Fifth-chroma temperaments: Difference between revisions

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The base observation is that it is in some sense optimal to find exactly four intervals between 6/5 and 5/4, which can be called:

* "supraminor", a general bucket/category for aggregating ratios like 35/29, 64/53, 29/24, 52/43, 23/19, 40/33, 17/14

* "subneutral", which principally represents ~39/32~50/41~11/9

* "subneutral", which principally represents ~39/32~50/41*~11/9

* "superneutral", which principally represents ~27/22~16/13

* "submajor", a general bucket/category for aggregating ratios like 21/17, 26/21, 31/25, 36/29, 41/33, 46/37

The lists of ratios for supraminor and submajor are not complete and a given edo tuning will not necessarily find contexts where all of these ratios make sense as interpretations, but the principle is that between 6/5 and 11/9 are many ratios that are various mediants of 6/5 and 11/9, and between 16/13 and 5/4 are many ratios that are various mediants of 16/13 and 5/4, so that both general areas represent places where the source of concordance (if any) is not necessarily obvious, so that any mediant therein can potentially be suggested with sufficiently forcing amounts of harmonic context (notes in a chord with approximate frequency ratios suggesting a certain otonal/harmonic series interpretation). Therefore, it represents a flexible melodic category able to represent a wide variety of tempered harmonies contextually.

:: <nowiki>*</nowiki> Of the 5 edos discussed, only 94edo does not temper out (50/41)/(39/32) = [[1600/1599]] = S40, corresponding to its inconsistently-flat mapping of 25/16; this is noteworthy only because the subneutral and superneutral categories generally are supposed to be unambiguous in their interpretation; an asterisk with 84edo's mapping of prime 11 is discussed later

The structure therefore assumes tempering out [[352/351]] = (11/9)/(39/32) = (32/27)/(13/11).

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 The base observation is that it is in some sense optimal to find exactly four intervals between 6/5 and 5/4, which can be called:
 * "supraminor", a general bucket/category for aggregating ratios like 35/29, 64/53, 29/24, 52/43, 23/19, 40/33, 17/14
-* "subneutral", which principally represents ~39/32~50/41~11/9
+* "subneutral", which principally represents ~39/32~50/41*~11/9
 * "superneutral", which principally represents ~27/22~16/13
 * "submajor", a general bucket/category for aggregating ratios like 21/17, 26/21, 31/25, 36/29, 41/33, 46/37
 The lists of ratios for supraminor and submajor are not complete and a given edo tuning will not necessarily find contexts where all of these ratios make sense as interpretations, but the principle is that between 6/5 and 11/9 are many ratios that are various mediants of 6/5 and 11/9, and between 16/13 and 5/4 are many ratios that are various mediants of 16/13 and 5/4, so that both general areas represent places where the source of concordance (if any) is not necessarily obvious, so that any mediant therein can potentially be suggested with sufficiently forcing amounts of harmonic context (notes in a chord with approximate frequency ratios suggesting a certain otonal/harmonic series interpretation). Therefore, it represents a flexible melodic category able to represent a wide variety of tempered harmonies contextually.
+:: <nowiki>*</nowiki> Of the 5 edos discussed, only 94edo does not temper out (50/41)/(39/32) = [[1600/1599]] = S40, corresponding to its inconsistently-flat mapping of 25/16; this is noteworthy only because the subneutral and superneutral categories generally are supposed to be unambiguous in their interpretation; an asterisk with 84edo's mapping of prime 11 is discussed later
 The structure therefore assumes tempering out [[352/351]] = (11/9)/(39/32) = (32/27)/(13/11).