Val: Difference between revisions
Jump to navigation
Jump to search
Wikispaces>genewardsmith **Imported revision 250563018 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 250564722 - Original comment: ** |
||
| Line 1: | Line 1: | ||
<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-09- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-09-04 00:06:58 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>250564722</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt></tt><br> | ||
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
<h4>Original Wikitext content:</h4> | <h4>Original Wikitext content:</h4> | ||
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"> | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"> | ||
**What are vals and what are they for?** | |||
A val is a map which sends just intonation intervals to integers. One | |||
A val is a map | sort of val is the kind which represents how exactly we'd like to describe the intervals in an equal or well-temperament as being tempered versions of more fundamental JI intervals, by sending just intonation intervals to certain numbers of steps in the equal temperament. It is also used to describe more general temperaments such as meantone temperament. It provides a way to describe temperaments by mapping JI intervals to and from a stack of tempered generator "steps," of which a traditional EDO is only one type (but of which something like the meantone chain of fifths, barring octave equivalence, could be another type). A val tells us which interval in that stack we're going to describe as the tempered 3/2, which interval we're going to describe as the tempered 5/4, and so forth. | ||
A val maps all intervals in this way by simply mapping each of the primes, hence indirectly mapping all of the | A val maps all intervals in some [[harmonic limit]] in this way by simply mapping each of the primes up to some prime p, and hence indirectly mapping all of the positive rational numbers in the prime limit, since every such positive rational number can be described as a product of primes. It's usually written in the notation <a b c d e f ... p|, where each successive column represents a successive prime, such as 2, 3, 5, 7, 11, 13... etc, in that order, up to some [[harmonic limit|prime limit p]]. | ||
For example, the 5-limit val <12 19 28| tells us that you | For example, the 5-limit val <12 19 28| tells us that you should view 12 generator steps as mapping to the octave 2/1, which means you're describing 12-EDO. In addition to saying that 12 steps of 12-equal represents a (possibly tempered) 2/1, it also states that you should view 19 steps of 12-equal as being a tempered 3/1, and 28 steps of 12-equal as being a tempered 5/1. | ||
If you would like to assume the perspective that the 10 step interval in 12-equal (representing 1000 cents) is a very tempered 7/4, then that means that 7/1, which is 7/4 with two octaves stacked on top, is equal to 10 steps + 12 steps + 12 steps = 34 steps. This decision can hence be represented by using the 7-limit <12 19 28 34| val. If for some strange reason you'd instead like to say that 900 cents is 7/4, then that would be represented by the <12 19 28 33| val. It's not recommended that you use silly vals like that, but the mathematics will allow you to do it if you want, kind of like how a brick will allow you to hit yourself in the face with it. | If you would like to assume the perspective that the 10 step interval in 12-equal (representing 1000 cents) is a very tempered 7/4, then that means that 7/1, which is 7/4 with two octaves stacked on top, is equal to 10 steps + 12 steps + 12 steps = 34 steps. This decision can hence be represented by using the 7-limit <12 19 28 34| val. If for some strange reason you'd instead like to say that 900 cents is 7/4, then that would be represented by the <12 19 28 33| val. It's not recommended that you use silly vals like that, but the mathematics will allow you to do it if you want, kind of like how a brick will allow you to hit yourself in the face with it. | ||
Vals form the basis for all of regular temperament theory. They are important because they provide a way to mathematically formalize the chosen JI perspective you'd like to take on an EDO. As such, they will allow you to harness the very powerful realm of mathematics to describe the implications of your own musical intuitions. Once you've figured out how the perspective you've chosen to take on an EDO can be represented in val form, you can figure out what commas that EDO tempers out, what [[comma pump|comma pumps]] are available in the EDO, what the most consonant chords in the EDO are, how to optimize the octave stretch of the EDO to minimize tuning error, how to mix your val with another val to generate a rank-2 temperament such as [[meantone]] or [[Porcupine|porcupine]] temperament, and other operations as of yet undiscovered. | Vals form the basis for all of regular temperament theory. They are important because they provide a way to mathematically formalize the chosen JI perspective you'd like to take on an EDO. As such, they will allow you to harness the very powerful realm of mathematics to describe the implications of your own musical intuitions; in the language of mathematics, the kind of map vals are is a "group homomorphism" and the [[regular mapping paradigm]] might have been called the homomorphism paradigm. Once you've figured out how the perspective you've chosen to take on an EDO can be represented in val form, you can figure out what commas that EDO tempers out, what [[comma pump|comma pumps]] are available in the EDO, what the most consonant chords in the EDO are, how to optimize the octave stretch of the EDO to minimize tuning error, how to mix your val with another val to generate a rank-2 temperament such as [[meantone]] or [[Porcupine|porcupine]] temperament, and other operations as of yet undiscovered. | ||
See also: [[Monzos and Interval Space]], [[Patent val]] | See also: [[Monzos and Interval Space]], [[Patent val]], [[Vals and Tuning Space]] | ||
</pre></div> | </pre></div> | ||
<h4>Original HTML content:</h4> | <h4>Original HTML content:</h4> | ||
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Vals</title></head><body | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Vals</title></head><body><br /> | ||
<strong>What are vals and what are they for?</strong> <br /> | |||
< | |||
<br /> | <br /> | ||
A val | A val is a map which sends just intonation intervals to integers. One <br /> | ||
sort of val is the kind which represents how exactly we'd like to describe the intervals in an equal or well-temperament as being tempered versions of more fundamental JI intervals, by sending just intonation intervals to certain numbers of steps in the equal temperament. It is also used to describe more general temperaments such as meantone temperament. It provides a way to describe temperaments by mapping JI intervals to and from a stack of tempered generator &quot;steps,&quot; of which a traditional EDO is only one type (but of which something like the meantone chain of fifths, barring octave equivalence, could be another type). A val tells us which interval in that stack we're going to describe as the tempered 3/2, which interval we're going to describe as the tempered 5/4, and so forth.<br /> | |||
<br /> | <br /> | ||
For example, the 5-limit val &lt;12 19 28| tells us that you | A val maps all intervals in some <a class="wiki_link" href="/harmonic%20limit">harmonic limit</a> in this way by simply mapping each of the primes up to some prime p, and hence indirectly mapping all of the positive rational numbers in the prime limit, since every such positive rational number can be described as a product of primes. It's usually written in the notation &lt;a b c d e f ... p|, where each successive column represents a successive prime, such as 2, 3, 5, 7, 11, 13... etc, in that order, up to some <a class="wiki_link" href="/harmonic%20limit">prime limit p</a>.<br /> | ||
<br /> | |||
For example, the 5-limit val &lt;12 19 28| tells us that you should view 12 generator steps as mapping to the octave 2/1, which means you're describing 12-EDO. In addition to saying that 12 steps of 12-equal represents a (possibly tempered) 2/1, it also states that you should view 19 steps of 12-equal as being a tempered 3/1, and 28 steps of 12-equal as being a tempered 5/1.<br /> | |||
<br /> | <br /> | ||
If you would like to assume the perspective that the 10 step interval in 12-equal (representing 1000 cents) is a very tempered 7/4, then that means that 7/1, which is 7/4 with two octaves stacked on top, is equal to 10 steps + 12 steps + 12 steps = 34 steps. This decision can hence be represented by using the 7-limit &lt;12 19 28 34| val. If for some strange reason you'd instead like to say that 900 cents is 7/4, then that would be represented by the &lt;12 19 28 33| val. It's not recommended that you use silly vals like that, but the mathematics will allow you to do it if you want, kind of like how a brick will allow you to hit yourself in the face with it.<br /> | If you would like to assume the perspective that the 10 step interval in 12-equal (representing 1000 cents) is a very tempered 7/4, then that means that 7/1, which is 7/4 with two octaves stacked on top, is equal to 10 steps + 12 steps + 12 steps = 34 steps. This decision can hence be represented by using the 7-limit &lt;12 19 28 34| val. If for some strange reason you'd instead like to say that 900 cents is 7/4, then that would be represented by the &lt;12 19 28 33| val. It's not recommended that you use silly vals like that, but the mathematics will allow you to do it if you want, kind of like how a brick will allow you to hit yourself in the face with it.<br /> | ||
<br /> | <br /> | ||
Vals form the basis for all of regular temperament theory. They are important because they provide a way to mathematically formalize the chosen JI perspective you'd like to take on an EDO. As such, they will allow you to harness the very powerful realm of mathematics to describe the implications of your own musical intuitions. Once you've figured out how the perspective you've chosen to take on an EDO can be represented in val form, you can figure out what commas that EDO tempers out, what <a class="wiki_link" href="/comma%20pump">comma pumps</a> are available in the EDO, what the most consonant chords in the EDO are, how to optimize the octave stretch of the EDO to minimize tuning error, how to mix your val with another val to generate a rank-2 temperament such as <a class="wiki_link" href="/meantone">meantone</a> or <a class="wiki_link" href="/Porcupine">porcupine</a> temperament, and other operations as of yet undiscovered.<br /> | Vals form the basis for all of regular temperament theory. They are important because they provide a way to mathematically formalize the chosen JI perspective you'd like to take on an EDO. As such, they will allow you to harness the very powerful realm of mathematics to describe the implications of your own musical intuitions; in the language of mathematics, the kind of map vals are is a &quot;group homomorphism&quot; and the <a class="wiki_link" href="/regular%20mapping%20paradigm">regular mapping paradigm</a> might have been called the homomorphism paradigm. Once you've figured out how the perspective you've chosen to take on an EDO can be represented in val form, you can figure out what commas that EDO tempers out, what <a class="wiki_link" href="/comma%20pump">comma pumps</a> are available in the EDO, what the most consonant chords in the EDO are, how to optimize the octave stretch of the EDO to minimize tuning error, how to mix your val with another val to generate a rank-2 temperament such as <a class="wiki_link" href="/meantone">meantone</a> or <a class="wiki_link" href="/Porcupine">porcupine</a> temperament, and other operations as of yet undiscovered.<br /> | ||
<br /> | <br /> | ||
See also: <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">Monzos and Interval Space</a>, <a class="wiki_link" href="/Patent%20val">Patent val</a></body></html></pre></div> | See also: <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">Monzos and Interval Space</a>, <a class="wiki_link" href="/Patent%20val">Patent val</a>, <a class="wiki_link" href="/Vals%20and%20Tuning%20Space">Vals and Tuning Space</a></body></html></pre></div> | ||
Revision as of 00:06, 4 September 2011
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author genewardsmith and made on 2011-09-04 00:06:58 UTC.
- The original revision id was 250564722.
- 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:
**What are vals and what are they for?** A val is a map which sends just intonation intervals to integers. One sort of val is the kind which represents how exactly we'd like to describe the intervals in an equal or well-temperament as being tempered versions of more fundamental JI intervals, by sending just intonation intervals to certain numbers of steps in the equal temperament. It is also used to describe more general temperaments such as meantone temperament. It provides a way to describe temperaments by mapping JI intervals to and from a stack of tempered generator "steps," of which a traditional EDO is only one type (but of which something like the meantone chain of fifths, barring octave equivalence, could be another type). A val tells us which interval in that stack we're going to describe as the tempered 3/2, which interval we're going to describe as the tempered 5/4, and so forth. A val maps all intervals in some [[harmonic limit]] in this way by simply mapping each of the primes up to some prime p, and hence indirectly mapping all of the positive rational numbers in the prime limit, since every such positive rational number can be described as a product of primes. It's usually written in the notation <a b c d e f ... p|, where each successive column represents a successive prime, such as 2, 3, 5, 7, 11, 13... etc, in that order, up to some [[harmonic limit|prime limit p]]. For example, the 5-limit val <12 19 28| tells us that you should view 12 generator steps as mapping to the octave 2/1, which means you're describing 12-EDO. In addition to saying that 12 steps of 12-equal represents a (possibly tempered) 2/1, it also states that you should view 19 steps of 12-equal as being a tempered 3/1, and 28 steps of 12-equal as being a tempered 5/1. If you would like to assume the perspective that the 10 step interval in 12-equal (representing 1000 cents) is a very tempered 7/4, then that means that 7/1, which is 7/4 with two octaves stacked on top, is equal to 10 steps + 12 steps + 12 steps = 34 steps. This decision can hence be represented by using the 7-limit <12 19 28 34| val. If for some strange reason you'd instead like to say that 900 cents is 7/4, then that would be represented by the <12 19 28 33| val. It's not recommended that you use silly vals like that, but the mathematics will allow you to do it if you want, kind of like how a brick will allow you to hit yourself in the face with it. Vals form the basis for all of regular temperament theory. They are important because they provide a way to mathematically formalize the chosen JI perspective you'd like to take on an EDO. As such, they will allow you to harness the very powerful realm of mathematics to describe the implications of your own musical intuitions; in the language of mathematics, the kind of map vals are is a "group homomorphism" and the [[regular mapping paradigm]] might have been called the homomorphism paradigm. Once you've figured out how the perspective you've chosen to take on an EDO can be represented in val form, you can figure out what commas that EDO tempers out, what [[comma pump|comma pumps]] are available in the EDO, what the most consonant chords in the EDO are, how to optimize the octave stretch of the EDO to minimize tuning error, how to mix your val with another val to generate a rank-2 temperament such as [[meantone]] or [[Porcupine|porcupine]] temperament, and other operations as of yet undiscovered. See also: [[Monzos and Interval Space]], [[Patent val]], [[Vals and Tuning Space]]
Original HTML content:
<html><head><title>Vals</title></head><body><br /> <strong>What are vals and what are they for?</strong> <br /> <br /> A val is a map which sends just intonation intervals to integers. One <br /> sort of val is the kind which represents how exactly we'd like to describe the intervals in an equal or well-temperament as being tempered versions of more fundamental JI intervals, by sending just intonation intervals to certain numbers of steps in the equal temperament. It is also used to describe more general temperaments such as meantone temperament. It provides a way to describe temperaments by mapping JI intervals to and from a stack of tempered generator "steps," of which a traditional EDO is only one type (but of which something like the meantone chain of fifths, barring octave equivalence, could be another type). A val tells us which interval in that stack we're going to describe as the tempered 3/2, which interval we're going to describe as the tempered 5/4, and so forth.<br /> <br /> A val maps all intervals in some <a class="wiki_link" href="/harmonic%20limit">harmonic limit</a> in this way by simply mapping each of the primes up to some prime p, and hence indirectly mapping all of the positive rational numbers in the prime limit, since every such positive rational number can be described as a product of primes. It's usually written in the notation <a b c d e f ... p|, where each successive column represents a successive prime, such as 2, 3, 5, 7, 11, 13... etc, in that order, up to some <a class="wiki_link" href="/harmonic%20limit">prime limit p</a>.<br /> <br /> For example, the 5-limit val <12 19 28| tells us that you should view 12 generator steps as mapping to the octave 2/1, which means you're describing 12-EDO. In addition to saying that 12 steps of 12-equal represents a (possibly tempered) 2/1, it also states that you should view 19 steps of 12-equal as being a tempered 3/1, and 28 steps of 12-equal as being a tempered 5/1.<br /> <br /> If you would like to assume the perspective that the 10 step interval in 12-equal (representing 1000 cents) is a very tempered 7/4, then that means that 7/1, which is 7/4 with two octaves stacked on top, is equal to 10 steps + 12 steps + 12 steps = 34 steps. This decision can hence be represented by using the 7-limit <12 19 28 34| val. If for some strange reason you'd instead like to say that 900 cents is 7/4, then that would be represented by the <12 19 28 33| val. It's not recommended that you use silly vals like that, but the mathematics will allow you to do it if you want, kind of like how a brick will allow you to hit yourself in the face with it.<br /> <br /> Vals form the basis for all of regular temperament theory. They are important because they provide a way to mathematically formalize the chosen JI perspective you'd like to take on an EDO. As such, they will allow you to harness the very powerful realm of mathematics to describe the implications of your own musical intuitions; in the language of mathematics, the kind of map vals are is a "group homomorphism" and the <a class="wiki_link" href="/regular%20mapping%20paradigm">regular mapping paradigm</a> might have been called the homomorphism paradigm. Once you've figured out how the perspective you've chosen to take on an EDO can be represented in val form, you can figure out what commas that EDO tempers out, what <a class="wiki_link" href="/comma%20pump">comma pumps</a> are available in the EDO, what the most consonant chords in the EDO are, how to optimize the octave stretch of the EDO to minimize tuning error, how to mix your val with another val to generate a rank-2 temperament such as <a class="wiki_link" href="/meantone">meantone</a> or <a class="wiki_link" href="/Porcupine">porcupine</a> temperament, and other operations as of yet undiscovered.<br /> <br /> See also: <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">Monzos and Interval Space</a>, <a class="wiki_link" href="/Patent%20val">Patent val</a>, <a class="wiki_link" href="/Vals%20and%20Tuning%20Space">Vals and Tuning Space</a></body></html>