User:Mike Battaglia/Generated Tone Systems
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author mbattaglia1 and made on 2013-01-24 11:53:21 UTC.
- The original revision id was 401114056.
- The revision comment was:
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.
Original Wikitext content:
=<span style="line-height: 1.5;">NOTE: this is unfinished and lots of the terminology will change, I'm just putting this here for now so I don't lose my work. don't worry about it yet and I'll fix it later today</span>= =<span style="line-height: 1.5;">Unmapped regular tuning systems</span>= <span style="line-height: 1.5;">Sometimes, when working with scales, it can be useful to consider those scale-theoretic properties which are independent of the tempered JI intervals the scale is considered to represent, and which are independent of the absolute tuning of a scale. Such considerations arise when working with [[MODMOS's]], for instance, when looking at the Graham complexity of an arbitrary MODMOS, when taking the [[Product word]] of two existing patterns of scale steps, and when attempting to explore higher-dimensional generalizations of MOS and MODMOS.</span> This perspective becomes particularly useful when working with scales <span style="line-height: 1.5;">which are completely generated by taking Z-linear combinations of an (ideally smaller) set of intervals. When working with such scales, it can be useful for us to conceive of them as existing within a larger unmapped **Abstract Regular Tuning System (ARTS)**. To define an abstract regular tuning system, we first define a </span>**<span style="line-height: 1.5;">Regular Tuning System (RTS) </span>**as <span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; line-height: 1.5;">a set of real numbers, representing logarithmic musical intervals, which also has the structure of being a free abelian group under addition. To consider the general structure of such scales independently of their tuning, we then define the </span>ARTS<span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; line-height: 1.5;"> as an equivalence class of regular tuning systems which are related by isomorphism.</span> <span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; line-height: 1.5;">The subsets of an ARTS are called **abstract scales**.</span> ==Motivation== <span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; line-height: 1.5;">In plain English, an ARTS is much like the JI lattice, except the "monzos" now refer to abstract unmapped generators rather than primes, and the "vals" now refer to mappings of these unmapped generators.</span> <span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; line-height: 1.5;">Mathematically, we note that there is exactly one ARTS for each free abelian group, and that all of the RTS's of the same rank correspond to the same ARTS. This is useful, not because we intend to abandon harmonic mappings or regular temperaments, but because when we prove theorems about scales which depend on nothing but their properties as a subset of the ARTS, these theorems will still apply to all of the mappings and tunings of the same scale.</span> ==<span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; line-height: 1.5;">Properties:</span>== There are a number of important properties that exist to classify the various scales of an ARTS, which we list below. **Periodicity:** an abstract scale of an ARTS can be said to be an **abstract periodic scale** if there exists some element p in S such that, for all elements e in S, e+n*p is also in S for every natural number n. **Rank:** the **rank** of an abstract periodic scale in an ARTS is the rank of the smallest subgroup containing that scale. **Epimorphicity:** <span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; line-height: 1.5;">A subset S of an ARTS A is called "abstractly epimorphic" if there exists an element h in Hom(A,Z) such that the restriction of h to S is a bijection.</span> <span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; line-height: 1.5;">**Unmapped Fokker Block:** The </span>
Original HTML content:
<html><head><title>Generated Tone Systems</title></head><body><!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="NOTE: this is unfinished and lots of the terminology will change, I'm just putting this here for now so I don't lose my work. don't worry about it yet and I'll fix it later today"></a><!-- ws:end:WikiTextHeadingRule:0 --><span style="line-height: 1.5;">NOTE: this is unfinished and lots of the terminology will change, I'm just putting this here for now so I don't lose my work. don't worry about it yet and I'll fix it later today</span></h1> <br /> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h1> --><h1 id="toc1"><a name="Unmapped regular tuning systems"></a><!-- ws:end:WikiTextHeadingRule:2 --><span style="line-height: 1.5;">Unmapped regular tuning systems</span></h1> <br /> <span style="line-height: 1.5;">Sometimes, when working with scales, it can be useful to consider those scale-theoretic properties which are independent of the tempered JI intervals the scale is considered to represent, and which are independent of the absolute tuning of a scale. Such considerations arise when working with <a class="wiki_link" href="/MODMOS%27s">MODMOS's</a>, for instance, when looking at the Graham complexity of an arbitrary MODMOS, when taking the <a class="wiki_link" href="/Product%20word">Product word</a> of two existing patterns of scale steps, and when attempting to explore higher-dimensional generalizations of MOS and MODMOS.</span><br /> <br /> This perspective becomes particularly useful when working with scales <span style="line-height: 1.5;">which are completely generated by taking Z-linear combinations of an (ideally smaller) set of intervals. When working with such scales, it can be useful for us to conceive of them as existing within a larger unmapped <strong>Abstract Regular Tuning System (ARTS)</strong>. To define an abstract regular tuning system, we first define a </span><strong><span style="line-height: 1.5;">Regular Tuning System (RTS) </span></strong>as <span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; line-height: 1.5;">a set of real numbers, representing logarithmic musical intervals, which also has the structure of being a free abelian group under addition. To consider the general structure of such scales independently of their tuning, we then define the </span>ARTS<span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; line-height: 1.5;"> as an equivalence class of regular tuning systems which are related by isomorphism.</span><br /> <br /> <span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; line-height: 1.5;">The subsets of an ARTS are called <strong>abstract scales</strong>.</span><br /> <br /> <!-- ws:start:WikiTextHeadingRule:4:<h2> --><h2 id="toc2"><a name="Unmapped regular tuning systems-Motivation"></a><!-- ws:end:WikiTextHeadingRule:4 -->Motivation</h2> <span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; line-height: 1.5;">In plain English, an ARTS is much like the JI lattice, except the "monzos" now refer to abstract unmapped generators rather than primes, and the "vals" now refer to mappings of these unmapped generators.</span><br /> <br /> <span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; line-height: 1.5;">Mathematically, we note that there is exactly one ARTS for each free abelian group, and that all of the RTS's of the same rank correspond to the same ARTS. This is useful, not because we intend to abandon harmonic mappings or regular temperaments, but because when we prove theorems about scales which depend on nothing but their properties as a subset of the ARTS, these theorems will still apply to all of the mappings and tunings of the same scale.</span><br /> <br /> <br /> <!-- ws:start:WikiTextHeadingRule:6:<h2> --><h2 id="toc3"><a name="Unmapped regular tuning systems-Properties:"></a><!-- ws:end:WikiTextHeadingRule:6 --><span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; line-height: 1.5;">Properties:</span></h2> There are a number of important properties that exist to classify the various scales of an ARTS, which we list below.<br /> <br /> <strong>Periodicity:</strong> an abstract scale of an ARTS can be said to be an <strong>abstract periodic scale</strong> if there exists some element p in S such that, for all elements e in S, e+n*p is also in S for every natural number n.<br /> <br /> <strong>Rank:</strong> the <strong>rank</strong> of an abstract periodic scale in an ARTS is the rank of the smallest subgroup containing that scale.<br /> <br /> <strong>Epimorphicity:</strong> <span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; line-height: 1.5;">A subset S of an ARTS A is called "abstractly epimorphic" if there exists an element h in Hom(A,Z) such that the restriction of h to S is a bijection.</span><br /> <br /> <span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; line-height: 1.5;"><strong>Unmapped Fokker Block:</strong> The </span></body></html>