Talk:Chromatic pairs

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Move temperament data?

It's strange that this page has a lot of temperaments that are not catalogued elsewhere. I think they should be moved to their respective (comma clan/family) page and then linked here so that this stops being a garbage can of 'temperaments not defined elsewhere'.

I'm also not really sure what the purpose of this page is because I think you can define chromatic pairs for *any* temperament.

- Sintel (talk) 01:30, 28 February 2022 (UTC)

Most of these would simply go to the subgroup temperaments page, which doesn't help much. Otherwise some of them have been logged already, and we've been adding more. I agree with your second argument tho. FloraC (talk) 12:26, 28 February 2022 (UTC)

I think temperament data should only be found on appropriate pages to avoid duplicates (especially non-identical ones), even if that means moving large parts to subgroup temperaments. If that page becomes overloaded, additional pages for popular subgroups may be created. Also, I think this page is useful as a list of scales, and the lead section could be improved to emphasize the properties of MOS scales which are implied by the definition of chromatic pairs. --Fredg999 (talk) 03:45, 12 March 2023 (UTC)

I agree that we should be consistent about the types of pages where this type of temperament data is found. And I agree that a good temporary solution would be to move the temperament data (most of this material) from here to the subgroup temperaments page. I also agree that reworking this page to focus on scales rather than temperaments would be better.

By the way, Fredg999: in your previous reply, [[Category:Lists_of_scales|list of scales]] did not render as "list of scales" linking to that Category page; it rendered as nothing. I don't have any idea why. --Cmloegcmluin (talk) 16:27, 13 March 2023 (UTC)

I forgot the additional : before Category, it should be fixed now. --Fredg999 (talk) 23:24, 13 March 2023 (UTC)

Chromatic pairs or chromatic triples?

If there are three types of scale (haplotonic, albitonic and chromatic scale), should not this page be called "chromatic triples"? Also, in some temperaments, there are fourth and even fifth types of scales like "mega-albitonic" and "mega-chromatic". CompactStar (talk) 03:25, 21 May 2023 (UTC)

I once came up with a stricter definition that I'll propose: a pair of mosses xL ys and (x+y)L xs, such that x < y, where the two mosses combined form a child mos of (x+y)L xs, or the albitonic mos's chromatic scale: usually (x+y)L (2x+y)s or (2x+y)L (x+y)s. Basically two scales that combine to form a chromatic scale. In the case of intermediate mosses, it's usually (if xL ys is instead the albitonic mos) xL ys, xL (x+y)s (mega-albitonic), and (2x+y)L xs (chromatic). Such sequences would not fall under the strict definition described, but would fall under a less strict definition that allows for intermediate scales. Worse still, some scale sequences have more than one haplotonic scale. Ganaram inukshuk (talk) 04:42, 21 May 2023 (UTC)

When Gene Ward Smith started this page in 2011, he only included albitonic and chromatic scales, hence chromatic pairs. Other categories were added over the years (I haven't checked if that was Smith's idea or someone else's), and I agree that it doesn't make much sense to continue talking about "pairs" now.

However, I think the concept as a whole is debatable, because it tries to generalize the traditional pentatonic, diatonic and chromatic scales in multiples ways at once and fails at doing so, or at least fails at doing so systematically, because MOS scales are diverse by nature, such that every difference from the traditional scales leads to breaking some assumptions about those. Ganaram inukshuk's proposition highlights some of these issues, but I'll go into more detail.

Before proceeding, it is good to keep in mind that each temperament is associated with an infinite sequence of MOS scales (assuming the generator is not a rational multiple of the period) obtained by stacking the generator repeatedly and checking which sizes lead to MOS patterns. Every generated 2-tone and 3-tone scale is a MOS scale, so every sequence starts with {2, 3} and then proceeds with either 4 (2+2) or 5 (2+3), and after that it starts getting quite diverse.

So, here are the main assumptions that inevitably fail at some point.

First, the number of notes is kept as close as possible to 5, 7 and 12 respectively. If there were specific boundaries for each category, in the same way that comma size categories are defined on the wiki, it would be more coherent already, but you wouldn't always have exactly one scale in each category for a given temperament/MOS family tree (sometimes none, sometimes two or three). This has lead to a patchwork of extra terms such as "mega albitonic" and "mini haplotonic" in order to fill the holes whenever the number of notes of intermediate scales didn't feel right for a given category. Even with these terms, the result remains arbitrary; for example, I would have expected Shoe[5] to be haplotonic and Shoe[6] to be mega haplotonic rather than mini haplotonic and haplotonic respectively.

Second, the pentatonic, diatonic and chromatic scales are three consecutive MOS sizes for a ~3/2-sized generator, i.e. 2L 3s is the parent MOS of 5L 2s, which is the parent MOS of 5L 7s/12edo/7L 5s. Keeping this structure while also keeping the number of notes similar sometimes becomes an impossible task, especially for temperaments with generators associated with more numerous MOS sizes. Barton is a good example, with 5 different MOS sizes between 5 and 13. This is probably what lead to the "mega" and "mini" scales in the first place.

Third, the sizes of the traditional scales follow the equation 5+7=12. This is a frequent occurrence in MOS family trees, as evoked above with 2+2=4 and 2+3=5, but it doesn't always work like this obviously. To be more precise, you can always deduce the next MOS size by adding two previously seen MOS sizes, but not necessarily using the last two specifically. For example, the next size after 5, 7 and 12 and be either 5+12=17 or 7+12=19, depending on the size of the generator. The first case illustrates that new MOS scales are not always found by adding the last two sizes. For reference, not every addition leads to a new MOS; for instance, 3+12=15 doesn't work unless you change the period to 1\3, which brings the equation down to 1+4=5, and obviously that equation is only valid for a certain generator range. I'll bring back the Barton example; you can make a valid 11+13=24 equation with its MOS sequence, but that's how the scales are currently labeled, most likely because an 11-tone scale doesn't match the "haplotonic" label in terms of number of notes.

Finally, while pentatonic, diatonic and chromatic seem well defined, there is the issue that building MOS scales from regular temperaments does not lead to a unique sequence of abstract MOS patterns, because there are multiple ways to tune the generator and that will affect the MOS patterns you get down the line, whenever the generator gets too close to the boundaries of the relevant MOS pattern's generator range. An example of this is the Dominant[12], a chromatic scale, whose generator can potentially fall on either side of 700¢, leading to either 7L 5s (<700 ¢) or 5L 7s (>700 ¢). That would imply two different chromatic scales, which contradicts the assumption that the chromatic scale is unique, or at least its step pattern is unique. This also means that even if you tried very hard to make the first three assumptions work, this one would cause exceptions for some temperaments, which makes it harder to think of these systematically.

In Ganaram inukshuk's proposition, the elements of the pair are the haplotonic and the albitonic scale (which, should I remind, is not Smith's original definition, should it matter), and they are related to a chromatic scale which contains at least one copy of each, and possibly multiple copies of one of them (the idea of "containing copies" comes from the recursive structure of MOS scales). That definition bakes in assumption 2, but does nothing about assumption 1. Assumption 3 is treated in the difference between the "strict" and the "weak" variants. Assumption 4 isn't treated either, but since it's only used to observe irregularities with edge cases, it's not as fundamental as the previous three. So by this proposition, I could call meantone[2] haplotonic, meantone[3] albitonic and meantone[5] chromatic. Maybe it would be wise to systematically skip 2 and 3, which are always mosses (and are rather trivial too) and skip right ahead to whatever size comes next. That would make it retro-compatible with common temperaments such as meantone, and it would sort of solve the issue with assumption 1.

To sum up, I think it's fundamentally flawed to try to apply all 4 assumptions baked into the "traditional mosses" to all other mosses, but should someone try, I would go with Ganaram inukshuk's proposition and add the starting point rule I stated above (always start with the first size after 3). This will inevitably lead to 6-tone chromatic scales in extreme cases and to a lot of weak chromatic pairs despite the existence of "strong chromatic pairs" at higher sizes (see Barton example above), but that's the kind of information loss to be expected when taking too many variables at once. It's the problems of temperament all over again! --Fredg999 (talk) 06:01, 21 May 2023 (UTC)

Since we can rarely apply all of Ganaram inukshuk's assumptions at once to non-diatonic MOSses, I suggest we should give priority to the first assumption, because the scales closest to 5, 7, and 12 notes are the ones most melodically similar to the pentatonic, diatonic, and chromatic scales. For example, for porcupine, (very improper) 1L 4s is the haplotonic scale, 1L 6s is the albitonic scale, and 7L 8s is the chromatic scale.

However, this approach would still require the intermediate terms–using the porcupine example again, the 1L 5s and 7L 1s scales would need to use terms like "mega-haplotonic" and "mega-albitonic". I don't think any naming system will ever be able to get rid of these types of terms, because, even if we forced the haplotonic, albitonic, and chromatic scales to be next to each other in the chain (e.g. 7, 8, and 15 notes for porcupine), we would still need to worry about "mini-haplotonic" and "mega-chromatic" scales. If these terms are inevitably required, we should come up with some standard definition of them, instead of using them in weird ways, e.g. how Shoe[5] is "mini-haplotonic" instead of "haplotonic", and Slendric has two haplotonic scales with 5 and 6 notes instead of the 6 note one being "mega-haplotonic". CompactStar (talk) 01:06, 25 May 2023 (UTC)

Sorry, I might not have been clear, I didn't mean to imply that these 4 assumptions were Ganaram inukshuk's; rather, they are most likely Gene Ward Smith's, assuming he's the one to have come up with the terms albitonic an such (although I think haplotonic came later), and I'm stating them as general properties one is likely to generalize out of the diatonic scale in general.

Anyway, I believe it's important to keep in mind that another way to think of "albitonic" is "what scale should go on the white keys of a piano-like keyboard", and similarly "haplotonic" describes the scale that goes on the black keys, such that the combination of all keys is the corresponding chromatic scale. In the porcupine example, you would use 1L 6s for haplotonic (7 notes), 7L 1s for albitonic (8 notes) and 7L 8s for chromatic (15 notes). This corresponds to the usual porcupine keyboard layout. I think the structure of decomposing a chromatic scale in two subscales is more important, especially since it is actually possible to preserve that property integrally, while the number of notes is fated to fall outside of the usual 5/7/12-note forms, so I don't think we should try to enforce it artificially. In fact, the 3rd assumption, which ensures that the chromatic scale's size is equal to the sum of the other two scales' sizes, could be used to solve otherwise weird cases, such as Barton, which would be decomposed as 11+13=24 instead of 5/7/11, even though it's very tempting to treat 5 and 7 as haplotonic and albitonic respectively; it wouldn't make sense to me to try building a piano-like layout with scales of size 5/7/11, but 11+13 would be an almost trivial generalization of the diatonic layout. --Fredg999 (talk) 02:58, 25 May 2023 (UTC)