Talk:Chromatic pairs: Difference between revisions

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[[User:CompactStar|CompactStar]] ([[User talk: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. [[User:Ganaram inukshuk|Ganaram inukshuk]] ([[User talk:Ganaram inukshuk|talk]]) 04:42, 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. [[User:Ganaram inukshuk|Ganaram inukshuk]] ([[User talk: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! --[[User:Fredg999|Fredg999]] ([[User talk:Fredg999|talk]]) 06:01, 21 May 2023 (UTC)

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 [[User:CompactStar|CompactStar]] ([[User talk: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. [[User:Ganaram inukshuk|Ganaram inukshuk]] ([[User talk:Ganaram inukshuk|talk]]) 04:42, 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. [[User:Ganaram inukshuk|Ganaram inukshuk]] ([[User talk: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! --[[User:Fredg999|Fredg999]] ([[User talk:Fredg999|talk]]) 06:01, 21 May 2023 (UTC)