Hyperpent and hypopent: Difference between revisions

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'''Hyperpent''' or hyperpental edo is an equal division of the octave which has a just fifth that is less than a syntonic comma sharp. It is juxtaposed by hypopent or hypopental edo tuning systems.

'''Hyperpent''' or hyperpental edo is an [[equal division of the octave]] which has a [[3/2|just fifth]] that is less than a [[81/80|syntonic comma]] sharp. It is juxtaposed by '''hypopent''' or hypopental edo tuning systems, which have a [[3/2|just fifth]] that is less than a [[81/80|syntonic comma]] flat.

== Background ==

"Hyper-" is the Greek prefix for high or above and "pent" is Greek for five. Hyperpent is loosely associated with [[superpyth]], but specifically refers to equal division of the octave that result in any fifth tempered sharp, while "hypo-" is the Greek prefix for low or below and "pent" is Greek for five. Hypopent is loosely associated with [[meantone]], but specifically refers to equal division of the octave that result in any fifth tempered flat.

"Hyper-" is the Greek prefix for high or above and "pent" is Greek for five. The term is loosely associated with "superpythagorean," but specifically refers to equal division of the octave that result in any fifth tempered sharp.

Large edos are generally amphipent, meaning both hyperpent and hypopent, because they contain multiple representations of the fifth that may be sharp and flat. Some edo's may contain no intervals in the span of a just fifth plus or minus a [[syntonic comma]], and thus are neither hyperpent nor hypopent, and are refered to as "anpent."

Large ~~edo's~~ are generally both hyperpent and hypopent, because they contain multiple representations of the fifth that may be sharp and flat. Some edo's may contain no intervals in the span of a just fifth plus or minus a syntonic comma, and thus are neither hyperpent nor hypopent.

== Partial List ==

Anpent (neither hypopent nor hyperpent)

2edo

[[2edo]], [[3edo]], [[4edo]], [[6edo]], [[8edo]], [[9edo]], [[11edo]], [[13edo]], [[16edo]], [[18edo]], [[23edo]]

3edo

4edo

6edo

8edo

9edo

11edo

13edo

16edo

18edo

23edo

Hyperpent

5edo

[[5edo]], [[10edo]], [[15edo]], [[17edo]], [[20edo]], [[22edo]], [[25edo]], [[27edo]], [[29edo]], [[30edo]], [[32edo]], [[34edo]], [[39edo]], [[41edo]], [[46edo]]

10edo

15edo

17edo

20edo

22edo

25edo

27edo

29edo

30edo

32edo

34edo

39edo

41edo

46edo

Hypopent

7edo

[[7edo]], [[12edo]], [[14edo]], [[19edo]], [[21edo]], [[24edo]], [[26edo]], [[28edo]], [[31edo]], [[33edo]], [[36edo]], [[38edo]], [[43edo]]

12edo

14edo

19edo

21edo

24edo

26edo

28edo

31edo

33edo

36edo

38edo

43edo

Amphipent (both hyperpent and hypopent)

35edo

[[35edo]], [[37edo]], [[40edo]], [[42edo]], [[44edo]], [[45edo]], [[47edo]]

37edo

40edo

42edo

44edo

45edo

47edo

== Bozu's analysis ==

If 5edo and 7edo are taken as the smallest hyperpent and hypopent edo's, respectively, other edo tuning systems (let's call it ''X''-edo with ''X'' equal divisions of the octave) can be determined to be hyperpent, hypopent, anpent, or amphipent by addition as follows (with a few exceptions):

* If ''X'' can be represented as a sum of 5's, Xedo is hyperpent.

* If ''X'' can be represented as a sum of 7's, Xedo is hypopent.

* If ''X'' can be represented as a sum of 5's and 7's, with more 5's than 7's, Xedo is hyperpent.

* If ''X'' can be represented as a sum of 5's and 7's, with more 7's than 5's, or an equal number of 7's and 5's, Xedo is hypopent.

* If ''X'' cannot be represented as a sum of 5's and 7's, Xedo is anpent.

* If ''X'' can be represented as a sum of 5's and 7's in multiple ways (for example, 35 can be represented as five sevens or as seven fives), ''X''-edo is amphipent.

~~== Analysis ==~~

Some exceptions are 37edo and 44edo, which are amphipent, even though 37 cannot be written as multiple sums of 5's and 7's, nor can 44.

~~If 5edo and 7edo~~ are ~~taken as the smallest hyperpent~~ and ~~hypopent edo's~~, ~~respectively, other edo tuning systems (let's call it "Xedo" with "X" equal divisions of the octave) can be determined to be hyperpent, hypopen, anpent, or~~ amphipent ~~by addition as follows (with a few exceptions):~~

~~If X can be represented as a sum of 5's~~, ~~Xedo is hyperpent.~~

~~If X can be represented as a sum of 7's, Xedo is hypopent.~~

~~If X can be represented as a sum of 5's and 7's, with more 5's than 7's, Xedo is hyperpent.~~

~~If X can be represented as a sum of 5's and 7's, with more 7's than 5's, or an equal number of 7's and 5's, Xedo is hypopent.~~

~~If X~~ cannot be ~~represented~~ as ~~a sum of 5's and 7's, Xedo is anpent.~~

~~If X can be represented as a sum~~ of 5's and 7's ~~in multiple ways (for example~~, 35 can ~~be represented as five sevens or as seven fives), Xedo is amphipent~~.

~~Some exceptions are 37edo and 44edo, which are amphipent, even though 37 cannot be written as multiple sums~~ of ~~5's and 7's, nor can 44.~~

[[Category:Lists of scales]]

[[Category:EDO theory pages]]

@@ Line 1: / Line 1: @@
-'''Hyperpent''' or hyperpental edo is an equal division of the octave which has a just fifth that is less than a syntonic comma sharp.  It is juxtaposed by hypopent or hypopental edo tuning systems.
+'''Hyperpent''' or hyperpental edo is an [[equal division of the octave]] which has a [[3/2|just fifth]] that is less than a [[81/80|syntonic comma]] sharp.  It is juxtaposed by '''hypopent''' or hypopental edo tuning systems, which have a [[3/2|just fifth]] that is less than a [[81/80|syntonic comma]] flat.
 == Background ==
+"Hyper-" is the Greek prefix for high or above and "pent" is Greek for five. Hyperpent is loosely associated with [[superpyth]], but specifically refers to equal division of the octave that result in any fifth tempered sharp, while "hypo-" is the Greek prefix for low or below and "pent" is Greek for five. Hypopent is loosely associated with [[meantone]], but specifically refers to equal division of the octave that result in any fifth tempered flat.
-"Hyper-" is the Greek prefix for high or above and "pent" is Greek for five.  The term is loosely associated with "superpythagorean," but specifically refers to equal division of the octave that result in any fifth tempered sharp.
+Large edos are generally amphipent, meaning both hyperpent and hypopent, because they contain multiple representations of the fifth that may be sharp and flat. Some edo's may contain no intervals in the span of a just fifth plus or minus a [[syntonic comma]], and thus are neither hyperpent nor hypopent, and are refered to as "anpent."
-Large edo's are generally both hyperpent and hypopent, because they contain multiple representations of the fifth that may be sharp and flat.  Some edo's may contain no intervals in the span of a just fifth plus or minus a syntonic comma, and thus are neither hyperpent nor hypopent.
 == Partial List ==
 Anpent (neither hypopent nor hyperpent)
-edo
+[[2edo]], [[3edo]], [[4edo]], [[6edo]], [[8edo]], [[9edo]], [[11edo]], [[13edo]], [[16edo]], [[18edo]], [[23edo]]
-edo
-edo
-edo
-edo
-edo
-edo
-edo
-edo
-edo
-edo
 Hyperpent
-edo
+[[5edo]], [[10edo]], [[15edo]], [[17edo]], [[20edo]], [[22edo]], [[25edo]], [[27edo]], [[29edo]], [[30edo]], [[32edo]], [[34edo]], [[39edo]], [[41edo]], [[46edo]]
-edo
-edo
-edo
-edo
-edo
-edo
-edo
-edo
-edo
-edo
-edo
-edo
-edo
-edo
 Hypopent
-edo
+[[7edo]], [[12edo]], [[14edo]], [[19edo]], [[21edo]], [[24edo]], [[26edo]], [[28edo]], [[31edo]], [[33edo]], [[36edo]], [[38edo]], [[43edo]]
-edo
-edo
-edo
-edo
-edo
-edo
-edo
-edo
-edo
-edo
-edo
-edo
 Amphipent (both hyperpent and hypopent)
-edo
+[[35edo]], [[37edo]], [[40edo]], [[42edo]], [[44edo]], [[45edo]], [[47edo]]
-edo
-edo
-edo
-edo
-edo
-edo
+== Bozu's analysis ==
+If 5edo and 7edo are taken as the smallest hyperpent and hypopent edo's, respectively, other edo tuning systems (let's call it ''X''-edo with ''X'' equal divisions of the octave) can be determined to be hyperpent, hypopent, anpent, or amphipent by addition as follows (with a few exceptions):
+* If ''X'' can be represented as a sum of 5's, Xedo is hyperpent.
+* If ''X'' can be represented as a sum of 7's, Xedo is hypopent.
+* If ''X'' can be represented as a sum of 5's and 7's, with more 5's than 7's, Xedo is hyperpent.
+* If ''X'' can be represented as a sum of 5's and 7's, with more 7's than 5's, or an equal number of 7's and 5's, Xedo is hypopent.
+* If ''X'' cannot be represented as a sum of 5's and 7's, Xedo is anpent.
+* If ''X'' can be represented as a sum of 5's and 7's in multiple ways (for example, 35 can be represented as five sevens or as seven fives), ''X''-edo is amphipent.
-== Analysis ==
+Some exceptions are 37edo and 44edo, which are amphipent, even though 37 cannot be written as multiple sums of 5's and 7's, nor can 44.
-If 5edo and 7edo are taken as the smallest hyperpent and hypopent edo's, respectively, other edo tuning systems (let's call it "Xedo" with "X" equal divisions of the octave) can be determined to be hyperpent, hypopen, anpent, or amphipent by addition as follows (with a few exceptions):
-If X can be represented as a sum of 5's, Xedo is hyperpent.
-If X can be represented as a sum of 7's, Xedo is hypopent.
-If X can be represented as a sum of 5's and 7's, with more 5's than 7's, Xedo is hyperpent.
-If X can be represented as a sum of 5's and 7's, with more 7's than 5's, or an equal number of 7's and 5's, Xedo is hypopent.
-If X cannot be represented as a sum of 5's and 7's, Xedo is anpent.
-If X can be represented as a sum of 5's and 7's in multiple ways (for example, 35 can be represented as five sevens or as seven fives), Xedo is amphipent.
-Some exceptions are 37edo and 44edo, which are amphipent, even though 37 cannot be written as multiple sums of 5's and 7's, nor can 44.
+[[Category:Lists of scales]]
+[[Category:EDO theory pages]]