Ringer scale
A ringer n scale is a detempering of an edo to a minimal complexity harmonic series scale with the goals of having the constant structure (CS) property while having as many consecutive harmonics (starting from 1) as possible, meaning that the set of all intervals present in the scale should have the maximal odd limit possible under that restriction, with remaining notes being given "filler harmonics" chosen subjectively based on taste/preference. Ringer scales are constructed using a specific val corresponding to n-edo – which will map every interval present to the same number of abstract "scale steps". This means 2/1 must be mapped to n scale steps. Note that the val is not required to be patent and that the most consistent val is not always the patent val and usually depends on the tendency towards sharpness or flatness of the corresponding edo. The name ringer comes from the tendency of these JI scales to "ring" a lot because of them generally being as low complexity as possible in the sense of odd-limit.
A ringer scale can be thought of as testing the very limits of what the constant structure property (and the corresponding val by proxy) is capable of for the harmonic series. Note that as the maximum number of consecutive harmonics that are possible to fit for a given edo is not always clear, we informally often call something we think is likely to be the maximum a ringer scale. If we suspect it might not be maximal we can say it might not be a proper ringer scale. If we know it is not maximal we can say it is an improper ringer scale. Improper ringer scales are often desirable as a result of user preference/customisation, but are not ringer scales because they do not achieve the goal of approximating as much of the low end of the harmonic series (without exclusion) as mathematically possible while preserving CS. These can be called "pseudoringer" scales if they still very much go for the aesthetic and complexity of a ringer scale while deviating from the corresponding ringer scale in a small number of ways.
An important consideration when building a ringer n scale is what odd harmonics to add once you have reached the maximum odd-limit. To figure out where to place odd harmonics imbetween simpler odd harmonics already present, you need to use a choice of val to see what adjacent harmonics in the scale are mapped to more than 1 abstract scale step. The goal then is to make it so that every adjacent pair of harmonics in the ringer n scale is mapped by the val to 1 abstract scale step. Note that a ringer scale is completely described by the set of odd harmonics present because of octave equivalence because ringer scales are periodic scales with period equal to an octave.
Perfect ringer scales
A perfect ringer n scale is one that by some val can map the first n odd harmonics to distinct numbers of steps up to octave equivalence. It is conjectured that these are the only perfect ringer scales:
Ringer 1: 1:2
Ringer 2: 2:3:4
Ringer 3: 3:4:5:6
Ringer 4: 4:5:6:7:8
Ringer 5: 5:6:7:8:9:10
Ringer 7: 7:8:9:10:11:12:13:14
Notice how all of these do not skip any harmonics while representing the harmonic series completely up to some odd-limit.
Also note that there is a "Perfect Pseudoringer 9" if we allow a pair of harmonics to be swapped/out of order in order to preserve the constant structure property. It is not known how many "Perfect Pseudoringers" there are.
Origin of ringer scales
The name ringer was chosen by tuning theorist Scott Dakota (who discovered and raised awareness of the concept) to refer to the property of these scales to "ring" extremely and about as much as might be possible for a JI scale because the odd-limit complexity of the intervals in such scales is near-minimal meaning they consume as much of the early harmonic series as possible. It is worth noting however that the appearance of a virtual fundamental depends strongly on which notes of the scale you play - an observation important to primodality. The concept of ringer scales was additionally further developed by tuning theorists Praveen Venkataramana and later user:Godtone.
Example: Ringer 15
As definitions can be confusing, it can help to work through an example. For 15edo, we can look at how many harmonics it can map without tempering the intervals between any of them. In other words, what is the largest (meaning lowest odd-limit) superparticular interval that it tempers? This can be checked with code (an interesting exercise) or can be checked by hand. The answer is 28/27, meaning that the 27th harmonic cannot be included unless we choose a different mapping. If we change the mapping of prime 3 to second best (using the 15b val) then 18/17 is tempered instead. Changing the mapping of 17 to untemper 18/17 would not help as that would cause 17/16 to be tempered, meaning we have to keep the patent val mapping for 3 to maximize odd limit.
If we change the mapping of prime 7 to second best (using the 15d val), which is our only other option for trying to untemper 28/27, then we get 21/20 tempered instead. Note that of the primes present in the prime factorization of 21/20, 2 is fixed, 3 we deduced must be fixed to patent and 7 is what we are changing, leaving only changing the mapping of prime 5 as a way out, which if we then use the 15cd val we get 15/14 tempered with no options left unless we use try to use a third-, fourth-, etc. mapping for primes, which we will assume for simplicity does not lead anywhere. If we change the mapping of both (using the 15bd val) we also get 15/14 tempered. Therefore it seems like the patent val gives the best performance, so we will continue the construction of Ringer 15 assuming that the 25-odd-limit is the highest it is capable of. The corresponding mode of the harmonic series is mode 13, giving us:
13:14:15:16:17:18:19:20:21:22:23:24:25:26
Then we can notice we are missing two notes (as 26-13=13) to make it a 15-note scale and thus a Ringer 15 scale, so we need to add two odd harmonics above 27 to complete it. Here there are multiple choices based on the val used and one's preference. The patent val way to complete the scale, which seems to be the lowest complexity and thus arguably the canonical one, is:
13:14:29/2:15:16:17:35/2:18:19:20:21:22:23:24:25:26
Where the n/2 notation means that we are adding an odd harmonic that is inbetween those two harmonics in some higher harmonic mode (mode of the harmonic series). For example, mode 5 is 5:6:7:8:9:10 so because 6+7=13, we have the 13th harmonic appearing in mode 5*2=10 of the harmonic series between 6*2=12 and 7*2=14, so relative to mode 5 its as if the 13th harmonic is the 13/2 = 6.5th harmonic in the context of 5:6:6.5:7:8:9:10 = 5:6:13/2:7:8:9:10. (In other words the /2 serves to make the harmonic appear in the same octave as the rest.)
Another ringer 15 scale, if one prefers to get a 33rd harmonic instead of a 35th, is:
13:14:29/2:15:16:33/2:17:18:19:20:21:22:23:24:25:26
This uses the 15g val meaning prime 17 is mapped to the second-best mapping in 15edo. This scale can be called "Ringer 15g" in order to distinguish it from "Ringer 15", which refers to the ringer 15 scale derived from the patent val of 15edo.
Problem of warts
When trying to find a maximal odd-limit for a ringer scale, there is a problem of a combinatorial explosion if we insist on checking every possible val to try to increase the odd-limit. (Note that using a second-, third-, etc. -best mapping of a prime is called "warting" that prime.) This is a difficult problem to solve as it means it is unclear whether a scale is as high odd-limit as it could possibly be while maintaining the constant structure property. A potential solution to this problem is to insist that we do not use a val that uses more than one wart for a prime in order to try to keep the val as accurate and faithful to the structure of JI as possible. This makes checking all vals computationally possible. However, there are serious cases, for example 167edo, where the "tendency" towards sharpness or flatness of an edo is so strong that we need more than one wart for a prime in order to fit the pattern and therefore potentially achieve a higher odd-limit, so this is only really a serious solution for smaller edos, and is a partial solution for larger edos that prefers edos that do not have any "tendency". This solution works for edos as big as 80edo, resulting in scales like Ringer 80, which is an important example as 80edo has a strong sharp tendency for its size, to the extent that it does not map 21/16 or 27/16 consistently. It also tends to work well for edos that are relatively "well-tuned" in the traditional LCJI-focused RTT sense.
An example of this problem is that there is in some sense a "perfect Ringer 9" scale but that it is not quite monotonic in that in order for the CS property to apply, you need to consider the harmonics as being in a specific order that is different from being ordered simply by size. Consider:
Non-monotonic (otherwise-)perfect Ringer 9: 9:10:11:12:13:14:16:15:17:18
The 17-limit val that confirms this scale is CS is ⟨9 15 22 26 32 34 38], which written as warts is 9bccdefgg. (Note that in this case, where there is two warts this corresponds to the patent val mapping for the prime already being sharp and being warted to be a step sharper. If we assume that every wart means "sharpen by one step from patent val" this val can be written rather curiously as 9bcdefg, which shows that this val is the one sharpening every applicable prime by one step above the patent val mapping.) One can confirm that the above is CS because if one traverses it step by step, every one-step interval is mapped to one EDOstep which by linearity (more precisely, epimorphicity) implies CS. Note that it is important to preserve the order of these intervals. 14:16 = 16/14 = 8/7 is mapped to one positive step, as is 16:15 = 15/16, as is 15:17 = 17/15. Similarly (or thus/by linearity), 14:15 = 15/14 is mapped to 2 steps, as is 16:17 = 17/16, as is 15:18 = 18/15 = 6/5.
Proof of CS by linearity of the epimorphic val
Because the CS property means that every occurrence of an interval must occur with the same number of steps, it suffices to show that every one-step interval is mapped to one step by the val that the Ringer scale is constructed with. (This val shows that the Ringer scale is epimorphic.)
In other words, if the Ringer scale's val maps every 1-scalestep interval to 1\N (where N is the notes-per-period) then by linearity of the val the scale is CS and the corresponding rank-1 temperament resulting from tempering the differences between all the 1-scalestep intervals is the 'logic' that it obeys.
However, conversely, a scale being CS does not imply that such a val exists! In almost all observed practical cases if a scale is CS there is some val, but it is possible to construct scales where, for example, one 1-scalestep interval is equal to the product of more than one other 1-scalestep intervals; that is, if we have 1-scalestep intervals {a, b, c, ...} then we can choose ab as a 1-scalestep interval as long as ab doesn't occur as a 2-scalestep interval anywhere in the scale, which is why at least one extra 1-scalestep interval c is necessary to separate instances of a and b. You can even choose b = a but you need to be careful to avoid CS-violating contradictions. For a concrete example, you can use {5/4, 9/8, 45/32, ...} as 1-scalestep intervals to generate a nonlinear CS scale as long as 45/32 does not occur as a 2-scalestep interval anywhere in your scale.
Sketch of the proof
Consider an N-note periodic scale with period P as being defined by a function [math]f: \mathbb{Z} \to \mathbb{Q}_{\gt 0}[/math] with [math]f(Nk) = P^k.[/math]
By the construction of a ringer scale, we are given some val map [math]m : \mathbb{Q}_{\gt 0} \to \mathbb{Z}[/math] that satisfies [math]m(f(k+1)/f(k)) = 1[/math] for all k in Z. (This can be checked by hand or by computer as we only need to check one period P's worth of 1-scalestep intervals.)
By induction this implies [math]m(f(k+s)/f(k)) = s[/math] because the intervals from k to k+1, from k+1 to k+2, ..., from k+s-1 to k+s all multiply together. This also implies [math]m(f(k))=k,[/math] proving f to be epimorphic, therefore CS (see proof in the article epimorphic scale). [math]\square[/math]
Ringer scales
This section will detail known ringers for edos smaller than 100. Because warts are limited when it comes to large primes, any primes past 43 are explicitly listed in the form [p, q, r, ...] rather than abbreviated (rather cryptically) as letters. A quick summary of all the warts up to 43 is:
b means 3 gets a next-best mapping, c means 5 gets a next-best mapping, d means 7 gets a next-best mapping and so on: e means 11, f means 13, g means 17, h means 19, i means 23, j means 29, k means 31, l means 37, m means 41, n means 43. (2 = a is not used as it must always be patent.)
There should be at least two forms listed. One will be in the form used for the example of Ringer 15. One will be in the minimum mode of the harmonic series that contains all harmonics. Both can be pasted directly into scale workshop using the enumerate chord feature or into other programs, but the latter form is useful in case a program does not support the other notation.
List of ringer scales
W.I.P: This list has a specific format. The first format stated is in the mode of the corresponding complete odd-limit achieved by the ringer, with filler harmonics (odds beyond that odd-limit) bolded and alternatives disambiguated by specifying corresponding warts (if any), while the second format(s) stated are of the minimal mode(s) containing only integer harmonics for software that does not accept k/2, k/4 formats/harmonic series chord enumerations.
Ringer 6cc: 5:6:7:8:17/2:9:10
9:10:12:14:16:17:18
Ringer 8: 7:8:17/2:9:10:11:12:13:14
9:10:11:12:13:14:16:17:18
Ringer 9: 8:9:19/2:10:11:12:13:14:15:16
10:11:12:13:14:15:16:18:19:20
Ringer 10: 10:21/2:11:12:13:14:15:16:17:18:20
11:12:13:14:15:16:17:18:20:21:22
Ringer 11: 8:9:37/4:19/2:10:11:12:[49/4 OR 51/4]:13:14:15:16
26:28:30:32:36:37:38:40:44:48:[49 OR 51]:52
Ringer 11df: 8:9:37/4:19/2:10:11:12:13:14:29/2:15:16
19:20:22:24:26:28:29:30:32:36:37:38
Ringer 12: 11:12:[25/2:13(f) OR 13:27/2]:14:15:16:17:18:19:20:21:22
14:15:16:17:18:19:20:21:22:24:[25:26 OR 26:27]:28
Ringer 13g: 9:10:21/2:11:23/2:12:13:14:29/2:15:16:33/2:17:18
17:18:20:21:22:23:24:26:28:29:30:32:33:34:36
Ringer 14cf: 12:13:27/2:14:15:[31/2 OR 63/4]:16:17:18:19:20:21:22:23:24
16:17:18:19:20:21:22:23:24:26:27:28:30:31:32
32:34:36:38:40:42:44:46:48:52:54:56:60:63:64
Ringer 15: 13:14:29/2:15:16:[33/2:17(g) OR 17:35/2]:18:19:20:21:22:23:24:25:26
18:19:20:21:22:23:24:25:26:28:29:30:32:[33:34 OR 34:35]:36
Ringer 17cffg: 13:14:29/2:15:16:33/2:17:18:19:[77/4 OR 79/4]:20:21:22:23:24:49/2:25:26
40:42:44:46:48:49:50:52:56:58:60:64:66:68:72:76:[77 OR 79]:80
Ringer 19: 17:18:37/2:19:20:21:43/2:22:23:24:25:26:27:28:29:30:31:32:33:34
22:23:24:25:26:27:28:29:30:31:32:33:34:36:37:38:40:42:43:44
Ringer 22h: 19:20:41/2:21:22:[45/2:23 OR 23:47/2(i)]:24:25:[51/2:26(f) OR 26:53/2]:27:28:29:30:31:32:33:34:35:36:37:38
27:28:29:30:31:32:33:34:35:36:37:38:40:41:42:44:[45:46 OR 46:47(i)]:48:50:[51:52(f) OR 52:53]:54
Ringer 24: 20:21:43/2:22:[45/2:23 OR 23:47/2(i)]:24:[97/4(y=97) OR 99/4]:25:26:27:28:[57/2:29 OR 29:59/2(j)]:30:31:32:33:34:35:36:37:38:39:40
50:52:54:56:[57:58 OR 58:59(j)]:60:62:64:66:68:70:72:74:76:78:80:84:86:88:[90:92 OR 92:94(i)]:96:[97(y=97) OR 99]:100
Ringer 26i: 22:23:47/2:24:25:51/2:26:27:55/2:28:29:30:[61/2:31 OR 31:63/2(k)]:32:33:34:35:36:37:38:39:40:41:42:43:44
32:33:34:35:36:37:38:39:40:41:42:43:44:46:47:48:50:51:52:54:55:56:58:60:[61:62 OR 62:63(k)]:64
Ringer 27egi: 23:24:49/2:25:26:53/2:27:28:[57/2:29(j) OR 29:59/2]:30:31:32:65/2:33:34:35:36:37:38:39:40:41:42:43:44:45:46
33:34:35:36:37:38:39:40:41:42:43:44:45:46:48:49:50:52:53:54:56:[57:58(j) OR 58:59]:60:62:64:65:66
Ringer 29g: 24:25:51/2:26:53/2:27:28:[57/2:29 OR 29(j):59/2(q=59)]:30:[61/2:31 OR 31:63/2(k)]:32:33:34:35:71/2:36:37:38:39:40:41:42:43:44:45:46:47:48
36:37:38:39:40:41:42:43:44:45:46:47:48:50:51:52:53:54:56:[57:58 OR 58(j):59(q=59)]:60:[61:62 OR 62:63(k)]:64:66:68:70:71:72
Ringer 31: 26:27:55/2:28:57/2:29:30:[61/2:31 OR 31:63/2(k)]:32:33:67/2:34:35:36:[73/2:37(l) OR 37:75/2]:38:39:40:41:42:43:44:45:46:47:48:49:50:51:52
38:39:40:41:42:43:44:45:46:47:48:49:50:51:52:54:55:56:57:58:60:[61:62 OR 62:63(k)]:64:66:67:68:70:72:[73:74(l) OR 74:75]:76
Ringer 39dfgijkl [(7, 13, 17, 23, 29, 31, 37,) 53, 59, 61, 73, 97]: 31:32:129/4:33:67/2:34:69/2:35:36:73/2:37:38:77/2:39:40:41:83/2:42:43:44:89/2:45:46:47:48:97/2:49:50:51:52:53:54:55:56:57:58:59:60:61:62
66:67:68:69:70:72:73:74:76:77:78:80:82:83:84:86:88:89:90:92:94:96:97:98:100:102:104:106:108:110:112:114:116:118:120:122:124:128:129:132
Ringer 41i: 34:35:71/2:36:73/2:37:38:77/2:39:40:81/2:41:42:[85/2:43 OR 43:87/2(n)]:44:45:91/2:46:47:48:49:50:101/2:51:52:53:54:55:56:57:58:59:60:61:62:63:64:65:66:67:68
52:53:54:55:56:57:58:59:60:61:62:63:64:65:66:67:68:70:71:72:73:74:76:77:78:80:81:82:84:[85:86 OR 86:87(n)]:88:90:91:92:94:96:98:100:101:102:104
Ringer 43: 35:36:73/2:37:75/2:38:39:79/2:40:41:83/2:42:43:87/2:44:45:91/2:46:47:48:97/2:49:50:51:52:53:54:109/2:55:56:57:58:59:60:61:62:63:64:65:66:67:68:69:70
55:56:57:58:59:60:61:62:63:64:65:66:67:68:69:70:72:73:74:75:76:78:79:80:82:83:84:86:87:88:90:91:92:94:96:97:98:100:102:104:106:108:109:110
Ringer 46hjm: 38:39:79/2:40:[81/2:41 OR 41:83/2]:42:85/2:43:44:89/2:45:46:93/2:47:48:49:99/2:50:51:52:53:107/2:54:55:56:113/2:57:58:59:60:61:62:63:64:65:66:67:68:69:70:71:72:73:74:75:76
57:58:59:60:61:62:63:64:65:66:67:68:69:70:71:72:73:74:75:76:78:79:80:[81:82 OR 82:83]:84:85:86:88:89:90:92:93:94:96:98:99:100:102:104:106:107:108:110:112:113:114
Ringer 50km [(31, 41,) 47, 61, 83]: 40:41:83/2:42:85/2:43:87/2:44:45:91/2:46:47:95/2:48:49:99/2:50:51:103/2:52:53:54:109/2:55:56:57:115/2:58:59:60:61:62:63:127/2:64:65:66:67:68:69:70:71:72:73:74:75:76:77:78:79:80
64:65:66:67:68:69:70:71:72:73:74:75:76:77:78:79:80:82:83:84:85:86:87:88:90:91:92:94:95:96:98:99:100:102:103:104:106:108:109:110:112:114:115:116:118:120:122:124:126:127:128
Ringer 53: 43:44:89/2:45:91/2:46:47:95/2:48:97/2:49:50:101/2:51:52:105/2:53:54:55:111/2:56:57:58:117/2:59:60:61:123/2:62:63:64:65:66:133/2:67:68:69:70:71:72:73:74:75:76:77:78:79:80:81:82:83:84:85:86
67:68:69:70:71:72:73:74:75:76:77:78:79:80:81:82:83:84:85:86:88:89:90:91:92:94:95:96:97:98:100:101:102:104:105:106:108:110:111:112:114:116:117:118:120:122:123:124:126:128:130:132:133:134
Ringer 58hiko: 47:48:97/2:49:99/2:50:51:103/2:52:53:107/2:54:109/2:55:56:113/2:57:58:[117/2:59(q=59) OR 59:119/2]:60:[121/2:61(rr=61) OR 61:123/2]:62:63:64:129/2:65:66:67:68:137/2:69:70:71:72:73:74:149/2:75:76:77:78:79:80:81:82:83:84:85:86:87:88:89:90:91:92:93:94
75:76:77:78:79:80:81:82:83:84:85:86:87:88:89:90:91:92:93:94:96:97:98:99:100:102:103:104:106:107:108:109:110:112:113:114:116:[117:118(q=59) OR 118:119]:120:[121:122(rr=61) OR 122:123]:124:126:128:129:130:132:134:136:137:138:140:142:144:146:148:149:150