Subgroup monzos and vals: Difference between revisions
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A '''subgroup monzo''' is a monzo whose elements refer to powers of arbitrary "basis elements" in JI (specified rationals such as 2, 3, 35, and 13/11 which are combined to form the set of intervals in a [[subgroup]]), rather than strictly to ordered primes. For cases where the basis elements are all prime numbers, a subgroup monzo can be seen as "abbreviating" entries from a standard monzo: for example, the monzo [0 0 -1 0 0 1⟩ (13/5) may be abbreviated to the subgroup monzo 5.13 [-1 1⟩. However, subgroup monzos can refer to intervals in subgroups whose basis elements are not primes, for example 2.3.13/5 [1 -1 1⟩ for 26/15. In that case, conversion to and from a subgroup monzo is a little more complicated, and is covered in the next section. | |||
A '''subgroup val''' is like a standard val, but the entries are the mappings of subgroup basis elements rather than strictly of primes (and can be derived from a standard val by applying the val to the basis intervals). Since converting to a subgroup val loses information, there is no clear way to convert a subgroup val back to a standard val. | |||
For short, a subgroup monzo may be referred to as an '''smonzo''', and a subgroup val may be referred to as an '''sval'''. | |||
== Conversion == | == Conversion == | ||
The simplest way to convert between standard monzos and subgroup monzos is to evaluate the monzo as a rational number, then start factoring out either primes or subgroup basis elements. To convert 2.3.13/5 [1 -1 1⟩ to a standard monzo, we first look at the interval itself, which is 26/15. You can see that there's 2*13 on the top, and then 3*5 on the bottom, and so the correct monzo is [1 -1 -1 0 0 1⟩. | |||
To convert back, knowing the subgroup 2.3.13/5, we evaluate this monzo as a fraction (26/15). Recognizing the factor of 13/5 is a little tricky, but you can rearrange the expression as (26/5)/3 to make it more visible. Once you factor it out you have 13/5 * 2 on top, and 3 on the bottom, which can be used to write the subgroup monzo [1 -1 1⟩. | |||
=== Without rationals === | |||
The following section goes over ways to convert between the two types of monzos without looking at the monzo's value as a rational. | |||
In order to convert a subgroup monzo to a standard monzo (without converting it to a rational number), look at the monzo for each basis element of the subgroup monzo. For example, 2.3.13/5 [1 -1 1⟩: | In order to convert a subgroup monzo to a standard monzo (without converting it to a rational number), look at the monzo for each basis element of the subgroup monzo. For example, 2.3.13/5 [1 -1 1⟩: | ||
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26/15 = [1 -1 -1 0 0 1⟩. | 26/15 = [1 -1 -1 0 0 1⟩. | ||
To go back from [1 -1 -1 0 0 1⟩ to our subgroup monzo, the most reasonable option is to repeatedly subtract and add the monzos for our subgroup basis elements until we reach the unison [0 0 0 0 0 0⟩. | |||
[1 -1 -1 0 0 1⟩ - [0 0 -1 0 0 1⟩ + [0 1 0 0 0 0⟩ - [1 0 0 0 0 0⟩ = [0 0 0 0 0 0⟩ | |||
If we keep track of how many times we subtract each basis monzo (negative for adding it), it's 1 for 2, -1 for 3, and 1 for 13/5. Thus, we re-derive our subgroup monzo [1 -1 1⟩. | |||
=== Linear algebra === | |||
Mathematically speaking, if '''m'''<sub>''G''</sub> is an smonzo of the subgroup ''G'', and if ''S'' is a [[subgroup basis matrix]] whose columns form a basis for the subgroup ''G'', then the corresponding monzo '''m''' is given by | Mathematically speaking, if '''m'''<sub>''G''</sub> is an smonzo of the subgroup ''G'', and if ''S'' is a [[subgroup basis matrix]] whose columns form a basis for the subgroup ''G'', then the corresponding monzo '''m''' is given by | ||