37edo

37edo, or 37 equal divisions of the octave, is the equal tuning featuring steps of (1200/37) ~= 32.4 cents, 37 of which stack to the perfect octave 2/1. It is notable for having two reasonable choices for mapping prime 3 (at ~714 and ~681 cents), but mapping the rest of the 19-limit rather accurately.

Theory

JI approximation

37edo's most notable feature is a very good representation of the no-3 13-limit and a reasonably good representation of the no-3 19-limit. When 3 is included, it is a "dual-3" system, meaning that not only are there two tunings of 3 that can be combined to create an accurate 9/8, but those tunings of 3 are also passably accurate on their own in many contexts, such as approximating 4:5:6. The sharp fifth functions as a diatonic fifth and the primary fifth in a regular temperament interpretation of the EDO. As a dual-3 system, it features the dual-3 diatonic scales 5L 1m 1s, resembling 38edo's diatonic but with an edostep removed from one of the small steps, and 5L 2m 8s "trackdye".

The dual fifths can be thought of as a common fifth and a "leap fifth" (inserted every three fifths to correct the approximation of Pythagorean intervals).

Approximation of prime harmonics in 37edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	0.0	+11.6	+2.9	+4.1	+0.0	+2.7	-7.7	-5.6	-12.1	+8.3	-9.9
	Relative (%)	0.0	+35.6	+8.9	+12.8	+0.1	+8.4	-23.6	-17.3	-37.2	+25.5	-30.5
Steps (reduced)		37 (0)	59 (22)	86 (12)	104 (30)	128 (17)	137 (26)	151 (3)	157 (9)	167 (19)	180 (32)	183 (35)

Thirds in 37edo

Cents	259	291	324	356	389	421	454
Quality against flat fifth	Subminor	Pentaminor	Supraminor	Submajor	Pentamajor	Supermajor
Just interpretation	33/28	13/11	17/14	16/13	5/4	14/11
Quality against sharp fifth	Subminor	Neominor	Pentaminor	Neutral	Pentamajor	Neomajor	Supermajor
Just interpretation	7/6	13/11	6/5	16/13	5/4	14/11	13/10, 9/7
2.9... interpretations				11/9		9/7
Steps	8	9	10	11	12	13	14

Fifth-generated thirds (mosdiatonic and antidiatonic) are bolded.

Chords

Due to 37edo's two distinct fifths, it contains two different sets of fifth-bounded triads, which might be initially understood by examining the tuning's diatonic and antidiatonic scales. The sharp fifth naturally generates arto and tendo triads representing the 2.3.5.13 subgroup, while the flat fifth naturally generates supraminor and submajor triads (note that the "submajor third" is actually the neutral third corresponding to the sharp fifth). By alternating the two fifths as in a dual-fifth system, classical major and minor triads can be generated.

Scales

As mentioned previously, 37edo supports not only the very hard mosdiatonic generated by its sharp fifth, and the soft antidiatonic generated by its flat fifth, but a "dual-3" diatonic resembling 38edo's mosdiatonic, generated by alternating the two fifths. It also has a Zarlino diatonic where, like in 15 and 22edo, the three different step sizes are equidistant, meaning a single accidental can be used to cover the Zarlino scale. (In fact, this makes it a MODMOS of Porcupine[7].

[Add groundfault scales here]

Regular temperaments

If the two versions of the third harmonic are treated as stacking to the 9th harmonic, then 37edo supports a form of "2.<3.>3.5" meantone, where two sharp fifths and two flat fifths stack to create a 5/1. Another important temperament 37edo supports using only the sharp fifth is Porcupine, which it shares with 15edo and 22edo and for which it is the first edo that features a reasonable extension to the 13-limit.

Notation

As an archy edo, 37edo may be notated with diatonic notation, or with KISS or diamond-mos notation using its flat antidiatonic fifth (which also has the advantage of accidentals only raising or lowering by one step). It may also be notated with a straddle-3 erac notation system, taking advantage of its straddle-3 diatonic — this has an advantage in that 5/4 is a major third — or by Porcupine zarlino notation, which has the same advantage.

g_

Because I find it so useful as a dual-3 tuning, I recommend naming intervals like a subset of 74edo or 111edo to avoid giving either fifth unfair emphasis.

Cents	Ground Name	Cents	Ground Name
0.0	Unison	1200.0	Octave
32.4	Diesis	1167.6	Inverted Diesis
64.9	Subminor Second, Small Semitone	1135.1	Supermajor Seventh
97.3	Middle Minor Second, Middle Semitone	1102.7	Middle Major Seventh
129.7	Supraminor (Small Neutral) Second, Large Semitone	1070.3	Submajor (Large Neutral) Seventh
162.2	Submajor (Large Neutral) Second	1037.8	Supraminor (Small Neutral) Seventh
194.6	Meantone	1005.4	Meanseventh
227.0	Supermajor Second	973.0	Subminor Seventh
259.5	Subminor Third	940.5	Supermajor Sixth
291.9	Grave Minor Third	908.1	Acute Major Sixth
324.3	Acute Minor Third	875.7	Grave Major Sixth
356.8	Submajor (Large Neutral) Third	843.2	Supraminor (Small Neutral) Sixth
389.2	Pental Major Third	810.8	Pental Major Sixth
421.6	Neomajor Third	778.4	Neominor Seventh
454.1	Semisixth	745.9	Semitenth
486.5	Grave Fourth	713.5	Acute Fifth
518.9	Acute Fourth	681.1	Grave Fifth
551.4	Superfourth	648.6	Subfifth
583.8	Small Tritone	616.2	Large Tritone

SS

37 primarily appeals to me as a tuning of archy and porcupine (that is, if its size didn't prevent me from reasonably using it). The following names use the sharp fifth (that gives those temperaments) in ADIN, and map nearmajor and nearminor to porcupine's 5-limit intervals. I find it justified to emphasize the diatonic fifth either way, because it has rather significant structural utility and is twice as accurate as the flat fifth.

Cents	Vector Name	Cents	Vector Name
0.0	unison	1200.0	octave
32.4	inframinor second	1167.6	ultramajor seventh
64.9	farminor second	1135.1	farmajor seventh
97.3	nearminor second	1102.7	nearmajor seventh
129.7	neutral second	1070.3	neutral seventh
162.2	nearmajor second	1037.8	nearminor seventh
194.6	farmajor second	1005.4	farminor seventh
227.0	ultramajor second	973.0	inframinor seventh
259.5	inframinor third	940.5	ultramajor sixth
291.9	farminor third	908.1	farmajor sixth
324.3	nearminor third	875.7	nearmajor sixth
356.8	neutral third	843.2	neutral sixth
389.2	nearmajor third	810.8	nearminor sixth
421.6	farmajor third	778.4	farminor sixth
454.1	ultramajor third	745.9	inframinor sixth
486.5	perfect fourth	713.5	perfect fifth
518.9	far fourth	681.1	far fifth
551.4	near fourth	648.6	near fifth
583.8	neutral fourth	616.2	neutral fifth

Ground's notes

From User:ground:

37edo is the tuning that I use the largest number of distinct scales in. Here are the ones I could think of:

• 5:2:1 trackdye 5L2m8s
  • Step tunings: (227¢) : (195¢) : 162¢ : 65¢ : 32¢
  • This is the quintessential Aberration scale, short for "tract-aberrated diatonic". I invented it just to make 37edo work, and I now believe it to be the most important scale it offers by far, basically 12edo with all the intervals that make 37edo so special. It contains all three types of semitones, all three types of whole tones, and both fifths. There are seven possible structures depending on which diatonic mode you choose to aberrate. It's relatively hard to use due to its inherent dual-fifth structure, but you can't go wrong with it.
• 5:3:2 blackdye 5L2m3s
  • Step tunings: (227¢) : 162¢ : 97¢ : 62¢, patent val (9/8) : 10/9 : 16/15 : 81/80
  • A subset of Ultrapyth[17] and Porcupine[15]. It's close to the m=s Blackwood degenerate tuning, and the patent val 10/9 is a Porcupine neutral second. The 65¢ interval is possibly usable as an aberrisma, but too wide for me. I think it makes sense to use Blackwood cues here to apply it as a sort of alternate semitone, although I don't have much experience doing this. I usually just avoid it and insert fragments of the neogothic blackdye when I want an aberrisma.
  • Acute Minor / Grave Major modes: The standard pental scales of 37edo. The harmony is quite nice, the semitone is familiarly sized, but the whole tones are very distorted, leading to a scale that makes it trivial to achieve a xenharmonic sound. This is one of my favorite things about 37edo.
  • Grave Minor / Acute Major modes: One way to improve septal melody over the diatonic scale. I prefer the sound of the neogothic blackdye for this, but this one has the possible advantage of using the 422¢ major third in grave Aeolian and Dorian, which is more third-like than the 454¢ alternative.
• 6:2:1 blackdye 5L2m3s
  • Step tunings: (227¢) : 195¢ : 65¢ : 32¢
  • A subset of Ultrapyth[12], one of the two simple neogothic blackdye scales, the other being 32edo's 5:2:1. The melody is something expected from septal diasem, which makes it desirable to infuse that quality into 37edo music. The aberrisma is medium-small, which is very versatile.
  • Acute Minor / Grave Major modes: The standard neogothic scales of 37edo. On top of the melodic properties, I also like neogothic minor chords, so I use this one about as much as the pental blackdye.
  • Grave Minor / Acute Major modes: My preferred way to insert subminor thirds into 37edo music. Just like in the pental blackdye, no mode exists that has the inframinor sixth and diatonic fifth over the tonic, so I don't use this scale in its entirety, instead opting to mix its structures with other modes and scales.
• 7:1 diatonic 5L2s
  • Step tunings: 227¢ : 32¢
  • Ultrapyth[7], or Oceanfront temperament in other words. Just like with neogothic blackdye, the other Oceanfront tuning is 32edo's 6:1. This is about as hard as you can push a diatonic scale before it stops making sense, as the 454¢ major third is an effectively just 13/10, the interordinal on the boundary between major thirds and subfourths. It still has a usable 7/6 however, due to the fifth being so sharp, and I often prefer 13/10 to 9/7 anyway because it's past the peak of "majorness". Diatonic context is powerful, so as long as the instrumentation allows the comma-sized semitones to be audible, chords and melodies still make sense.
• 6:3:1 pinedye R/L 5L2m1s
  • Step tunings: (227¢) : 195¢ : 97¢ : 32¢
  • Unlike softer tunings like 27edo's 4:3:1, this doesn't much resemble the pental JI tuning. Instead it's closer to 25edo's 4:2:1, a compressed 12edo diatonic that uses both the flat and sharp fifth. The compressed thirds are close to 13/11 and 5/4, an excellent combination to my preference.
• 5:4:2 diasem "armosem" R/L 5L2m2s
  • Step tunings: (227¢) : 162¢ : 130¢ : 65¢
  • This is 37edo's only tuning of diasem, an equivalent to tempering pental blackdye with the flat fifth instead of the sharp diatonic one. Like pinedye, it combines ~13/11 and ~5/4, but with no alternate fifth. It's an optimization of Mavila armotonic, hence the name. It's the only scale on this list to feature a 130¢ step so prominently. Because the s step is 2\37, it can be split in half for a related 5:4:1 diaslen 5L2m4s.
• 6:1 archaeotonic 6L1s and 5:1 6L7s
  • Step tunings: (227¢) : 195¢ : 32¢ and (195¢) : 162¢ : 32¢
  • These are the scales of the whole-tone Didacus temperament. Normally I would just use archaeotonic, but 37edo's tuning is only remarkable when extended to higher limits.
• 7:5 onyx 1L6s and 5:2 pine 7L1s
  • Step tunings: 227¢ : 162¢ and (227¢) : 162¢ : 65¢
  • I don't use much Porcupine, but when I do, this is basically the perfect tuning. A better 6/5 than 27edo, a better 3/2 than 15edo, and a better 5/4 and 11/8 than both. Porcupine structures occasionally show up when I write chord progressions, which is always cool because I like neutral seconds.
• 7:3 smitonic 4L3s
  • Step tunings: 227¢ : 97¢
  • I've long been an enjoyer of the 5|1 "Vivecan" mode of smitonic. It's one of the few scales that I prefer to tune soft rather than hard, but tunings that are basic or slightly harder allow access to the very good 2.7.11 Orgone temperament. Being used to 11edo, I prefer this to the slightly harder 26edo tuning. I've been learning to accept ~11/7 as a fifth, extended from 11edo's ~14/9.
• 5:2 slentonic 5L6s
  • Step tunings: (227¢) : 162¢ : 65¢
  • This might be the weirdest one. I normally want Slendric for this scale structure rather than Laconic/Lywellen/Gorgo/Shoe, but I've been using this to test the boundaries of my perception for drastically compressed and stretched diatonic-coded intervals. 8/7 isn't a minor third, but it might possibly be usable as one in this scale, for example.
• Mosh
• Mosh3s
• 11/8-generated temperament (low-complexity 2.11.13, more accurate than Lovecraft)
• Ammonite oneirotonic (useful for the 8edo-like chords it generates, as this is the way its two neutral seconds stack repeatedly to make an octave)
• "Borcupat" (a joke name for my neogothic Porcupine-based scale, explanation needed)
• Shallowtone[16] (or [12] for a smaller ternary scale more analogous to deeptone)

My quasi-diatonic chord set in 37edo is like

• neominor [[7th]] (basic minor)
• compressed 6:7:9 (basic subminor which I don't use nearly enough)
• compressed 9:12:16:19 with stretched 9:12 (basic 47b9 ground chord)
• pental major/minor [[7th]] (basic major, alternate minor)
• pentagoth major (alternate major)
• ~10:13:15 (like major with sus quality)
• various approximations of sus4 [7th]
• ~[4]:5:6:7 (basic dominant/diminished)
• 16:19:22 (alternate diminished)
• 8:11:13 (basic superfourth/subfifth chord, tempered retroversion of 13:16:19)

Triad inversions are considered the same chord. Major and minor can be swapped (triads or tetrads retroverted), but these are situation-specific.

37edo has a nearly perfect logarithmically stretched gentle triad, by virtue of dividing the sharp fifth into 22 equal parts, via the following mathematical coincidence:

$1200{\log }_2(\frac{13}{11}\cdot \frac{14}{11})\cdot \frac{13}{22}=417.606\approx 1200{\log }_2\left(\frac{14}{11}\right)=417.508$

$1200{\log }_2(\frac{13}{11}\cdot \frac{14}{11})\cdot \frac{9}{22}=289.112\approx 1200{\log }_2\left(\frac{13}{11}\right)=289.210$

I don’t know if this is audibly significant, but at least it’s a cool justification for specifically 37edo as a neogothic blackdye tuning. It is also a good tuning of "hiberidian"

74edo scales:

• 12:5:2 Diasem (eridial meantone)
• 12:4:3 Diasem (Animist detemper of semiquartal, supermajor third is ~9/7 and subfourth is ~13/10)
• Half-octave Porcupine (9/7) and Oceanfront (17/14), see Diaschismic straddle temperaments

111edo scales:

• Blackdye and diasem in a very accurate 7-limit
• 5L2m3s4t
• Cohemimabila (extreme Sixix)
• Vengeance

Multiples

74edo

74edo is best known as a nearly optimal tuning of Grosstone, an extension of Septimal Meantone. Its flat 3 and 37edo's sharp 3 stack to make a sharp 9, which fills some gaps in 37edo's sharp-tending 13-limit. It is an excellent tuning of Diaschismic straddle temperaments, adding a half-octave period which turns Porcupine into Stearn, a very efficient 2.3.7 restriction of Hedgehog. It is a decent Monarch tuning.

Approximation of prime harmonics in 74edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31	37
Error	Absolute (¢)	0.0	-4.7	+2.9	+4.1	+0.0	+2.7	-7.7	-5.6	+4.2	-8.0	+6.3	-8.1
	Relative (%)	0.0	-28.7	+17.7	+25.6	+0.2	+16.7	-47.2	-34.7	+25.6	-49.1	+38.9	-50.0
Steps (reduced)		74 (0)	117 (43)	172 (24)	208 (60)	256 (34)	274 (52)	302 (6)	314 (18)	335 (39)	359 (63)	367 (71)	385 (15)

111edo

111edo is a multiple of 37edo that adds an accurate prime 3 to 37edo's 13-limit, making it significant to RTT. It is notable for being an Aberschismic system (tempering out 5120/5103, equating 81/80 and 64/63) and a Buzzard tuning with exact interordinal intervals. It is highly accurate in the 17-limit and consistent in the 21-odd-limit, although notably sharp-tending. It contains good tunings of Cohemimabila and Hibernal.

Approximation of prime harmonics in 111edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31	37
Error	Absolute (¢)	0.0	+0.7	+2.9	+4.1	+0.0	+2.7	+3.2	+5.2	-1.2	-2.6	+0.9	-2.7
	Relative (%)	0.0	+6.9	+26.6	+38.4	+0.3	+25.1	+29.2	+48.0	-11.5	-23.6	+8.4	-24.9
Steps (reduced)		111 (0)	176 (65)	258 (36)	312 (90)	384 (51)	411 (78)	454 (10)	472 (28)	502 (58)	539 (95)	550 (106)	578 (23)

g_

With no regard for structure, 37edo is an extremely special set of notes, which transcends structure. If you assume that approximating successive primes is good, this is a good tuning. If you think straddling 3 causes things to bend in a desirable way, this is the perfect tuning. After years of experience, I think 37edo is best described as 2.<<3.>3.5.7.11.13.17.19.

74 doesn't change much over 37, just adding a flat 3 and a reasonable 23, but the sharp 9 makes a big difference in available intervals, especially 9/7, which I think is the most obvious gap in 37edo. I think that Diaschismic Porcupine/Semioceanfront are by far the most important new structures, not just to 74 as a multiple of 37, but 74 in general.

A more granular erac subgroup in 74edo is 2.<3.>>>3.<<<5.>5.<<<7.>7.11.<<<13.>13.<<17.>>17.<19.>>>19.<<<23.>23 (this distribution is about optimal for overall accuracy of erac representations, but an interval directly between <<<<3 and >>3 as in 37edo would normally be <3).

I use 37edo in the piano roll with six channels to reach 222, the superset of 74 and 111, which are a weird combination. 74 is a similar extension in spirit of 37, filling in missing intervals and continuing to straddle 3. On the other hand, 111 is a microtempered Buzzard edo (53+58) where the 17-limit JI lattice is more useful than straddling in most cases. 222 just adds extra intervals including a better flat 5 and 7 which I don't expect to use often.