Bird's eye view of temperaments by accuracy: Difference between revisions
m →Buzzard: correction & add more room for harmony |
change bound to 4c because the intent is max of around 3.5c error, not of strictly less than 3.5c |
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'''Importantly:''' the "accuracy" classification of a temperament is the ''maximum error allowed'' on ''all'' intervals. If there is some intervals in the corresponding (thr)odd-limit that violate this bound slightly even in an optimized tuning, they should be noted as "'''Bound-violating intervals:'''" under the header for easy comprehension and consideration, but this should not be abused to attempt to reclassify temperaments, as there should only be one or two such intervals at most, and ideally zero. The main exception that justifies more than two bound-violating interval pairs is when a single odd is responsible for all of them (EG odd 7 in [[#Buzzard]]). Therefore, most temperaments in a category are more accurate than the bound suggests. | '''Importantly:''' the "accuracy" classification of a temperament is the ''maximum error allowed'' on ''all'' intervals. If there is some intervals in the corresponding (thr)odd-limit that violate this bound slightly even in an optimized tuning, they should be noted as "'''Bound-violating intervals:'''" under the header for easy comprehension and consideration, but this should not be abused to attempt to reclassify temperaments, as there should only be one or two such intervals at most, and ideally zero. The main exception that justifies more than two bound-violating interval pairs is when a single odd is responsible for all of them (EG odd 7 in [[#Buzzard]]). Therefore, most temperaments in a category are more accurate than the bound suggests. | ||
The bounds are: '''exotemperament''' (>~18c), '''low or very low''' (<~18c), '''medium or low''' (<12c), '''high or medium''' (<7c), '''very high or high''' (< | The bounds are: '''exotemperament''' (>~18c), '''low or very low''' (<~18c), '''medium or low''' (<12c), '''high or medium''' (<7c), '''very high or high''' (<4c), '''microtemperament''' (<1c). | ||
The two name system is for two reasons: to account for people's varying preferences and terminology for accuracies and to make the system of categories symmetrical, with [[exotemperament]]s and [[microtemperament]]s as the extrema. | The two name system is for two reasons: to account for people's varying preferences and terminology for accuracies and to make the system of categories symmetrical, with [[exotemperament]]s and [[microtemperament]]s as the extrema. | ||
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* 12 for {3, 7, 9, 21} ([[5L 8s]]) | * 12 for {3, 7, 9, 21} ([[5L 8s]]) | ||
The 2.3.7 part of [[#Buzzard]] is not as accurate as everything else in the 13-limit; specifically, its interval of 7 barely violates the | The 2.3.7 part of [[#Buzzard]] is not as accurate as everything else in the 13-limit; specifically, its interval of 7 barely violates the 4{{cent}} bound, however it makes up for it by being much simpler (mapping-wise) so that it is interesting as a 2.3.7-subgroup temperament that splits [[3/1]] into four equal parts, each representing a sharp [[~]][[21/16]], which defines it in the 2.3.7 subgroup. [[53edo]], [[58edo]] and [[111edo]] are good tunings. | ||
= '''Very high or high accuracy (< | = '''Very high or high accuracy (<4c)''' = | ||
The bound is the approximate [[JND|melodic JND (Just-Noticeable-Difference)]], though note that this doesn't mean that damage/mistuning is ''imperceptible'' in these temperaments as the harmonic JND can often be significantly smaller, depending largely on context, timbre and who is listening/who you ask. | The bound is the approximate [[JND|melodic JND (Just-Noticeable-Difference)]], though note that this doesn't mean that damage/mistuning is ''imperceptible'' in these temperaments as the harmonic JND can often be significantly smaller, depending largely on context, timbre and who is listening/who you ask. | ||
== 5-limit focus == | == 5-limit focus == | ||