Ringer scale

A Ringer n scale is a minimal complexity* constant structure neji periodic scale with a period of an octave which has n notes (AKA scale degrees) per octave. (*What "minimal complexity" means is discussed later in this article.) The fact that it has a constant structure implies there is at least one val - corresponding to n edo - that 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.

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 scale degree.

Minimal complexity

The most striking feature of a Ringer scale is that it is "minimal complexity" in the sense that the maximum full odd-limit must be achieved, meaning as many odd harmonics (up to octave equivalence) must be present in the scale as possible without missing any, which 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. As this maximum is not always easy to find, often informally we call something that 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 as possible.

Perfect Ringer scale

A perfect Ringer n scale is one that by some val can map the first n odd harmonics (up to octave equivalence) to distinct number of steps. It is likely that only a small finite number of perfect Ringer s. Here are some known ones (to be expanded as more are found):

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.

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]:15:16:17:[35]:18:19:20:21:22:23:24:25:26

Where the [n] notation means that we are adding an odd harmonic that is imbetween those two harmonics in some higher harmonic mode (mode of the harmonic series).

Another Ringer 15 scale, if one prefers to get a 33rd harmonic instead of a 35th, is:

13:14:[29]:15:16:[33]: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.

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.

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 41 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 41 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. (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. The latter can be pasted directly into scale workshop using the enumerate chord feature or into other programs like scala.

List of Ringer scales

Ringer 15p: 13:14:[29]:15:16:17:[35]:18:19:20:21:22:23:24:25:26

18:19:20:21:22:23:24:25:26:28:29:30:32:34:35:36

Ringer 15g: 13:14:[29]:15:16:[33]:17:18:19:20:21:22:23:24:25:26

17:18:19:20:21:22:23:24:25:26:28:29:30:32:33:34