Stacking: Difference between revisions
No edit summary |
No edit summary |
||
| Line 6: | Line 6: | ||
== Mathematical definition == | == Mathematical definition == | ||
{{Wikipedia|Free abelian group}} | |||
Stacking is the group operation on a free abelian group of musical intervals. | Stacking is the group operation on a free abelian group of musical intervals. | ||
In the context of group theory, any space of [[Interval|intervals]] created by multiplicatively stacking arbitrarily many (or negatively many) of a given set of [[Generator|generators]] is considered a '''free abelian group''' under stacking. | In the context of group theory, any space of [[Interval|intervals]] created by multiplicatively stacking arbitrarily many (or negatively many) of a given set of [[Generator|generators]] is considered a '''free abelian group''' under stacking. Where the set of generators is finite, it is called a '''finitely generated free abelian group'''. | ||
The more explicit definition for this follows. | The more explicit definition for this follows. | ||
| Line 30: | Line 31: | ||
* In the case of tunings such as [[Equal-step tuning|equal temperaments]], there is only one generator. | * In the case of tunings such as [[Equal-step tuning|equal temperaments]], there is only one generator. | ||
* The number of generators is the [[rank]] of the temperament, so that equal temperaments are rank-1, temperaments with a generator and period are rank-2, and so on. | * The number of generators is the [[rank]] of the temperament, so that equal temperaments are rank-1, temperaments with a generator and period are rank-2, and so on. | ||
[[Category:Method]] | [[Category:Method]] | ||