Wedgie: Difference between revisions

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A '''wedgie''' is an object that uniquely characterizes a [[regular temperament]] regardless of choice of [[period]] ~~vs.~~ [[generator~~]] or of [[equave~~]], which can therefore provide some illuminating information beyond the [[mapping|mapping matrix]]~~, which specifies a particular equave~~. A wedgie takes the form ⟨⟨…⟨ ''w''1 ''w''2 … ''w''''n'' ]]…], with ''n'' entries listed in between multiple [[val]] brackets (double brackets for rank-2, triple brackets for rank-3, and so on). Wedgies can be thought of as a generalization of vals, called ''multivals'', so that a val is a wedgie for a rank-1 temperament. Each element conveys information about the structure of a set of primes in the temperament, containing a number of primes equivalent to the temperament's rank.

A '''wedgie''' is a primarily mathematical object that uniquely characterizes a [[regular temperament]] regardless of choice of [[period]] and [[generator]], which can therefore provide some illuminating information beyond the [[mapping|mapping matrix]]. A wedgie takes the form ⟨⟨…⟨ ''w''1 ''w''2 … ''w''''n'' ]]…], with ''n'' entries listed in between multiple [[val]] brackets (double brackets for rank-2, triple brackets for rank-3, and so on). Wedgies can be thought of as a generalization of vals, called ''multivals'', so that a val is a wedgie for a rank-1 temperament. Each element conveys information about the structure of a set of primes in the temperament, containing a number of primes equivalent to the temperament's rank.

While wedgies have served as a canonical way to represent temperaments mathematically due to their generator-agnostic nature, their musical utility is not obvious – [[mapping|mappings]] would often be of more immediate use to musicians.

== How to read a wedgie ==

Following intuitions from [[ploidacot]], one way to characterize ~~a temperament~~ is how many ~~equal~~ parts ~~it splits~~ the perfect fifth ~~(3/2) into. For an example, meantone does not split it at all, so we say it is ''monocot''. We also say it is ''haploid'', since it does not split~~ the octave ~~(2/1) at all~~.

Following intuitions from [[ploidacot]], one way to characterize temperaments is by how many parts they split important intervals into – for example, the perfect fifth and the octave.

~~The elements~~ of a ~~wedgie each represent the number of parts into which a particular [[subgroup]] is split by the~~ temperament, or ~~the number~~ of ~~distinct sets~~ of ~~notes~~ within the temperament ~~linked~~ by ~~motions within~~ that subgroup. ~~They can be thought of as~~ a ~~generalization of the ploidacot information to all possible combinations of~~ [[~~prime harmonic~~|~~primes~~]] ~~within~~ a temperament, and ~~format that information in a concise manner; as it turns out~~, this is enough to uniquely characterize the temperament.

For any ''n''-prime subgroup of a rank-''n'' temperament, there exist a finite number (1 or more) of copies of that subgroup within the temperament. Think of these as "universes" that are connected exclusively by intervals of that subgroup and may be travelled between by using intervals outside the subgroup. For example, a temperament that is [[Ploidacot/Diploid dicot|diploid dicot]] – dividing the octave into two parts and also dividing the perfect fifth into two parts – will have a 2.3 wedgie entry of 4 (since there are four distinct copies of the 3-limit – the basic 3-limit, offset by a neutral third, offset by a semioctave, and offset by both). Each wedgie entry counts the number of copies of its corresponding subgroup. Because any temperament can be defined by splitting some interval and assigning the parts just interpretations, this is enough to uniquely characterize the temperament.

~~For example, take the~~ wedgie for meantone: {{multival| 1 4 4 }}. Each entry corresponds to a pair of primes: 2.3, 2.5, and 3.5. The first entry of the wedgie is the ploidacot signatures multiplied together, which in this case is 1, ~~telling us that~~ the octave finds 2 at one step, and the fifth finds 3 at one generator minus one octave (which, since there are no even splits, still counts as 1).

=== Example 1: meantone ===

The wedgie for meantone is {{multival| 1 4 4 }}. Each entry corresponds to a pair of primes: 2.3, 2.5, and 3.5. The first entry of the wedgie is the ploidacot signatures multiplied together, which in this case is 1, because the octave finds 2 at one step, and the fifth finds 3 at one generator minus one octave (which, since there are no even splits, still counts as 1).

For an intuition as to why multiplication is performed, consider [[hemipyth]] (or ''diploid dicot''), which divides both the octave and perfect fifth into two parts. In hemipyth, you can use [[radical interval|radical]] steps to traverse between four different "universes" connected by 3-limit intervals: base Pythagorean, offset by a semioctave, offset by a neutral third, and offset by both. The wedgie entry essentially counts these universes.

For 2.5, the procedure generalizes, with the entry, 4, being the number of steps 2 and 5 are divided into respectively multiplied together; that is, the number of parts into which the 2.5 subgroup is split. But since we already know 2 is divided into only one octave, this must mean 5 is split into four parts. In fact, 5 is found at four fifths up.

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For the final entry, which is for the 3.5 subgroup, we have another 4. But this time, we are thinking tritave-equivalently now, so we will be reaching 5/3. This is the number of parts 3 and 5/3 are divided into respectively, multiplied together. 3 is reached by going up one 3/2 and one 2/1, but no splitting is happening, so the factor of 4 must come from 5/3, which is indeed reached by four 3/2's.

=== Example 2: father ===

For another example, take father, which has the wedgie {{multival| 1 -1 -4 }}.

Here, we again have a 1 as our entry for 2.3, meaning that the temperament is haploid monocot, or in other words that 2/1 is unsplit and 3/2 is one generator.

Thus, going into our second entry, -1 for 2.5. Again, since we already know our temperament is haploid, the negative sign must come from the mapping for 5. But what could a negative sign possibly mean in a wedgie? Well, to reach the 5th harmonic (specifically, 5/4) in father, you go up a perfect ''fourth''. Since we are thinking octave-equivalently, this can be seen as going ''down'' a perfect fifth.

Thus, going into our second entry, -1 for 2.5. Again, since we already know our temperament is haploid, the negative sign must come from the mapping for 5. But what could a negative sign possibly mean in a wedgie? Well, to reach the 5th harmonic (specifically, 5/4) in father, you go up a perfect ''fourth''. Since we are thinking octave-equivalently, this can be seen as going ''down'' a perfect fifth. This means there are "-1 copies of 2.5" in father. This may seem weird, but {{multival| 1 1 ...}} would imply that 5/4 is the same interval as 3/2, which is not what father does. Essentially, as part of the counting, two numbers are being multiplied together, and one of those numbers is negative when you go down instead of up to find the interval.

Finally, for 3.5, we have the entry -4. Again, we are tritave-equivalent and 3/1 is simply found by an octave and a fifth, so we will be finding 5/3 by splitting it into four parts. 5/3 is equated to 16/9 in father, which is found by going up two octaves and down two fifths. This might seem like only a split into two, but keep in mind - we are in tritave-equivalent territory. Octaves are the tritave complement of fifths. So instead of going up two octaves, we can instead simply go down two more fifths to reach 5/3. And there we have it – 5/3 is split into four parts, which each contain a negative generator.

=== Example 3: blackwood ===

For our final example, we will consider blackwood {{multival| 0 5 8 }}.

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And then the final entry, for 3.5, is 8. Again, 5 (i.e. 5/3) is found at one generator, but 3/1 is split into 8 parts by blackwood temperament. So, {{nowrap| 1 × 8 {{=}} 8 }}.

For wedgies of temperaments of larger prime subgroups, the number of entries is increased, so a rank-2 temperament of 7-limit JI would have 6 entries, for 2.3, 2.5, 2.7, 3.5, 3.7, and 5.7. Note that the new septimal entries are inserted between the entries for the 5-limit. If it helps, think of arranging all the entries in a grid, where rows represent the first prime, and columns represent the second, and reading them off one by one.

=== Generalizations ===

For wedgies of temperaments of larger prime subgroups, the number of entries is increased, so a rank-2 temperament of 7-limit JI would have 6 entries, for 2.3, 2.5, 2.7, 3.5, 3.7, and 5.7. Note that the new septimal entries are inserted between the entries for the 5-limit (so that the 5-limit entries of a 7-limit wedgie are . If it helps, think of arranging all the entries in a grid, where rows represent the first prime, and columns represent the second, and reading them off one by one.

For wedgies of higher-rank temperaments, the number of primes per entry is increased, so that for a rank-3 temperament of the 7-limit, all possible combinations of 3 primes (2.3.5, 2.3.7, 2.5.7, and 3.5.7) would be covered.

== ~~The form~~ of a wedgie ==

== Form of a wedgie ==

The notation being used previously, {{multival| x y z }}, is formally a shorthand for a matrix form, written

The notation being used previously, {{multival| ''x'' ''y'' ''z'' }}, is formally a shorthand for a matrix form, written

$$

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$$

For a wedgie on a 4-prime subgroup, the structure of {{multival| a b c d e f }} is actually

For a wedgie on a 4-prime subgroup, the structure of {{multival| ''a'' ''b'' ''c'' ''d'' ''e'' ''f'' }} is actually

$$

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== Wedgies and contorsion ==

== Conversion ==

=== Mapping matrix to wedgie ===

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# normalize for a positive first entry

# unneeded if the mapping is in canonical form

if wedgie[0] < 0:

wedgie *= -1

@@ Line 1: / Line 1: @@
 {{Beginner|Plücker coordinates}}
-A '''wedgie''' is an object that uniquely characterizes a [[regular temperament]] regardless of choice of [[period]] vs. [[generator]] or of [[equave]], which can therefore provide some illuminating information beyond the [[mapping|mapping matrix]], which specifies a particular equave. A wedgie takes the form ⟨⟨…⟨ ''w''<sub>1</sub> ''w''<sub>2</sub> … ''w''<sub>''n''</sub> ]]…], with ''n'' entries listed in between multiple [[val]] brackets (double brackets for rank-2, triple brackets for rank-3, and so on). Wedgies can be thought of as a generalization of vals, called ''multivals'', so that a val is a wedgie for a rank-1 temperament. Each element conveys information about the structure of a set of primes in the temperament, containing a number of primes equivalent to the temperament's rank.
+A '''wedgie''' is a primarily mathematical object that uniquely characterizes a [[regular temperament]] regardless of choice of [[period]] and [[generator]], which can therefore provide some illuminating information beyond the [[mapping|mapping matrix]]. A wedgie takes the form ⟨⟨…⟨ ''w''<sub>1</sub> ''w''<sub>2</sub> … ''w''<sub>''n''</sub> ]]…], with ''n'' entries listed in between multiple [[val]] brackets (double brackets for rank-2, triple brackets for rank-3, and so on). Wedgies can be thought of as a generalization of vals, called ''multivals'', so that a val is a wedgie for a rank-1 temperament. Each element conveys information about the structure of a set of primes in the temperament, containing a number of primes equivalent to the temperament's rank.
+While wedgies have served as a canonical way to represent temperaments mathematically due to their generator-agnostic nature, their musical utility is not obvious – [[mapping|mappings]] would often be of more immediate use to musicians.
 == How to read a wedgie ==
-Following intuitions from [[ploidacot]], one way to characterize a temperament is how many equal parts it splits the perfect fifth (3/2) into. For an example, meantone does not split it at all, so we say it is ''monocot''. We also say it is ''haploid'', since it does not split the octave (2/1) at all.
+Following intuitions from [[ploidacot]], one way to characterize temperaments is by how many parts they split important intervals into – for example, the perfect fifth and the octave.
-The elements of a wedgie each represent the number of parts into which a particular [[subgroup]] is split by the temperament, or the number of distinct sets of notes within the temperament linked by motions within that subgroup. They can be thought of as a generalization of the ploidacot information to all possible combinations of [[prime harmonic|primes]] within a temperament, and format that information in a concise manner; as it turns out, this is enough to uniquely characterize the temperament.
+For any ''n''-prime subgroup of a rank-''n'' temperament, there exist a finite number (1 or more) of copies of that subgroup within the temperament. Think of these as "universes" that are connected exclusively by intervals of that subgroup and may be travelled between by using intervals outside the subgroup. For example, a temperament that is [[Ploidacot/Diploid dicot|diploid dicot]] – dividing the octave into two parts and also dividing the perfect fifth into two parts – will have a 2.3 wedgie entry of 4 (since there are four distinct copies of the 3-limit – the basic 3-limit, offset by a neutral third, offset by a semioctave, and offset by both).  Each wedgie entry counts the number of copies of its corresponding subgroup. Because any temperament can be defined by splitting some interval and assigning the parts just interpretations, this is enough to uniquely characterize the temperament.
-For example, take the wedgie for meantone: {{multival| 1 4 4 }}. Each entry corresponds to a pair of primes: 2.3, 2.5, and 3.5. The first entry of the wedgie is the ploidacot signatures multiplied together, which in this case is 1, telling us that the octave finds 2 at one step, and the fifth finds 3 at one generator minus one octave (which, since there are no even splits, still counts as 1).
+=== Example 1: meantone ===
+The wedgie for meantone is {{multival| 1 4 4 }}. Each entry corresponds to a pair of primes: 2.3, 2.5, and 3.5. The first entry of the wedgie is the ploidacot signatures multiplied together, which in this case is 1, because the octave finds 2 at one step, and the fifth finds 3 at one generator minus one octave (which, since there are no even splits, still counts as 1).
-For an intuition as to why multiplication is performed, consider [[hemipyth]] (or ''diploid dicot''), which divides both the octave and perfect fifth into two parts. In hemipyth, you can use [[radical interval|radical]] steps to traverse between four different "universes" connected by 3-limit intervals: base Pythagorean, offset by a semioctave, offset by a neutral third, and offset by both. The wedgie entry essentially counts these universes.
 For 2.5, the procedure generalizes, with the entry, 4, being the number of steps 2 and 5 are divided into respectively multiplied together; that is, the number of parts into which the 2.5 subgroup is split. But since we already know 2 is divided into only one octave, this must mean 5 is split into four parts. In fact, 5 is found at four fifths up.
@@ Line 16: / Line 17: @@
 For the final entry, which is for the 3.5 subgroup, we have another 4. But this time, we are thinking tritave-equivalently now, so we will be reaching 5/3. This is the number of parts 3 and 5/3 are divided into respectively, multiplied together. 3 is reached by going up one 3/2 and one 2/1, but no splitting is happening, so the factor of 4 must come from 5/3, which is indeed reached by four 3/2's.
+=== Example 2: father ===
 For another example, take father, which has the wedgie {{multival| 1 -1 -4 }}.
 Here, we again have a 1 as our entry for 2.3, meaning that the temperament is haploid monocot, or in other words that 2/1 is unsplit and 3/2 is one generator.
-Thus, going into our second entry, -1 for 2.5. Again, since we already know our temperament is haploid, the negative sign must come from the mapping for 5. But what could a negative sign possibly mean in a wedgie? Well, to reach the 5th harmonic (specifically, 5/4) in father, you go up a perfect ''fourth''. Since we are thinking octave-equivalently, this can be seen as going ''down'' a perfect fifth.
+Thus, going into our second entry, -1 for 2.5. Again, since we already know our temperament is haploid, the negative sign must come from the mapping for 5. But what could a negative sign possibly mean in a wedgie? Well, to reach the 5th harmonic (specifically, 5/4) in father, you go up a perfect ''fourth''. Since we are thinking octave-equivalently, this can be seen as going ''down'' a perfect fifth. This means there are "-1 copies of 2.5" in father. This may seem weird, but {{multival| 1 1 ...}} would imply that 5/4 is the same interval as 3/2, which is not what father does. Essentially, as part of the counting, two numbers are being multiplied together, and one of those numbers is negative when you go down instead of up to find the interval.
 Finally, for 3.5, we have the entry -4. Again, we are tritave-equivalent and 3/1 is simply found by an octave and a fifth, so we will be finding 5/3 by splitting it into four parts. 5/3 is equated to 16/9 in father, which is found by going up two octaves and down two fifths. This might seem like only a split into two, but keep in mind - we are in tritave-equivalent territory. Octaves are the tritave complement of fifths. So instead of going up two octaves, we can instead simply go down two more fifths to reach 5/3. And there we have it – 5/3 is split into four parts, which each contain a negative generator.
+=== Example 3: blackwood ===
 For our final example, we will consider blackwood {{multival| 0 5 8 }}.
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 And then the final entry, for 3.5, is 8. Again, 5 (i.e. 5/3) is found at one generator, but 3/1 is split into 8 parts by blackwood temperament. So, {{nowrap| 1 × 8 {{=}} 8 }}.
-For wedgies of temperaments of larger prime subgroups, the number of entries is increased, so a rank-2 temperament of 7-limit JI would have 6 entries, for 2.3, 2.5, 2.7, 3.5, 3.7, and 5.7. Note that the new septimal entries are inserted between the entries for the 5-limit. If it helps, think of arranging all the entries in a grid, where rows represent the first prime, and columns represent the second, and reading them off one by one.
+=== Generalizations ===
+For wedgies of temperaments of larger prime subgroups, the number of entries is increased, so a rank-2 temperament of 7-limit JI would have 6 entries, for 2.3, 2.5, 2.7, 3.5, 3.7, and 5.7. Note that the new septimal entries are inserted between the entries for the 5-limit (so that the 5-limit entries of a 7-limit wedgie are . If it helps, think of arranging all the entries in a grid, where rows represent the first prime, and columns represent the second, and reading them off one by one.
 For wedgies of higher-rank temperaments, the number of primes per entry is increased, so that for a rank-3 temperament of the 7-limit, all possible combinations of 3 primes (2.3.5, 2.3.7, 2.5.7, and 3.5.7) would be covered.
-== The form of a wedgie ==
+== Form of a wedgie ==
-The notation being used previously, {{multival| x y z }}, is formally a shorthand for a matrix form, written
+The notation being used previously, {{multival| ''x'' ''y'' ''z'' }}, is formally a shorthand for a matrix form, written
 $$
@@ Line 47: / Line 51: @@
 $$
-For a wedgie on a 4-prime subgroup, the structure of {{multival| a b c d e f }} is actually
+For a wedgie on a 4-prime subgroup, the structure of {{multival| ''a'' ''b'' ''c'' ''d'' ''e'' ''f'' }} is actually
 $$
@@ Line 67: / Line 71: @@
 == Wedgies and contorsion ==
 {{Todo|inline=1| expand }}
 == Conversion ==
 === Mapping matrix to wedgie ===
@@ Line 84: / Line 87: @@
      # normalize for a positive first entry
+    # unneeded if the mapping is in canonical form
      if wedgie[0] < 0:
          wedgie *= -1