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) 07:00, 9 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.
Added: Lucius Chiaraviglio (talk) 11:04, 16 February 2025 (UTC)
Still need comma for back-extension to 5th harmonic and maybe back extension to 7th harmonic.
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:
| 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) | |
| 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) | |
| 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) | |
| 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) | |
| 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) | |
| 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) | |
| 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) | |
| 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) | |
| 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) | |
| 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) | |
| 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) | |
| 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) | |
| 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) | |
| 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) | |
| 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) | |
| 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) | |
| 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) | |
| 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) | |
| 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) | |
| 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) | |
| 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) | |
| 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) | |
| 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) | |
| 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) | |
| 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) | |
| 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) | |
| 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) | |
| 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) | |
| 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) | |
| 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) | |
| 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) | |
| 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) | |
| 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) 07:31, 10 April 2025 (UTC)
Musical Mad Science Musings on Diatonicized Sixth-Tone Sub-Chromaticism(?)
The 36edo equivalent of Diatonicized Chromaticism is 17L 2s. So I've been giving a bit of thought to how to start constructing a temperament (or set thereof) that uses this scale. This is in a very rudimentary stage, but so far I have observed:
- 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 unacceptably large primes, 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. 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.
- 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.
- It is noteworthy (more detail needed) that harmonics 3 and 23 are very stable over the tuning spectrum of this scale (at least for EDO values up into the mid double digits), 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 tuning spectrum to get rather bad at the extremes; the 5th harmonic is definitely not stable, and would need different extensions for at least the hard and soft halves of the tuning spectrum; commas including powers of 5 should be avoided in the core of the associated temperament, while commas including powers of 3 and 23 (and possibly 13 — need further checking to be sure, but this is looking less good than 3 and 23) seem like they would be good choices for the core. (Coming in the future: Checking this further.)
- Tentatively assigning the generator as 23/16 ~ 13/9, tempering out 208/207. 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 tuning spectrum, even for the for the hard half of the tuning spectrum (closer to just 13/9, including having the best 3rd harmonic within the tuning spectrum). Maybe splitting the tuning spectrum of 17L 2s into 2 or more temperaments is in order? Maybe the 5th harmonic is stable enough for the soft half of the 17L 2s tuning spectrum (closer to just 23/16, but even closer to the just barely out-of-reach 36/25)? And maybe the 7th and 17th harmonics are stable enough for the middle of the 17L 2s tuning spectrum? (Coming in the future: Checking this further; may need to insert some more supporting material above.)
Added: Lucius Chiaraviglio (talk) 08:20, 4 April 2025 (UTC)
Last modified: Lucius Chiaraviglio (talk) 07:42, 8 April 2025 (UTC)
Table of odd harmonics for various EDO values supporting 17L 2s
This table (actually a collection of tables for now) is for tracking trends in odd harmonics along the tuning spectrum of 17L 2s; it is intended to match the organization of the corresponding scale tree, except for omitting the right-most column other than the top and bottom extremes:
| 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) | |
| 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) | |
| 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) | |
| 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) | |
| 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) | |
| 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) | |
| 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) | |
| 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) | |
| 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) | |
| 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) | |
| 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) | |
| 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) | |
| 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) | |
| 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) | |
| 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) | |
| 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) | |
| 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) | |
| 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) | |
| 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 preliminary observations of scrolling through the above table group, 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 presumably its mapping would break if I put in the rest of the right-most column of the MOS spectrum table. And there the 5th harmonic seems very much usable in the soft end of the scale tuning spectrum 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 meantone extension; it goes 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 harder than 36edo (basic), and the 13th harmonic just barely misses being rock-solid in this zone (just barely breaks on 125edo, for which 125f would be not bad). Although those harmonics would also appear less solid if I included the rest of the MOS tuning spectrum.
Added: Lucius Chiaraviglio (talk) 07:42, 8 April 2025 (UTC) Last modified: Lucius Chiaraviglio (talk) 07:31, 10 April 2025 (UTC)