Vals and tuning space: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2011-09-03 19:29:33 UTC</tt>.

: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2011-09-03 19:32:54 UTC</tt>.

: The original revision id was <tt>~~250541350~~</tt>.

: The original revision id was <tt>250541612</tt>.

: The revision comment was: <tt></tt>

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

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A val, intuitively speaking, provides a way to map intervals in an EDO back to JI. It tells us, when we look at an EDO like 12-equal, how exactly we'd like to describe the intervals in an EDO as being tempered versions of more fundamental JI intervals. It tells us which interval we're going to describe as the tempered 3/2, which interval we're going to describe as the tempered 5/4, etc.

~~Sometimes there's more than one sensible way to map the intervals in an EDO back to JI, and hence in that situation there's more than one val you could apply to an EDO.~~ A val maps the intervals in an EDO back to JI by describing the mapping for each of the primes. By mapping the primes, you hence indirectly map all of the rationals, since every rational number can be described as a product of primes. It's usually written in the notation <a b c d e f ... |, where each column represents prime 2, 3, 5, 7, 11, 13... etc, in that order, up to some [[harmonic limit|prime limit]].

A val maps the intervals in an EDO back to JI by describing the mapping for each of the primes. By mapping the primes, you hence indirectly map all of the rationals, since every rational number can be described as a product of primes. It's usually written in the notation <a b c d e f ... |, where each column represents prime 2, 3, 5, 7, 11, 13... etc, in that order, up to some [[harmonic limit|prime limit]].

For example, the 5-limit val <12 19 28| tells us that you'd like to view 12 steps of 12-equal as representing a tempered 2/1, 19 steps of 12-equal as representing a tempered 3/1, and 28 steps of 12-equal as representing 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.

For example, the 5-limit val <12 19 28| tells us that you'd like to view 12 steps of 12-equal as representing a tempered 2/1, 19 steps of 12-equal as representing a tempered 3/1, and 28 steps of 12-equal as representing 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. 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.

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A val, intuitively speaking, provides a way to map intervals in an EDO back to JI. It tells us, when we look at an EDO like 12-equal, how exactly we'd like to describe the intervals in an EDO as being tempered versions of more fundamental JI intervals. It tells us which interval we're going to describe as the tempered 3/2, which interval we're going to describe as the tempered 5/4, etc.

~~Sometimes there's more than one sensible way to map the intervals in an EDO back to JI, and hence in that situation there's more than one val you could apply to an EDO.~~ A val maps the intervals in an EDO back to JI by describing the mapping for each of the primes. By mapping the primes, you hence indirectly map all of the rationals, since every rational number can be described as a product of primes. It's usually written in the notation &lt;a b c d e f ... |, where each column represents prime 2, 3, 5, 7, 11, 13... etc, in that order, up to some <a class="wiki_link" href="/harmonic%20limit">prime limit</a>.

A val maps the intervals in an EDO back to JI by describing the mapping for each of the primes. By mapping the primes, you hence indirectly map all of the rationals, since every rational number can be described as a product of primes. It's usually written in the notation &lt;a b c d e f ... |, where each column represents prime 2, 3, 5, 7, 11, 13... etc, in that order, up to some <a class="wiki_link" href="/harmonic%20limit">prime limit</a>.

For example, the 5-limit val &lt;12 19 28| tells us that you'd like to view 12 steps of 12-equal as representing a tempered 2/1, 19 steps of 12-equal as representing a tempered 3/1, and 28 steps of 12-equal as representing 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 &lt;12 19 28 34| val.

For example, the 5-limit val &lt;12 19 28| tells us that you'd like to view 12 steps of 12-equal as representing a tempered 2/1, 19 steps of 12-equal as representing a tempered 3/1, and 28 steps of 12-equal as representing 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 &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.

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.

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2011-09-03 19:29:33 UTC</tt>.<br>
+: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2011-09-03 19:32:54 UTC</tt>.<br>
-: The original revision id was <tt>250541350</tt>.<br>
+: The original revision id was <tt>250541612</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>
@@ Line 10: / Line 10: @@
 A val, intuitively speaking, provides a way to map intervals in an EDO back to JI. It tells us, when we look at an EDO like 12-equal, how exactly we'd like to describe the intervals in an EDO as being tempered versions of more fundamental JI intervals. It tells us which interval we're going to describe as the tempered 3/2, which interval we're going to describe as the tempered 5/4, etc.
-Sometimes there's more than one sensible way to map the intervals in an EDO back to JI, and hence in that situation there's more than one val you could apply to an EDO. A val maps the intervals in an EDO back to JI by describing the mapping for each of the primes. By mapping the primes, you hence indirectly map all of the rationals, since every rational number can be described as a product of primes. It's usually written in the notation &lt;a b c d e f ... |, where each column represents prime 2, 3, 5, 7, 11, 13... etc, in that order, up to some [[harmonic limit|prime limit]].
+A val maps the intervals in an EDO back to JI by describing the mapping for each of the primes. By mapping the primes, you hence indirectly map all of the rationals, since every rational number can be described as a product of primes. It's usually written in the notation &lt;a b c d e f ... |, where each column represents prime 2, 3, 5, 7, 11, 13... etc, in that order, up to some [[harmonic limit|prime limit]].
-For example, the 5-limit val &lt;12 19 28| tells us that you'd like to view 12 steps of 12-equal as representing a tempered 2/1, 19 steps of 12-equal as representing a tempered 3/1, and 28 steps of 12-equal as representing 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 &lt;12 19 28 34| val.
+For example, the 5-limit val &lt;12 19 28| tells us that you'd like to view 12 steps of 12-equal as representing a tempered 2/1, 19 steps of 12-equal as representing a tempered 3/1, and 28 steps of 12-equal as representing 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 &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.
 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.
@@ Line 64: / Line 64: @@
   A val, intuitively speaking, provides a way to map intervals in an EDO back to JI. It tells us, when we look at an EDO like 12-equal, how exactly we'd like to describe the intervals in an EDO as being tempered versions of more fundamental JI intervals. It tells us which interval we're going to describe as the tempered 3/2, which interval we're going to describe as the tempered 5/4, etc.&lt;br /&gt;
 &lt;br /&gt;
-Sometimes there's more than one sensible way to map the intervals in an EDO back to JI, and hence in that situation there's more than one val you could apply to an EDO. A val maps the intervals in an EDO back to JI by describing the mapping for each of the primes. By mapping the primes, you hence indirectly map all of the rationals, since every rational number can be described as a product of primes. It's usually written in the notation &amp;lt;a b c d e f ... |, where each column represents prime 2, 3, 5, 7, 11, 13... etc, in that order, up to some &lt;a class="wiki_link" href="/harmonic%20limit"&gt;prime limit&lt;/a&gt;.&lt;br /&gt;
+A val maps the intervals in an EDO back to JI by describing the mapping for each of the primes. By mapping the primes, you hence indirectly map all of the rationals, since every rational number can be described as a product of primes. It's usually written in the notation &amp;lt;a b c d e f ... |, where each column represents prime 2, 3, 5, 7, 11, 13... etc, in that order, up to some &lt;a class="wiki_link" href="/harmonic%20limit"&gt;prime limit&lt;/a&gt;.&lt;br /&gt;
 &lt;br /&gt;
-For example, the 5-limit val &amp;lt;12 19 28| tells us that you'd like to view 12 steps of 12-equal as representing a tempered 2/1, 19 steps of 12-equal as representing a tempered 3/1, and 28 steps of 12-equal as representing 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 &amp;lt;12 19 28 34| val.&lt;br /&gt;
+For example, the 5-limit val &amp;lt;12 19 28| tells us that you'd like to view 12 steps of 12-equal as representing a tempered 2/1, 19 steps of 12-equal as representing a tempered 3/1, and 28 steps of 12-equal as representing 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 &amp;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 &amp;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.&lt;br /&gt;
 &lt;br /&gt;
 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 &lt;a class="wiki_link" href="/comma%20pump"&gt;comma pumps&lt;/a&gt; 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 &lt;a class="wiki_link" href="/meantone"&gt;meantone&lt;/a&gt; or &lt;a class="wiki_link" href="/Porcupine"&gt;porcupine&lt;/a&gt; temperament, and other operations as of yet undiscovered.&lt;br /&gt;