311edo: Difference between revisions

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The 41-limit add-73 add-89 add-101 add-109 add-113 123-odd-limit is represented very close to completely consistently, and as aforementioned, the 77-odd-limit subset of that odd-limit is perfectly consistent, to which a variety of odds can be added that keep perfect consistency, but for comprehensiveness and practical use as a temperament approximating the low-to-mid end of the harmonic series, we consider a larger odd-limit than that which seeks to be more complete.

There is 884 interval pairs in that [[odd-limit]] (the [[41-limit]] add-73 add-89 add-101 add-109 add-113 123-odd-limit), where "pairs" refers to that each interval has an [[octave complement]] with equal and opposite error.

There are 884 interval pairs in that [[odd-limit]] (the [[41-limit]] add-73 add-89 add-101 add-109 add-113 123-odd-limit), where "pairs" refers to that each interval has an [[octave complement]] with equal and opposite error.

That odd-limit can be described explicitly as the~~...~~

That odd limit can be described explicitly as the

{1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 45, 49, 51, 55, 57, 63, 65, 69, 73, 75, 77, 81, 85, 87, 89, 91, 93, 95, 99, 101, 105, 109, 111, 113, 115, 117, 119, 121, 123}-odd-limit,

{1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 45, 49, 51, 55, 57, 63, 65, 69, 73, 75, 77, 81, 85, 87, 89, 91, 93, 95, 99, 101, 105, 109, 111, 113, 115, 117, 119, 121, 123}-odd-limit

~~...~~or equivalently, as the [[diamond function]] applied to that set of odds.

or equivalently, as the [[diamond function]] applied to that set of odds.

We can also express that odd-limit as the 123-odd-limit minus only the following twelve prime odds: {43, 47, 53, 59, 61, 67, 71, 79, 83, 97, 103, 107}.

Of those 884 interval pairs, only 42 interval pairs (<4.8%) are inconsistent ~~(meaning~~ not mapped to the nearest interval of [[311edo]] but to the second-nearest interval).

Of those 884 interval pairs, only 42 interval pairs (< 4.8%) are inconsistent, not mapped to the nearest interval of 311edo but to the second-nearest interval.

~~Preferring~~ the ~~interval~~ of the ~~pair that is less than 600{{cent}} = 1\2~~, these intervals, from smallest to largest, are:

Reduced to the lower half of the octave, these intervals, from smallest to largest, are:

101/100, 100/99, 82/81, 121/119, 119/117, 95/93, 87/85, 124/119, 85/81, 101/95, 100/93, 85/78, 93/85, 119/108, 93/82, 81/70, 138/119, 136/117, 99/85, 117/100, 95/81, 119/101, 101/85, 81/68, 140/117, 119/99, 117/95, 85/69, 100/81, 108/85, 119/93, 85/66, 156/119, 93/70, 162/119, 93/68, 119/87, 85/62, 117/85, 140/101, 164/117, 170/121

(and their octave complements.)

and their octave complements.

Of them, only 6 interval pairs (119/117, 85/81, 93/85, 101/85, 119/93, 117/85) are more than 10% inconsistent, which is to say, all 36 of the other inconsistent intervals have less than 60% of a step of [[311edo]] of error relative to where they are mapped in [[311edo]] by the patent val, which is to say less than 3/5 = 60% [[relative interval error]]~~; specifically~~, ~~strictly less than 1200{{cent}}/311*3/5 =~~ 2.3{{cent}}.

Of them, only 6 interval pairs (119/117, 85/81, 93/85, 101/85, 119/93, 117/85) are more than 10% inconsistent, which is to say, all 36 of the other inconsistent intervals have less than 60% of a step of 311edo of error relative to where they are mapped in 311edo by the patent val, which is to say less than 3/5 = 60% [[relative interval error]], which is equal to 2.3{{cent}}.

The 6 highest-error intervals mentioned instead have less than 2/3 = 66% [[relative interval error]].

The 6 highest-error intervals mentioned instead have less than 2/3 = 67% [[relative interval error]].

The below table was generated by ~~writing~~ a simple Python 3 script to print it in plaintext using [[User:Godtone#~~My_Python_3_code~~|~~this~~ code]] to simplify certain steps.

The below table was generated by a simple Python 3 script to print it in plaintext using [[User: Godtone #My Python 3 code|Godtone's code]] to simplify certain steps.

It should be noted that while almost all intervals shown in the table are intervals of the 123-odd-limit restricted to the aforementioned prime subgroup, the [[square-particular]]s up to [[1681/1680|S41 = (41/40)/(42/41)]] were added manually for completeness and reference in understanding the mapping of the [[41-odd-limit]] by [[311edo]]. Therefore, the very beginning of the table (from 0\311 to 3\311 inclusive) is the only part that is not algorithmically generated.

It should be noted that while almost all intervals shown in the table are intervals of the 123-odd-limit restricted to the aforementioned prime subgroup, the [[square-particular]]s up to [[1681/1680|S41 = (41/40)/(42/41)]] were added manually for completeness and reference in understanding the mapping of the [[41-odd-limit]] by 311edo. Therefore, the very beginning of the table (from 0\311 to 3\311 inclusive) is the only part that is not algorithmically generated.

{| class="mw-collapsible mw-collapsed wikitable center-1 center-2 center-3"

|+ style=white-space:nowrap | Table of 311edo intervals

~~| style="text-align:center;" | genes<ref group="">''genes'' is named after [[Gene Ward Smith]]; as documented [http://www.tonalsoft.com/enc/g/gene.aspx here]</ref>~~

! Genes*

~~| style="text-align:center;" | cents~~

! Cents

~~| style="text-align:center;" | marks~~

! Marks

~~| style="text-align:left;" | approximate intervals (/2^n and 2^n/ in bold & linked, inconsistent in italics, else, if [[23-limit]], linked)~~

! Approximate Intervals†

|-

| 0

Line 1,615:

| '''2/1'''

|}

<~~references group=""~~ />

<nowiki>*</nowiki> ''gene'' is the [[interval size measure]] for 311edo, named after [[Gene Ward Smith]]<br>

† odd harmonics and subharmonics are in bold and linked, inconsistent intervals in italics, all [[23-limit]] intervals linked)

== Notation ==

@@ Line 22: / Line 22: @@
 The 41-limit add-73 add-89 add-101 add-109 add-113 123-odd-limit is represented very close to completely consistently, and as aforementioned, the 77-odd-limit subset of that odd-limit is perfectly consistent, to which a variety of odds can be added that keep perfect consistency, but for comprehensiveness and practical use as a temperament approximating the low-to-mid end of the harmonic series, we consider a larger odd-limit than that which seeks to be more complete.
-There is 884 interval pairs in that [[odd-limit]] (the [[41-limit]] add-73 add-89 add-101 add-109 add-113 123-odd-limit), where "pairs" refers to that each interval has an [[octave complement]] with equal and opposite error.
+There are 884 interval pairs in that [[odd-limit]] (the [[41-limit]] add-73 add-89 add-101 add-109 add-113 123-odd-limit), where "pairs" refers to that each interval has an [[octave complement]] with equal and opposite error.
-That odd-limit can be described explicitly as the...
+That odd limit can be described explicitly as the
-{1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 45, 49, 51, 55, 57, 63, 65, 69, 73, 75, 77, 81, 85, 87, 89, 91, 93, 95, 99, 101, 105, 109, 111, 113, 115, 117, 119, 121, 123}-odd-limit,
+{1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 45, 49, 51, 55, 57, 63, 65, 69, 73, 75, 77, 81, 85, 87, 89, 91, 93, 95, 99, 101, 105, 109, 111, 113, 115, 117, 119, 121, 123}-odd-limit
-...or equivalently, as the [[diamond function]] applied to that set of odds.
+or equivalently, as the [[diamond function]] applied to that set of odds.
 We can also express that odd-limit as the 123-odd-limit minus only the following twelve prime odds: {43, 47, 53, 59, 61, 67, 71, 79, 83, 97, 103, 107}.
-Of those 884 interval pairs, only 42 interval pairs (<4.8%) are inconsistent (meaning not mapped to the nearest interval of [[311edo]] but to the second-nearest interval).
+Of those 884 interval pairs, only 42 interval pairs (< 4.8%) are inconsistent, not mapped to the nearest interval of 311edo but to the second-nearest interval.
-Preferring the interval of the pair that is less than 600{{cent}} = 1\2, these intervals, from smallest to largest, are:
+Reduced to the lower half of the octave, these intervals, from smallest to largest, are:
 /100, 100/99, 82/81, 121/119, 119/117, 95/93, 87/85, 124/119, 85/81, 101/95, 100/93, 85/78, 93/85, 119/108, 93/82, 81/70, 138/119, 136/117, 99/85, 117/100, 95/81, 119/101, 101/85, 81/68, 140/117, 119/99, 117/95, 85/69, 100/81, 108/85, 119/93, 85/66, 156/119, 93/70, 162/119, 93/68, 119/87, 85/62, 117/85, 140/101, 164/117, 170/121
-(and their octave complements.)
+and their octave complements.
-Of them, only 6 interval pairs (119/117, 85/81, 93/85, 101/85, 119/93, 117/85) are more than 10% inconsistent, which is to say, all 36 of the other inconsistent intervals have less than 60% of a step of [[311edo]] of error relative to where they are mapped in [[311edo]] by the patent val, which is to say less than 3/5 = 60% [[relative interval error]]; specifically, strictly less than 1200{{cent}}/311*3/5 = 2.3{{cent}}.
+Of them, only 6 interval pairs (119/117, 85/81, 93/85, 101/85, 119/93, 117/85) are more than 10% inconsistent, which is to say, all 36 of the other inconsistent intervals have less than 60% of a step of 311edo of error relative to where they are mapped in 311edo by the patent val, which is to say less than 3/5 = 60% [[relative interval error]], which is equal to 2.3{{cent}}.
-The 6 highest-error intervals mentioned instead have less than 2/3 = 66% [[relative interval error]].
+The 6 highest-error intervals mentioned instead have less than 2/3 = 67% [[relative interval error]].
-The below table was generated by writing a simple Python 3 script to print it in plaintext using [[User:Godtone#My_Python_3_code|this code]] to simplify certain steps.
+The below table was generated by a simple Python 3 script to print it in plaintext using [[User: Godtone #My Python 3 code|Godtone's code]] to simplify certain steps.
-It should be noted that while almost all intervals shown in the table are intervals of the 123-odd-limit restricted to the aforementioned prime subgroup, the [[square-particular]]s up to [[1681/1680|S41 = (41/40)/(42/41)]] were added manually for completeness and reference in understanding the mapping of the [[41-odd-limit]] by [[311edo]]. Therefore, the very beginning of the table (from 0\311 to 3\311 inclusive) is the only part that is not algorithmically generated.
+It should be noted that while almost all intervals shown in the table are intervals of the 123-odd-limit restricted to the aforementioned prime subgroup, the [[square-particular]]s up to [[1681/1680|S41 = (41/40)/(42/41)]] were added manually for completeness and reference in understanding the mapping of the [[41-odd-limit]] by 311edo. Therefore, the very beginning of the table (from 0\311 to 3\311 inclusive) is the only part that is not algorithmically generated.
 {| class="mw-collapsible mw-collapsed wikitable center-1 center-2 center-3"
 |+ style=white-space:nowrap | Table of 311edo intervals
-| style="text-align:center;" | genes<ref group="">''genes'' is named after [[Gene Ward Smith]]; as documented [http://www.tonalsoft.com/enc/g/gene.aspx here]</ref>
+! Genes*
-| style="text-align:center;" | cents
+! Cents
-| style="text-align:center;" | marks
+! Marks
-| style="text-align:left;" | approximate intervals (/2^n and 2^n/ in bold & linked, inconsistent in italics, else, if [[23-limit]], linked)
+! Approximate Intervals†
 |-
 | 0
@@ Line 1,615: / Line 1,615: @@
 | '''2/1'''
 |}
-<references group="" />
+<nowiki>*</nowiki> ''gene'' is the [[interval size measure]] for 311edo, named after [[Gene Ward Smith]]<br>
+† odd harmonics and subharmonics are in bold and linked, inconsistent intervals in italics, all [[23-limit]] intervals linked)
 == Notation ==