-1.37	-1.56	-0.06	+0.39	-0.85	+1.10	-0.60	+0.43	-1.61	+0.42	+1.27	+0.11	+0.40	+1.57	-1.09	+0.37	+1.38	+1.63	+0.81	-1.32	-0.78	-1.97	-0.87
	Relative (%)	-29.9	-7.1	+9.6	+40.1	-39.7	-32.6	-37.1	-1.4	+9.3	-20.3	+26.1	-14.3	+10.2	-38.3	+10.0	+30.3	+2.5	+9.6	+37.5	-26.0	+8.8	+33.0	+38.8	+19.3	-31.4	-18.5	-46.9	-20.7
Steps (reduced)		453 (167)	664 (92)	803 (231)	907 (49)	989 (131)	1058 (200)	1117 (259)	1169 (25)	1215 (71)	1256 (112)	1294 (150)	1328 (184)	1360 (216)	1389 (245)	1417 (273)	1443 (13)	1467 (37)	1490 (60)	1512 (82)	1532 (102)	1552 (122)	1571 (141)	1589 (159)	1606 (176)	1622 (192)	1638 (208)	1653 (223)	1668 (238)

125edo (L=7, s=3, BrightGen is 66\125; patent ~13/9 = 67\125, 'f' wart usable; patent ~23/16 = 65\125; patent ~49/34 = 66\125; patent ~62/43 = 66\125; patent ~75/52 = 65\125, 'f' wart usable; patent ~384/265 = 67\125; patent ~59049/40960 = 65\125;)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	-1.16	-2.31	+0.77	-2.31	-4.12	+4.27	-3.47	+0.64	+0.09	-0.38	-4.27	-4.63	-3.47	-2.38	-2.64	+4.33	-1.54	-1.74	+3.12	+2.94	-2.72	-4.62	-3.11	+1.55	-0.51	+0.10	+3.17	-1.07
	Relative (%)	-12.0	-24.1	+8.1	-24.1	-42.9	+44.5	-36.1	+6.7	+0.9	-4.0	-44.5	-48.2	-36.1	-24.8	-27.5	+45.1	-16.0	-18.2	+32.5	+30.6	-28.3	-48.2	-32.4	+16.1	-5.3	+1.0	+33.0	-11.1
Steps (reduced)		198 (73)	290 (40)	351 (101)	396 (21)	432 (57)	463 (88)	488 (113)	511 (11)	531 (31)	549 (49)	565 (65)	580 (80)	594 (94)	607 (107)	619 (119)	631 (6)	641 (16)	651 (26)	661 (36)	670 (45)	678 (53)	686 (61)	694 (69)	702 (77)	709 (84)	716 (91)	723 (98)	729 (104)

339edo (L=19, s=8, BrightGen is 179\339; patent ~13/9 = _\339; patent ~23/16 = _\339; patent ~49/34 = _\339; patent ~62/43 = _\339; patent ~75/52 = _\339; patent ~384/265 = _\339; patent ~59049/40960 = _\339)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	-1.07	-0.47	+1.09	+1.40	+0.89	-1.59	-1.54	+1.24	-0.17	+0.02	-1.73	-0.95	+0.33	+0.51	-1.67	-0.18	+0.61	-0.02	+0.88	-0.74	+1.76	+0.93	-0.02	-1.37	+0.17	+0.83	+0.42	-1.24
	Relative (%)	-30.2	-13.4	+30.7	+39.5	+25.3	-44.9	-43.6	+35.0	-4.7	+0.4	-48.8	-26.7	+9.3	+14.4	-47.3	-5.0	+17.3	-0.5	+24.9	-21.0	+49.6	+26.2	-0.6	-38.7	+4.8	+23.5	+11.9	-35.0
Steps (reduced)		537 (198)	787 (109)	952 (274)	1075 (58)	1173 (156)	1254 (237)	1324 (307)	1386 (30)	1440 (84)	1489 (133)	1533 (177)	1574 (218)	1612 (256)	1647 (291)	1679 (323)	1710 (15)	1739 (44)	1766 (71)	1792 (97)	1816 (121)	1840 (145)	1862 (167)	1883 (188)	1903 (208)	1923 (228)	1942 (247)	1960 (265)	1977 (282)

214edo (L=12, s=5, BrightGen is 113\214; patent ~13/9 = 114\214; patent ~23/16 = 112\214; patent ~49/34 = 113\214; patent ~62/43 = 113\214; patent ~75/52 = 113\214; patent ~384/265 = 114\214; patent ~59049/40960 = 111\214)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	-1.02	+0.60	+1.27	-2.04	-1.79	+0.59	-0.42	+1.59	-0.32	+0.25	-0.24	+1.20	+2.55	+2.20	-1.11	+2.80	+1.87	+0.99	-0.43	+2.71	-1.24	-1.44	+1.78	+2.54	+0.57	+1.26	-1.18	-1.34
	Relative (%)	-18.2	+10.7	+22.6	-36.4	-31.8	+10.6	-7.5	+28.3	-5.6	+4.4	-4.2	+21.5	+45.4	+39.2	-19.8	+50.0	+33.3	+17.7	-7.6	+48.4	-22.1	-25.7	+31.8	+45.2	+10.1	+22.5	-21.1	-23.8
Steps (reduced)		339 (125)	497 (69)	601 (173)	678 (36)	740 (98)	792 (150)	836 (194)	875 (19)	909 (53)	940 (84)	968 (112)	994 (138)	1018 (162)	1040 (184)	1060 (204)	1080 (10)	1098 (28)	1115 (45)	1131 (61)	1147 (77)	1161 (91)	1175 (105)	1189 (119)	1202 (132)	1214 (144)	1226 (156)	1237 (167)	1248 (178)

303edo (L=17, s=7, BrightGen is 160\303; patent ~13/9 = _\303; patent ~23/16 = _\303; patent ~49/34 = _\303; patent ~62/43 = _\303; patent ~75/52 = _\303; patent ~384/265 = _\303; patent ~59049/40960 = _\303)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	-0.96	+1.81	+1.47	-1.93	-0.82	-0.92	+0.84	+1.98	-0.48	+0.51	+1.43	-0.35	+1.07	+0.13	-0.48	-1.79	-0.68	-1.84	-1.89	-1.34	-0.63	-0.12	-0.16	-1.02	+1.01	+1.74	+0.98	-1.45
	Relative (%)	-24.4	+45.6	+37.1	-48.7	-20.8	-23.3	+21.2	+49.9	-12.2	+12.8	+36.1	-8.8	+26.9	+3.2	-12.1	-45.1	-17.3	-46.4	-47.7	-33.8	-15.8	-3.1	-4.0	-25.7	+25.5	+44.0	+24.8	-36.6
Steps (reduced)		480 (177)	704 (98)	851 (245)	960 (51)	1048 (139)	1121 (212)	1184 (275)	1239 (27)	1287 (75)	1331 (119)	1371 (159)	1407 (195)	1441 (229)	1472 (260)	1501 (289)	1528 (13)	1554 (39)	1578 (63)	1601 (86)	1623 (108)	1644 (129)	1664 (149)	1683 (168)	1701 (186)	1719 (204)	1736 (221)	1752 (237)	1767 (252)

89edo (L=5, s=2, BrightGen is 47\89; patent ~13/9 = 47\89; patent ~23/16 = 47\89; patent ~49/34 = 47\89; patent ~62/43 = 47\89; patent ~75/52 = 48\89; patent ~384/265 = 47\89; patent ~59049/40960 = 46\89) — Semihard 17L 2s

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	-0.83	+4.70	+1.96	-1.66	+1.49	-4.57	+3.87	+2.91	-0.88	+1.13	+5.43	-4.09	-2.49	-4.86	+1.03	+0.66	+6.66	+4.84	-5.40	+2.40	+0.84	+3.03	-4.83	+3.92	+2.08	+2.90	+6.19	-1.72
	Relative (%)	-6.2	+34.8	+14.5	-12.3	+11.1	-33.9	+28.7	+21.6	-6.6	+8.4	+40.3	-30.3	-18.5	-36.0	+7.7	+4.9	+49.4	+35.9	-40.1	+17.8	+6.2	+22.5	-35.8	+29.1	+15.4	+21.5	+45.9	-12.7
Steps (reduced)		141 (52)	207 (29)	250 (72)	282 (15)	308 (41)	329 (62)	348 (81)	364 (8)	378 (22)	391 (35)	403 (47)	413 (57)	423 (67)	432 (76)	441 (85)	449 (4)	457 (12)	464 (19)	470 (25)	477 (32)	483 (38)	489 (44)	494 (49)	500 (55)	505 (60)	510 (65)	515 (70)	519 (74)

320edo (L=18, s=7, BrightGen is 169\320; patent ~13/9 = _\320; patent ~23/16 = _\320; patent ~49/34 = _\320; patent ~62/43 = _\320; patent ~75/52 = _\320; patent ~384/265 = _\320; patent ~59049/40960 = _\320)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	-0.71	-0.06	-1.33	-1.41	-0.07	-0.53	-0.77	+0.04	-1.26	+1.72	+1.73	-0.13	+1.63	+1.67	-1.29	-0.77	-1.39	-0.09	-1.23	-1.56	-1.52	-1.47	-1.76	+1.10	-0.66	+0.25	-0.13	+1.78
	Relative (%)	-18.8	-1.7	-35.4	-37.6	-1.8	-14.1	-20.5	+1.2	-33.7	+45.8	+46.0	-3.4	+43.6	+44.6	-34.3	-20.6	-37.1	-2.5	-32.9	-41.7	-40.5	-39.3	-46.8	+29.3	-17.6	+6.5	-3.5	+47.5
Steps (reduced)		507 (187)	743 (103)	898 (258)	1014 (54)	1107 (147)	1184 (224)	1250 (290)	1308 (28)	1359 (79)	1406 (126)	1448 (168)	1486 (206)	1522 (242)	1555 (275)	1585 (305)	1614 (14)	1641 (41)	1667 (67)	1691 (91)	1714 (114)	1736 (136)	1757 (157)	1777 (177)	1797 (197)	1815 (215)	1833 (233)	1850 (250)	1867 (267)

231edo (L=13, s=5, BrightGen is 122\231; patent ~13/9 = 123\231; patent ~23/16 = 121\231; patent ~49/34 = 121\231; patent ~62/43 = 122\231; patent ~75/52 = 121\231; patent ~384/265 = 124\231; patent ~59049/40960 = 121\231)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	-0.66	-1.90	-2.59	-1.31	-0.67	+1.03	-2.55	-1.06	-1.41	+1.95	+0.30	+1.40	-1.97	-1.01	-2.18	-1.32	+0.70	-1.99	+0.37	+2.11	-2.43	+1.98	-0.57	+0.01	-1.72	-0.78	-2.57	-2.07
	Relative (%)	-12.6	-36.5	-49.9	-25.3	-12.9	+19.8	-49.2	-20.4	-27.1	+37.5	+5.7	+26.9	-37.9	-19.4	-41.9	-25.5	+13.6	-38.4	+7.2	+40.5	-46.7	+38.2	-11.0	+0.2	-33.0	-15.0	-49.4	-39.8
Steps (reduced)		366 (135)	536 (74)	648 (186)	732 (39)	799 (106)	855 (162)	902 (209)	944 (20)	981 (57)	1015 (91)	1045 (121)	1073 (149)	1098 (174)	1122 (198)	1144 (220)	1165 (10)	1185 (30)	1203 (48)	1221 (66)	1238 (83)	1253 (98)	1269 (114)	1283 (128)	1297 (142)	1310 (155)	1323 (168)	1335 (180)	1347 (192)

373edo (L=21, s=8, BrightGen is 197\373; patent ~13/9 = _\373; patent ~23/16 = _\373; patent ~49/34 = _\373; patent ~62/43 = _\373; patent ~75/52 = _\373; patent ~384/265 = _\373; patent ~59049/40960 = _\373)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	-0.61	-0.25	-0.46	-1.23	-1.18	-0.85	-0.87	+1.21	-1.53	-1.08	-0.93	-0.51	+1.37	-0.09	+0.27	+1.42	-0.72	-0.41	-1.46	-1.18	+0.01	-1.48	+0.45	-0.92	+0.60	+1.56	-1.44	+1.07
	Relative (%)	-19.1	-7.9	-14.3	-38.2	-36.8	-26.4	-27.0	+37.6	-47.7	-33.4	-28.9	-15.8	+42.7	-2.7	+8.5	+44.1	-22.3	-12.6	-45.5	-36.7	+0.3	-46.1	+13.8	-28.7	+18.5	+48.6	-44.7	+33.2
Steps (reduced)		591 (218)	866 (120)	1047 (301)	1182 (63)	1290 (171)	1380 (261)	1457 (338)	1525 (33)	1584 (92)	1638 (146)	1687 (195)	1732 (240)	1774 (282)	1812 (320)	1848 (356)	1882 (17)	1913 (48)	1943 (78)	1971 (106)	1998 (133)	2024 (159)	2048 (183)	2072 (207)	2094 (229)	2116 (251)	2137 (272)	2156 (291)	2176 (311)

142edo (L=8, s=3, BrightGen is 75\142; patent ~13/9 = 75\142; patent ~23/16 = 74\142; patent ~49/34 = 76\142; patent ~62/43 = 74\142, 'k' wart usable; patent ~75/52 = 76\142, 'f' wart usable; patent ~384/265 = 76\142; patent ~59049/40960 = 74\142)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	-0.55	+2.42	+3.01	-1.09	-2.02	-3.91	+1.87	-3.55	-1.74	+2.46	-2.92	-3.61	-1.64	+1.41	-4.19	-2.57	-3.03	+2.18	+4.00	+1.92	+3.98	+1.33	+2.10	-2.44	-4.09	-3.08	+0.40	-2.28
	Relative (%)	-6.5	+28.6	+35.6	-12.9	-23.9	-46.2	+22.2	-42.0	-20.6	+29.1	-34.6	-42.8	-19.4	+16.7	-49.6	-30.4	-35.8	+25.8	+47.3	+22.8	+47.0	+15.7	+24.8	-28.9	-48.4	-36.5	+4.7	-27.0
Steps (reduced)		225 (83)	330 (46)	399 (115)	450 (24)	491 (65)	525 (99)	555 (129)	580 (12)	603 (35)	624 (56)	642 (74)	659 (91)	675 (107)	690 (122)	703 (135)	716 (6)	728 (18)	740 (30)	751 (41)	761 (51)	771 (61)	780 (70)	789 (79)	797 (87)	805 (95)	813 (103)	821 (111)	828 (118)

337edo (L=19, s=7, BrightGen is 178\337; patent ~13/9 = _\337; patent ~23/16 = _\337; patent ~49/34 = _\337; patent ~62/43 = _\337; patent ~75/52 = _\337; patent ~384/265 = _\337; patent ~59049/40960 = _\337)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	-0.47	-1.74	-0.28	-0.94	+0.61	-0.17	+1.35	-1.69	+1.60	-0.75	-1.57	+0.07	-1.41	-0.50	+1.55	+0.14	+1.54	+1.47	-0.64	-1.76	+1.24	+0.87	+0.37	-0.56	+1.40	-1.10	-1.13	+1.13
	Relative (%)	-13.2	-49.0	-7.9	-26.5	+17.2	-4.8	+37.8	-47.5	+44.8	-21.1	-44.0	+2.0	-39.7	-14.0	+43.6	+3.9	+43.2	+41.4	-18.1	-49.5	+34.9	+24.6	+10.4	-15.7	+39.3	-30.9	-31.8	+31.6
Steps (reduced)		534 (197)	782 (108)	946 (272)	1068 (57)	1166 (155)	1247 (236)	1317 (306)	1377 (29)	1432 (84)	1480 (132)	1524 (176)	1565 (217)	1602 (254)	1637 (289)	1670 (322)	1700 (15)	1729 (44)	1756 (71)	1781 (96)	1805 (120)	1829 (144)	1851 (166)	1872 (187)	1892 (207)	1912 (227)	1930 (245)	1948 (263)	1966 (281)

195edo (L=11, s=4, BrightGen is 103\195; patent ~13/9 = 104\195;, 'f' wart usable but requires 'e' wart for 11th harmonic; patent ~23/16 = 102\195; patent ~49/34 = 102\195; patent ~62/43 = 103\195; patent ~75/52 = 103\195; patent ~384/265 = 104\195; patent ~59049/40960 = 102\195)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	-0.42	+1.38	-2.67	-0.83	+2.53	+2.55	+0.96	-0.34	-2.13	+3.07	-0.58	+2.76	-1.25	-1.88	-0.42	+2.11	-1.29	+0.96	+2.13	+1.71	-0.75	+0.55	-0.89	+0.81	-0.76	+0.34	-2.25	-2.54
	Relative (%)	-6.8	+22.4	-43.4	-13.5	+41.1	+41.4	+15.6	-5.5	-34.6	+49.8	-9.5	+44.8	-20.3	-30.6	-6.8	+34.3	-21.0	+15.7	+34.7	+27.7	-12.2	+8.9	-14.5	+13.2	-12.3	+5.6	-36.5	-41.4
Steps (reduced)		309 (114)	453 (63)	547 (157)	618 (33)	675 (90)	722 (137)	762 (177)	797 (17)	828 (48)	857 (77)	882 (102)	906 (126)	927 (147)	947 (167)	966 (186)	984 (9)	1000 (25)	1016 (41)	1031 (56)	1045 (70)	1058 (83)	1071 (96)	1083 (108)	1095 (120)	1106 (131)	1117 (142)	1127 (152)	1137 (162)

248edo (L=14, s=5, BrightGen is 131\248; patent ~13/9 = _\248; patent ~23/16 = _\248; patent ~49/34 = _\248; patent ~62/43 = _\248; patent ~75/52 = _\248; patent ~384/265 = _\248; patent ~59049/40960 = _\248)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	-0.34	+0.78	-1.08	-0.68	+0.29	+1.41	+0.44	+1.50	-2.35	-1.43	+0.76	+1.57	-1.03	+1.07	+1.74	-0.05	-0.30	+0.27	+1.07	+1.58	+1.39	+0.10	+2.24	-2.17	+1.15	+2.30	+1.08	+2.14
	Relative (%)	-7.1	+16.2	-22.4	-14.1	+6.1	+29.1	+9.1	+30.9	-48.6	-29.5	+15.7	+32.4	-21.2	+22.1	+35.9	-1.0	-6.2	+5.6	+22.0	+32.7	+28.6	+2.0	+46.2	-44.8	+23.9	+47.6	+22.3	+44.3
Steps (reduced)		393 (145)	576 (80)	696 (200)	786 (42)	858 (114)	918 (174)	969 (225)	1014 (22)	1053 (61)	1089 (97)	1122 (130)	1152 (160)	1179 (187)	1205 (213)	1229 (237)	1251 (11)	1272 (32)	1292 (52)	1311 (71)	1329 (89)	1346 (106)	1362 (122)	1378 (138)	1392 (152)	1407 (167)	1421 (181)	1434 (194)	1447 (207)

53edo (L=3, s=1, BrightGen is 28\53; patent ~13/9 = 28\53; patent ~23/16 = 28\53; patent ~49/34 = 28\53; patent ~62/43 = 28\53; patent ~75/52 = 28\53; patent ~384/265 = 28\53; patent ~59049/40960 = 28\53) — Hard 17L 2s

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	-0.07	-1.41	+4.76	-0.14	-7.92	-2.79	-1.48	+8.25	-3.17	+4.69	+5.69	-2.82	-0.20	-10.71	+9.68	-7.99	+3.35	-2.29	-2.86	+1.13	+9.24	-1.54	-8.90	+9.52	+8.18	+9.51	-9.33	-3.24
	Relative (%)	-0.3	-6.2	+21.0	-0.6	-35.0	-12.3	-6.5	+36.4	-14.0	+20.7	+25.1	-12.4	-0.9	-47.3	+42.8	-35.3	+14.8	-10.1	-12.6	+5.0	+40.8	-6.8	-39.3	+42.0	+36.1	+42.0	-41.2	-14.3
Steps (reduced)		84 (31)	123 (17)	149 (43)	168 (9)	183 (24)	196 (37)	207 (48)	217 (5)	225 (13)	233 (21)	240 (28)	246 (34)	252 (40)	257 (45)	263 (51)	267 (2)	272 (7)	276 (11)	280 (15)	284 (19)	288 (23)	291 (26)	294 (29)	298 (33)	301 (36)	304 (39)	306 (41)	309 (44)

229edo (L=13, s=4, BrightGen is 121\229; patent ~13/9 = _\229; patent ~23/16 = _\229; patent ~49/34 = _\229; patent ~62/43 = _\229; patent ~75/52 = _\229; patent ~384/265 = _\229; patent ~59049/40960 = _\229)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	+0.23	+1.46	+0.61	+0.46	-1.10	-2.10	+1.69	-0.15	+1.18	+0.83	+0.55	-2.32	+0.69	-2.50	+2.56	-0.87	+2.07	+0.18	-1.87	+0.63	+2.02	+1.92	-0.00	+1.21	+0.08	+1.60	+0.36	+1.41
	Relative (%)	+4.4	+27.8	+11.6	+8.7	-21.0	-40.1	+32.2	-2.9	+22.5	+15.9	+10.4	-44.3	+13.1	-47.8	+48.9	-16.6	+39.4	+3.5	-35.7	+12.1	+38.5	+36.6	-0.1	+23.1	+1.5	+30.6	+6.9	+26.8
Steps (reduced)		363 (134)	532 (74)	643 (185)	726 (39)	792 (105)	847 (160)	895 (208)	936 (20)	973 (57)	1006 (90)	1036 (120)	1063 (147)	1089 (173)	1112 (196)	1135 (219)	1155 (10)	1175 (30)	1193 (48)	1210 (65)	1227 (82)	1243 (98)	1258 (113)	1272 (127)	1286 (141)	1299 (154)	1312 (167)	1324 (179)	1336 (191)

176edo (L=10, s=3, BrightGen is 93\176; patent ~13/9 = 93\176; patent ~23/16 = 92\176; patent ~49/34 = 93\176; patent ~62/43 = 93\176; patent ~75/52 = 94\176; patent ~384/265 = 94\176; patent ~59049/40960 = 93\176)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	+0.32	+2.32	-0.64	+0.64	+0.95	-1.89	+2.64	-2.68	+2.49	-0.33	-1.00	-2.17	+0.95	-0.03	+0.42	+1.27	+1.68	+0.93	-1.57	+0.48	-0.15	+2.96	+2.68	-1.29	-2.36	-0.78	+3.28	+2.80
	Relative (%)	+4.7	+34.1	-9.4	+9.3	+14.0	-27.7	+38.7	-39.3	+36.5	-4.8	-14.7	-31.9	+14.0	-0.5	+6.1	+18.7	+24.6	+13.6	-23.1	+7.1	-2.3	+43.4	+39.2	-18.9	-34.7	-11.4	+48.1	+41.1
Steps (reduced)		279 (103)	409 (57)	494 (142)	558 (30)	609 (81)	651 (123)	688 (160)	719 (15)	748 (44)	773 (69)	796 (92)	817 (113)	837 (133)	855 (151)	872 (168)	888 (8)	903 (23)	917 (37)	930 (50)	943 (63)	955 (75)	967 (87)	978 (98)	988 (108)	998 (118)	1008 (128)	1018 (138)	1027 (147)

299edo (L=17, s=5, BrightGen is 158\299; patent ~13/9 = _\299; patent ~23/16 = _\299; patent ~49/34 = _\299; patent ~62/43 = _\299; patent ~75/52 = _\299; patent ~384/265 = _\299; patent ~59049/40960 = _\299)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	+0.39	-1.03	-1.60	+0.77	-1.49	-1.73	-0.64	-0.61	-0.52	-1.22	+1.83	+1.95	+1.16	+1.86	-1.22	-1.10	+1.38	+1.50	-1.35	+0.37	-1.82	-0.26	+0.71	+0.81	-0.22	+1.41	+1.50	-0.14
	Relative (%)	+9.6	-25.7	-39.9	+19.2	-37.0	-43.1	-16.0	-15.1	-13.0	-30.3	+45.5	+48.7	+28.9	+46.4	-30.5	-27.4	+34.4	+37.3	-33.5	+9.2	-45.3	-6.4	+17.8	+20.2	-5.5	+35.2	+37.3	-3.4
Steps (reduced)		474 (175)	694 (96)	839 (241)	948 (51)	1034 (137)	1106 (209)	1168 (271)	1222 (26)	1270 (74)	1313 (117)	1353 (157)	1389 (193)	1422 (226)	1453 (257)	1481 (285)	1508 (13)	1534 (39)	1558 (63)	1580 (85)	1602 (107)	1622 (127)	1642 (147)	1661 (166)	1679 (184)	1696 (201)	1713 (218)	1729 (234)	1744 (249)

123edo (L=7, s=2, BrightGen is 65\123; patent ~13/9 = 65\123; patent ~23/16 = 64\123, 'i' wart is usable; patent ~49/34 = 64\123; patent ~62/43 = 65\123; patent ~75/52 = 66\123; patent ~384/265 = 65\123; patent ~59049/40960 = 65\123)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	+0.48	+3.93	-2.97	+0.97	+4.78	-1.50	+4.41	+2.36	-4.83	-2.49	-3.88	-1.90	+1.45	+4.57	-3.57	-4.49	+0.96	+2.31	-1.02	+0.21	-4.20	-4.86	-2.09	+3.81	+2.85	+4.54	-1.05	-4.35
	Relative (%)	+5.0	+40.3	-30.5	+9.9	+49.0	-15.4	+45.2	+24.2	-49.5	-25.5	-39.8	-19.4	+14.9	+46.8	-36.6	-46.0	+9.8	+23.7	-10.4	+2.1	-43.1	-49.8	-21.4	+39.1	+29.2	+46.6	-10.7	-44.5
Steps (reduced)		195 (72)	286 (40)	345 (99)	390 (21)	426 (57)	455 (86)	481 (112)	503 (11)	522 (30)	540 (48)	556 (64)	571 (79)	585 (93)	598 (106)	609 (117)	620 (5)	631 (16)	641 (26)	650 (35)	659 (44)	667 (52)	675 (60)	683 (68)	691 (76)	698 (83)	705 (90)	711 (96)	717 (102)

316edo (L=18, s=5, BrightGen is 167\316; patent ~13/9 = _\316; patent ~23/16 = _\316; patent ~49/34 = _\316; patent ~62/43 = _\316; patent ~75/52 = _\316; patent ~384/265 = _\316; patent ~59049/40960 = _\316)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	+0.58	+1.03	-0.47	+1.15	-0.69	-1.29	+1.60	+1.37	-1.31	+0.11	-1.69	-1.74	+1.73	-0.46	+1.80	-0.11	+0.56	-0.71	-0.71	+0.05	+1.14	-1.62	-0.95	-0.94	-1.85	-0.09	+0.34	-0.73
	Relative (%)	+15.2	+27.1	-12.4	+30.4	-18.0	-33.9	+42.3	+36.2	-34.5	+2.8	-44.6	-45.9	+45.6	-12.2	+47.4	-2.9	+14.7	-18.7	-18.7	+1.4	+30.0	-42.6	-25.0	-24.8	-48.6	-2.3	+9.0	-19.3
Steps (reduced)		501 (185)	734 (102)	887 (255)	1002 (54)	1093 (145)	1169 (221)	1235 (287)	1292 (28)	1342 (78)	1388 (124)	1429 (165)	1467 (203)	1503 (239)	1535 (271)	1566 (302)	1594 (14)	1621 (41)	1646 (66)	1670 (90)	1693 (113)	1715 (135)	1735 (155)	1755 (175)	1774 (194)	1792 (212)	1810 (230)	1827 (247)	1843 (263)

193edo (L=11, s=3, BrightGen is 102\193; patent ~13/9 = 102\193; patent ~23/16 = 101\193;; patent ~49/34 = 102\193; patent ~62/43 = 102\193; patent ~75/52 = 102\193; patent ~384/265 = 104\193; patent ~59049/40960 = 103\193)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	+0.64	-0.82	+1.12	+1.27	+2.05	-1.15	-0.19	+0.74	+0.93	+1.76	-0.30	-1.64	+1.91	+2.55	-0.99	+2.69	+0.30	-2.64	-0.51	-0.05	-1.67	+0.45	-0.22	+2.24	+1.38	-3.04	+1.23	+1.57
	Relative (%)	+10.2	-13.2	+18.1	+20.4	+33.0	-18.5	-3.0	+12.0	+15.0	+28.3	-4.7	-26.4	+30.7	+41.0	-16.0	+43.2	+4.8	-42.4	-8.3	-0.8	-26.9	+7.2	-3.6	+36.1	+22.2	-48.9	+19.8	+25.2
Steps (reduced)		306 (113)	448 (62)	542 (156)	612 (33)	668 (89)	714 (135)	754 (175)	789 (17)	820 (48)	848 (76)	873 (101)	896 (124)	918 (146)	938 (166)	956 (184)	974 (9)	990 (25)	1005 (40)	1020 (55)	1034 (69)	1047 (82)	1060 (95)	1072 (107)	1084 (119)	1095 (130)	1105 (140)	1116 (151)	1126 (161)

263edo (L=15, s=4, BrightGen is 139\263; patent ~13/9 = _\263; patent ~23/16 = _\263; patent ~49/34 = _\263; patent ~62/43 = _\263; patent ~75/52 = _\263; patent ~384/265 = _\263; patent ~59049/40960 = _\263)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	+0.71	+1.52	-1.53	+1.41	+0.77	-0.98	+2.23	-0.01	-0.94	-0.82	+1.38	-1.52	+2.12	+1.60	+0.21	+1.48	-0.01	-0.39	-0.28	-0.17	-0.49	-1.63	+0.65	+1.51	+0.69	-2.02	-2.27	-0.23
	Relative (%)	+15.5	+33.3	-33.4	+31.0	+16.9	-21.6	+48.8	-0.3	-20.5	-17.9	+30.3	-33.4	+46.5	+35.1	+4.6	+32.4	-0.1	-8.6	-6.1	-3.6	-10.8	-35.7	+14.3	+33.1	+15.2	-44.3	-49.8	-5.0
Steps (reduced)		417 (154)	611 (85)	738 (212)	834 (45)	910 (121)	973 (184)	1028 (239)	1075 (23)	1117 (65)	1155 (103)	1190 (138)	1221 (169)	1251 (199)	1278 (226)	1303 (251)	1327 (12)	1349 (34)	1370 (55)	1390 (75)	1409 (94)	1427 (112)	1444 (129)	1461 (146)	1477 (162)	1492 (177)	1506 (191)	1520 (205)	1534 (219)

70edo (L=4, s=1, BrightGen is 37\70; patent ~13/9 = 37\70; patent ~23/16 = 37\70; patent ~49/34 = 38\70; patent ~62/43 = 37\70; patent ~75/52 = 38\70; patent ~384/265 = 37\70; patent ~59049/40960 = 37\70) — Superhard 17L 2s

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	+0.90	+7.97	+8.32	+1.80	-2.75	-0.53	-8.27	-2.10	-6.08	-7.92	+6.01	-1.20	+2.71	-1.01	+3.54	-1.84	-0.85	+5.80	+0.37	-0.49	+2.77	-7.37	+3.06	-0.51	-1.20	+0.78	+5.23	-5.18
	Relative (%)	+5.3	+46.5	+48.5	+10.5	-16.0	-3.1	-48.2	-12.2	-35.5	-46.2	+35.1	-7.0	+15.8	-5.9	+20.6	-10.8	-5.0	+33.8	+2.2	-2.9	+16.1	-43.0	+17.9	-3.0	-7.0	+4.6	+30.5	-30.2
Steps (reduced)		111 (41)	163 (23)	197 (57)	222 (12)	242 (32)	259 (49)	273 (63)	286 (6)	297 (17)	307 (27)	317 (37)	325 (45)	333 (53)	340 (60)	347 (67)	353 (3)	359 (9)	365 (15)	370 (20)	375 (25)	380 (30)	384 (34)	389 (39)	393 (43)	397 (47)	401 (51)	405 (55)	408 (58)

227edo (L=13, s=3, BrightGen is 120\227; patent ~13/9 = _\227; patent ~23/16 = _\227; patent ~49/34 = _\227; patent ~62/43 = _\227; patent ~75/52 = _\227; patent ~384/265 = _\227; patent ~59049/40960 = _\227)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	+1.13	-0.41	-1.43	+2.26	-1.54	+0.00	+0.72	+0.77	-1.48	-0.30	+0.80	-0.82	-1.90	+1.26	+2.10	-0.41	-1.84	+2.40	+1.13	-0.87	+1.26	+1.85	+0.57	+2.44	+1.90	-1.26	-1.95	-0.35
	Relative (%)	+21.4	-7.8	-27.0	+42.7	-29.1	+0.0	+13.6	+14.6	-28.0	-5.6	+15.1	-15.5	-35.9	+23.8	+39.7	-7.7	-34.7	+45.4	+21.4	-16.4	+23.8	+34.9	+10.8	+46.1	+35.9	-23.8	-36.9	-6.6
Steps (reduced)		360 (133)	527 (73)	637 (183)	720 (39)	785 (104)	840 (159)	887 (206)	928 (20)	964 (56)	997 (89)	1027 (119)	1054 (146)	1079 (171)	1103 (195)	1125 (217)	1145 (10)	1164 (29)	1183 (48)	1200 (65)	1216 (81)	1232 (97)	1247 (112)	1261 (126)	1275 (140)	1288 (153)	1300 (165)	1312 (177)	1324 (189)

157edo (L=9, s=2, BrightGen is 83\157; patent ~13/9 = 83\157; patent ~23/16 = 82\157; patent ~49/34 = 83\157; patent ~62/43 = 83\157; patent ~75/52 = 84\157; patent ~384/265 = 84\157; patent ~59049/40960 = 84\157)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	+1.23	+3.50	+1.87	+2.46	-1.00	+0.24	-2.92	+2.05	+0.58	+3.10	-1.52	-0.65	+3.69	+2.27	+1.46	+0.23	-2.27	+0.89	+1.47	-1.04	+0.58	-1.69	-0.54	+3.75	+3.28	-2.17	+2.50	+1.81
	Relative (%)	+16.1	+45.7	+24.5	+32.2	-13.1	+3.1	-38.2	+26.8	+7.5	+40.6	-19.9	-8.5	+48.3	+29.7	+19.1	+3.0	-29.7	+11.6	+19.2	-13.6	+7.6	-22.1	-7.0	+49.1	+42.9	-28.4	+32.7	+23.6
Steps (reduced)		249 (92)	365 (51)	441 (127)	498 (27)	543 (72)	581 (110)	613 (142)	642 (14)	667 (39)	690 (62)	710 (82)	729 (101)	747 (119)	763 (135)	778 (150)	792 (7)	805 (20)	818 (33)	830 (45)	841 (56)	852 (67)	862 (77)	872 (87)	882 (97)	891 (106)	899 (114)	908 (123)	916 (131)

244edo (L=14, s=3, BrightGen is 129\244; patent ~13/9 = _\244; patent ~23/16 = _\244; patent ~49/34 = _\244; patent ~62/43 = _\244; patent ~75/52 = _\244; patent ~384/265 = _\244; patent ~59049/40960 = _\244)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	+1.32	+2.21	+0.03	-2.27	-0.50	+0.46	-1.38	-1.68	-2.43	+1.35	+1.23	-0.50	-0.95	-1.71	+0.87	+0.83	+2.24	-0.52	+1.78	-1.19	-0.04	-0.06	-1.57	+0.05	-0.35	+1.91	+1.71	-1.11
	Relative (%)	+26.9	+45.0	+0.5	-46.2	-10.1	+9.3	-28.1	-34.1	-49.4	+27.5	+25.1	-10.1	-19.3	-34.7	+17.6	+16.8	+45.5	-10.7	+36.2	-24.3	-0.9	-1.2	-32.0	+1.1	-7.2	+38.7	+34.8	-22.5
Steps (reduced)		387 (143)	567 (79)	685 (197)	773 (41)	844 (112)	903 (171)	953 (221)	997 (21)	1036 (60)	1072 (96)	1104 (128)	1133 (157)	1160 (184)	1185 (209)	1209 (233)	1231 (11)	1252 (32)	1271 (51)	1290 (70)	1307 (87)	1324 (104)	1340 (120)	1355 (135)	1370 (150)	1384 (164)	1398 (178)	1411 (191)	1423 (203)

87edo (L=5, s=1, BrightGen is 46\87; patent ~13/9 = 46 \87; patent ~23/16 = 46\87; patent ~49/34 = 45\87; patent ~62/43 = 46\87; patent ~75/52 = 46\87; patent ~384/265 = 47\87; patent ~59049/40960 = 47\87)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	+1.49	-0.11	-3.31	+2.99	+0.41	+0.85	+1.39	+5.39	+5.94	-1.82	+6.21	-0.21	+4.48	+4.91	-0.21	+1.90	-3.42	-3.07	+2.34	-1.48	-1.17	+2.88	-3.44	-6.62	+6.88	-4.54	+0.30	-6.36
	Relative (%)	+10.8	-0.8	-24.0	+21.7	+2.9	+6.2	+10.1	+39.1	+43.0	-13.2	+45.0	-1.5	+32.5	+35.6	-1.5	+13.8	-24.8	-22.2	+17.0	-10.7	-8.5	+20.9	-24.9	-48.0	+49.9	-32.9	+2.2	-46.1
Steps (reduced)		138 (51)	202 (28)	244 (70)	276 (15)	301 (40)	322 (61)	340 (79)	356 (8)	370 (22)	382 (34)	394 (46)	404 (56)	414 (66)	423 (75)	431 (83)	439 (4)	446 (11)	453 (18)	460 (25)	466 (31)	472 (37)	478 (43)	483 (48)	488 (53)	494 (59)	498 (63)	503 (68)	507 (72)

191edo (L=11, s=2, BrightGen is 101\191; patent ~13/9 = _\191; patent ~23/16 = _\191; patent ~49/34 = _\191; patent ~62/43 = _\191; patent ~75/52 = _\191; patent ~384/265 = _\191; patent ~59049/40960 = _\191)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	+1.71	-3.07	-1.29	-2.86	+1.56	+1.36	-1.36	+1.85	-2.23	+0.42	-0.00	+0.15	-1.15	+0.79	-1.58	-3.01	+1.93	-0.04	+3.07	-1.84	-2.62	+0.35	+0.46	-2.57	-2.72	-0.21	-1.51	-0.52
	Relative (%)	+27.2	-48.8	-20.5	-45.6	+24.9	+21.6	-21.6	+29.5	-35.4	+6.7	-0.0	+2.3	-18.4	+12.6	-25.1	-47.9	+30.7	-0.6	+48.8	-29.2	-41.7	+5.6	+7.4	-41.0	-43.3	-3.3	-24.0	-8.2
Steps (reduced)		303 (112)	443 (61)	536 (154)	605 (32)	661 (88)	707 (134)	746 (173)	781 (17)	811 (47)	839 (75)	864 (100)	887 (123)	908 (144)	928 (164)	946 (182)	963 (8)	980 (25)	995 (40)	1010 (55)	1023 (68)	1036 (81)	1049 (94)	1061 (106)	1072 (117)	1083 (128)	1094 (139)	1104 (149)	1114 (159)

104edo (L=6, s=1, BrightGen is 55\104; patent ~13/9 = 55\104; patent ~23/16 = 54\104; patent ~49/34 = 55\104; patent ~62/43 = 55\104; patent ~75/52 = 54\104; patent ~384/265 = 56\104; patent ~59049/40960 = 57\104)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	+1.89	-5.54	+0.40	+3.78	+2.53	+1.78	-3.65	-1.11	+2.49	+2.30	-5.20	+0.45	+5.67	-2.65	-2.73	+4.42	-5.14	+2.50	+3.67	-2.14	-3.83	-1.76	+3.72	+0.81	+0.78	+3.42	-3.02	+4.38
	Relative (%)	+16.4	-48.1	+3.5	+32.8	+21.9	+15.4	-31.7	-9.6	+21.6	+19.9	-45.0	+3.9	+49.2	-23.0	-23.6	+38.3	-44.5	+21.7	+31.8	-18.5	-33.2	-15.3	+32.3	+7.0	+6.8	+29.6	-26.1	+37.9
Steps (reduced)		165 (61)	241 (33)	292 (84)	330 (18)	360 (48)	385 (73)	406 (94)	425 (9)	442 (26)	457 (41)	470 (54)	483 (67)	495 (79)	505 (89)	515 (99)	525 (5)	533 (13)	542 (22)	550 (30)	557 (37)	564 (44)	571 (51)	578 (58)	584 (64)	590 (70)	596 (76)	601 (81)	607 (87)

121edo (L=7, s=1, BrightGen is 64\121; patent ~13/9 = _\121; patent ~23/16 = _\121; patent ~49/34 = _\121; patent ~62/43 = _\121; patent ~75/52 = _\121; patent ~384/265 = _\121; patent ~59049/40960 = _\121)

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	+2.18	+0.46	+3.07	+4.35	+4.05	+2.45	+2.64	+4.14	+0.01	-4.67	-3.48	+0.93	-3.39	+1.83	-4.54	-3.69	+3.54	-3.41	+4.62	-2.62	+4.18	+4.82	-1.04	-3.77	-3.60	-0.78	+4.52	+2.18
	Relative (%)	+22.0	+4.7	+31.0	+43.9	+40.9	+24.7	+26.6	+41.7	+0.1	-47.0	-35.1	+9.3	-34.1	+18.4	-45.8	-37.2	+35.7	-34.4	+46.6	-26.4	+42.2	+48.6	-10.5	-38.0	-36.3	-7.8	+45.5	+22.0
Steps (reduced)		192 (71)	281 (39)	340 (98)	384 (21)	419 (56)	448 (85)	473 (110)	495 (11)	514 (30)	531 (47)	547 (63)	562 (78)	575 (91)	588 (104)	599 (115)	610 (5)	621 (16)	630 (25)	640 (35)	648 (43)	657 (52)	665 (60)	672 (67)	679 (74)	686 (81)	693 (88)	700 (95)	706 (101)

17edo (L=1, s=0, BrightGen is 9\17; patent ~13/9 = 9\17; patent ~23/16 = 9\17; patent ~49/34 = 10\17, 'g' wart usable but requires 'c' wart for 5th harmonic; patent ~62/43 = 9\17; patent ~75/52 = 8\17; patent ~384/265 = 10\17; c val ~384/265 = 9\17; patent ~59049/40960 = 10\17; c val ~59049/40960 = 9\17) — Collapsed 17L 2s

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57
Error	Absolute (¢)	+3.93	-33.37	+19.41	+7.85	+13.39	+6.53	-29.45	-34.37	-15.16	+23.34	+7.02	+3.84	+11.78	+29.25	-15.62	+17.32	-13.96	+31.01	+10.46	-5.53	-17.40	-25.52	-30.21	-31.77	-30.44	-26.45	-19.98	-11.23
	Relative (%)	+5.6	-47.3	+27.5	+11.1	+19.0	+9.3	-41.7	-48.7	-21.5	+33.1	+9.9	+5.4	+16.7	+41.4	-22.1	+24.5	-19.8	+43.9	+14.8	-7.8	-24.7	-36.2	-42.8	-45.0	-43.1	-37.5	-28.3	-15.9
Steps (reduced)		27 (10)	39 (5)	48 (14)	54 (3)	59 (8)	63 (12)	66 (15)	69 (1)	72 (4)	75 (7)	77 (9)	79 (11)	81 (13)	83 (15)	84 (16)	86 (1)	87 (2)	89 (4)	90 (5)	91 (6)	92 (7)	93 (8)	94 (9)	95 (10)	96 (11)	97 (12)	98 (13)	99 (14)

(Need a way to combine the collection of tables into a single table for better readability.)

In detailed observations of scrolling through the above table group (which has not yet been updated to include the extra column from the 17L 2s scale tree, I started noticing interesting things, like how the harmonic/subharmonics of the generator have unstable mapping (because no simple ratio with a reasonable sized numerator and denominator fits into this zone), but the 3rd harmonic is nearly rock-solid (and 112b is a respectable if overly-complex quarter-comma meantone approximation), although its mapping causes strange effects in the right-most column of the MOS spectrum table, causing ~13/9 to map to the bright generator in several EDOs in the soft half of the scale tree and to map 1 step sharp of the bright generator in several EDOs in the hard half of the spectrum. And the mapping of the 53rd harmonic is reasonably solid as long as the right-most column of the scale tree is entirely left out (will need to leave out the right-most 2 columns after this is updated to the expanded scale tree); ~384/265 does not do as well as expected as a generator. And there the 5th harmonic seems very much usable in the soft end of the scale scale tree as long as the EDO sizes don't get too large (and even then, sometimes it is still okay), which looks to me like enabling a 2.3.5.23.53 meantone extension. The 5th and 53rd harmonics go all over the place in the hard end, but there the 25th harmonic shines and is rock-solid as long as you don't go softer than 36edo (basic), and the 13th harmonic jis fairly rock-solid in this zone (and some of the exceptions are candidates for rescue by applying an 'f' wart); in the soft half of the spectrum, the 13th harmonic always maps to 1 step too many for ~13/9 to be usable as the bright generator, and an 'f' wart would worsen consistency with nearby harmonics, except strangely in most of the right-most column of the the 17L 2s scale tree. The generator ~59049/40960 = ~|-13 10 -1⟩ of the established Alphatricot family only works for a narrow band in the hard to super-hard region of the 17L 2s scale tree, plus 17c. The generator ~49/34 = |-1 0 0 2 0 0 -1⟩ has a just value not too far off from the middle of the 17L 2s scale tree, so it works over a fairly wide range of EDO values within this scale tree, but suffers from the 7th and 17th harmonics not covarying as well as would be needed for general applicability, as well as having 2 powers of 7, which precludes use of a 'd' wart to fix a fault with the mapping of the 7th harmonic. The generator ~75/52 = |-2 1 2 0 0 -1⟩ also has a just value not too far off from the middle of the 17L 2s scale tree, so it works over a fairly wide range of EDO values within this scale tree, but suffers from the 3rd, 5th, and 13th harmonics not covarying as well as would be needed for general applicability, as well as having 2 powers of 5, which precludes use of a 'c' wart to fix a fault with the mapping of the 5th harmonic. Finally, the ratio ~62/43 (bright generator) or ~43/31 (dark generator) has amazing stability — even though the mapping of its component harmonics is pretty unstable, they co-vary sufficiently well that a wart is needed for proper mapping in only a few places, and in each case the wart does not appear to hurt consistency, at least at a brief inspection.

Added: Lucius Chiaraviglio (talk) 07:42, 8 April 2025 (UTC)
Last modified: Lucius Chiaraviglio (talk) 09:25, 21 June 2025 (UTC)