Temperament addition: Difference between revisions

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'''Temperament arithmetic''' is the general name for either the '''temperament sum''' or the '''temperament difference''', which are two closely related operations on [[regular temperaments]]. Basically, to do temperament arithmetic means to match up the entries of temperament vectors and then add or subtract them individually. The result is a new temperament that has similar properties to the original temperaments.

For example, the sum of [[12-ET]] and [[7-ET]] is [[19-ET]] because {{map|12 19 28}} + {{map|7 11 16}} = {{map|(12+7) (19+11) (28+16)}} = {{map|19 30 44}}, and the difference of 12-ET and 7-ET is 5-ET because {{map|12 19 28}} - {{map|7 11 16}} = {{map|(12-7) (8-11) (12-16)}} = {{map|5 8 12}}. We can write these using [[wart notation]] as 12p + 7p = 19p and 12p - 7p = 5p, respectively. The similarity in these temperaments can be seen in how, like both 12-ET and 7-ET, 19-ET also ~~supports~~ [[meantone temperament]].

For example, the sum of [[12-ET]] and [[7-ET]] is [[19-ET]] because {{map|12 19 28}} + {{map|7 11 16}} = {{map|(12+7) (19+11) (28+16)}} = {{map|19 30 44}}, and the difference of 12-ET and 7-ET is 5-ET because {{map|12 19 28}} - {{map|7 11 16}} = {{map|(12-7) (8-11) (12-16)}} = {{map|5 8 12}}. We can write these using [[wart notation]] as 12p + 7p = 19p and 12p - 7p = 5p, respectively. The similarity in these temperaments can be seen in how, like both 12-ET and 7-ET, 19-ET (their sum) and 5-ET (their difference) both also support [[meantone temperament]].

Temperament sums and differences can also be found using commas; for example meantone + porcupine = tetracot because {{vector|4 -4 1}} + {{vector|1 -5 3}} = {{vector|(4+1) (-4+-5) (1+3)}} = {{vector|5 -9 4}} and meantone - porcupine = dicot because {{vector|4 -4 1}} - {{vector|1 -5 3}} = {{vector|(4-1) (-4--5) (1-3)}} = {{vector|3 1 -2}}. We could write this in ratio form — replacing addition with multiplication and subtraction with division — as 80/81 × 250/243 = 20000/19683 and 80/81 ÷ 250/243 = 25/24, respectively. The similarity in these temperaments can be seen in how all of them are supported by 7-ET.

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 '''Temperament arithmetic''' is the general name for either the '''temperament sum''' or the '''temperament difference''', which are two closely related operations on [[regular temperaments]]. Basically, to do temperament arithmetic means to match up the entries of temperament vectors and then add or subtract them individually. The result is a new temperament that has similar properties to the original temperaments.
-For example, the sum of [[12-ET]] and [[7-ET]] is [[19-ET]] because {{map|12 19 28}} + {{map|7 11 16}} = {{map|(12+7) (19+11) (28+16)}} = {{map|19 30 44}}, and the difference of 12-ET and 7-ET is 5-ET because {{map|12 19 28}} - {{map|7 11 16}} = {{map|(12-7) (8-11) (12-16)}} = {{map|5 8 12}}. We can write these using [[wart notation]] as 12p + 7p = 19p and 12p - 7p = 5p, respectively. The similarity in these temperaments can be seen in how, like both 12-ET and 7-ET, 19-ET also supports [[meantone temperament]].
+For example, the sum of [[12-ET]] and [[7-ET]] is [[19-ET]] because {{map|12 19 28}} + {{map|7 11 16}} = {{map|(12+7) (19+11) (28+16)}} = {{map|19 30 44}}, and the difference of 12-ET and 7-ET is 5-ET because {{map|12 19 28}} - {{map|7 11 16}} = {{map|(12-7) (8-11) (12-16)}} = {{map|5 8 12}}. We can write these using [[wart notation]] as 12p + 7p = 19p and 12p - 7p = 5p, respectively. The similarity in these temperaments can be seen in how, like both 12-ET and 7-ET, 19-ET (their sum) and 5-ET (their difference) both also support [[meantone temperament]].
 Temperament sums and differences can also be found using commas; for example meantone + porcupine = tetracot because {{vector|4 -4 1}} + {{vector|1 -5 3}} = {{vector|(4+1) (-4+-5) (1+3)}} = {{vector|5 -9 4}} and meantone - porcupine = dicot because {{vector|4 -4 1}} - {{vector|1 -5 3}} = {{vector|(4-1) (-4--5) (1-3)}} = {{vector|3 1 -2}}. We could write this in ratio form — replacing addition with multiplication and subtraction with division — as 80/81 × 250/243 = 20000/19683 and 80/81 ÷ 250/243 = 25/24, respectively. The similarity in these temperaments can be seen in how all of them are supported by 7-ET.