Tempered monzos and vals: Difference between revisions

(One intermediate revision by one other user not shown)

Line 1:

~~{{Breadcrumb|Monzo|}}{{Breadcrumb|Val|}}~~'''Tempered monzos''' and '''tempered vals''' are like regular [[monzo]]s and [[val]]s, except they work in a space of tempered intervals (for example, the intervals found in meantone) rather than in [[just intonation]]. A tval specifies a tuning or further temperament of the [[generator]]s of a temperament. For example, the 31edo tval for meantone is {{val| 31 49 }}, assume the generators are ~2 and ~3. A tempered monzo (for short, "'''tmonzo'''") specifies a particular tempered interval in terms of stacking the temperament's generators. For example, the tmonzo form of the major third in meantone is {{monzo| -6 4 }}. Taking the {{w|dot product}} of these (multiplying corresponding elements and adding up the results) yields the tuning of the major third in 31edo, 10\31.

'''Tempered monzos''' and '''tempered vals''' are like regular [[monzo]]s and [[val]]s, except they work in a space of tempered intervals (for example, the intervals found in meantone) rather than in [[just intonation]]. A tempered val (for short, "'''tval'''") specifies a tuning or further temperament of the [[generator]]s of a temperament. For example, the 31edo tval for meantone is {{val| 31 49 }}, assume the generators are ~2 and ~3. A tempered monzo (for short, "'''tmonzo'''") specifies a particular tempered interval in terms of stacking the temperament's generators. For example, the tmonzo form of the major third in meantone is {{monzo| -6 4 }}. Taking the {{w|dot product}} of these (multiplying corresponding elements and adding up the results) yields the tuning of the major third in 31edo, 10\31.

Tmonzos are rather straightforward, and ~~tempered vals (for short, "'''~~tvals~~'''")~~ act on tmonzos in the same way that vals act on monzos, in that they map from tmonzos to a scalar representing a certain number of steps. A similar concept is a generator [[tuning map]], which maps from tmonzos to tunings in cents (or another logarithmic measure).

Tmonzos are rather straightforward, and tvals act on tmonzos in the same way that vals act on monzos, in that they map from tmonzos to a scalar representing a certain number of steps. A similar concept is a generator [[tuning map]], which maps from tmonzos to tunings in cents (or another logarithmic measure).

Note that there is no restriction on which bases tmonzos can be written in (i.e. what intervals should be considered the generators of the temperament), but one option is to use the basis corresponding to the [[temperament mapping matrix|mapping matrix]] for the temperament which is in [[normal lists #Normal val list|normal val list]] form.

@@ Line 1: / Line 1: @@
-{{Breadcrumb|Monzo|}}{{Breadcrumb|Val|}}'''Tempered monzos''' and '''tempered vals''' are like regular [[monzo]]s and [[val]]s, except they work in a space of tempered intervals (for example, the intervals found in meantone) rather than in [[just intonation]]. A tval specifies a tuning or further temperament of the [[generator]]s of a temperament. For example, the 31edo tval for meantone is {{val| 31 49 }}, assume the generators are ~2 and ~3. A tempered monzo (for short, "'''tmonzo'''") specifies a particular tempered interval in terms of stacking the temperament's generators. For example, the tmonzo form of the major third in meantone is {{monzo| -6 4 }}. Taking the {{w|dot product}} of these (multiplying corresponding elements and adding up the results) yields the tuning of the major third in 31edo, 10\31.
+'''Tempered monzos''' and '''tempered vals''' are like regular [[monzo]]s and [[val]]s, except they work in a space of tempered intervals (for example, the intervals found in meantone) rather than in [[just intonation]]. A tempered val (for short, "'''tval'''") specifies a tuning or further temperament of the [[generator]]s of a temperament. For example, the 31edo tval for meantone is {{val| 31 49 }}, assume the generators are ~2 and ~3. A tempered monzo (for short, "'''tmonzo'''") specifies a particular tempered interval in terms of stacking the temperament's generators. For example, the tmonzo form of the major third in meantone is {{monzo| -6 4 }}. Taking the {{w|dot product}} of these (multiplying corresponding elements and adding up the results) yields the tuning of the major third in 31edo, 10\31.
-Tmonzos are rather straightforward, and tempered vals (for short, "'''tvals'''") act on tmonzos in the same way that vals act on monzos, in that they map from tmonzos to a scalar representing a certain number of steps. A similar concept is a generator [[tuning map]], which maps from tmonzos to tunings in cents (or another logarithmic measure).
+Tmonzos are rather straightforward, and tvals act on tmonzos in the same way that vals act on monzos, in that they map from tmonzos to a scalar representing a certain number of steps. A similar concept is a generator [[tuning map]], which maps from tmonzos to tunings in cents (or another logarithmic measure).
 Note that there is no restriction on which bases tmonzos can be written in (i.e. what intervals should be considered the generators of the temperament), but one option is to use the basis corresponding to the [[temperament mapping matrix|mapping matrix]] for the temperament which is in [[normal lists #Normal val list|normal val list]] form.