Hemififths: Difference between revisions

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'''Hemififths''' is the [[~~temperament~~]] [[tempering out]] the breedsma, [[2401/2400]], and the hemifamity comma, [[5120/5103]], and as the ~~name suggests, uses a neutral third~~ as ~~a generator~~. '''~~Hemif~~''' is ~~the no-5 subgroup version of hemififths~~. ~~It is supported by~~ [[41edo|41-]], [[58edo|58-]], and [[99edo~~|99et~~]].

'''Hemififths''' is a [[regular temperament|temperament]] that uses a neutral third as a [[generator]], just as the name suggests. A stack of 13 generators represents [[7/4]] and a stack of 25 generators represents [[5/4]], [[tempering out]] the breedsma, [[2401/2400]], and the hemifamity comma, [[5120/5103]]. It extends fairly naturally to the 11- and 13-limit by treating the generator as [[11/9]][[~]][[16/13]]. The no-5 subgroup [[restriction]], called '''hemif''', is also notable. Possible tunings include [[41edo|41-]], [[58edo|58-]], and [[99edo]].

Hemififths was named by [[Gene Ward Smith]] in 2004<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_10541.html Yahoo! Tuning Group (Archive) | ''Names for important high-complexity temperaments'']</ref>.

See [[Breedsmic temperaments #Hemififths]] and [[No-fives subgroup temperaments#Hemif]] for more technical data.

See [[Breedsmic temperaments #Hemififths]] and [[No-fives subgroup temperaments #Hemif]] for more technical data.

== Interval chain ==

In the following table, odd harmonics 1–21 are labeled in '''bold'''.

In the following table, odd harmonics 1–21 and their inversions are labeled in '''bold'''.

{| class="wikitable center-1 right-2"

|-

! rowspan="2" | &#~~35;~~

! rowspan="2" | #

! rowspan="2" | Cents*

! colspan="2" | Approximate ~~Ratios~~

! colspan="2" | Approximate ratios

! rowspan="2" | [[Ups and downs notation|Ups and downs notation]]**

! rowspan="2" | [[Ups and downs notation|Ups and downs notation]]**

|-

! 7-limit

! 13-limit ~~Extension~~

! 13-limit extension

|-

| 0

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| ^A3 = \A4

|}

<nowiki />* In 7-limit CTE tuning, {{nowrap|generator {{=}} 351.445¢,|P5 {{=}} 702.89¢|~~and~~ c {{=}} 2.89¢}}

<nowiki/>* In 7-limit CTE tuning, {{nowrap|generator {{=}} 351.445¢ }}, {{nowrap|P5 {{=}} 702.89¢}} and {{nowrap|c {{=}} 2.89¢}}

<nowiki />** Enharmonic equivalences: vvA1 and v\m2. Cents: {{nowrap|^1 {{=}} 50¢ + 3.5c}} and {{nowrap|/1 {{=}} 50¢ ~~−~~ 8.5c}}

<nowiki/>** Enharmonic equivalences: vvA1 and v\m2. Cents: {{nowrap|^1 {{=}} 50¢ + 3.5c}} and {{nowrap|/1 {{=}} 50¢ − 8.5c}}

== Notation ==

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Below is tabulated how to notate the prime harmonics with an arrow representing a syntonic~septimal comma (thus ^C = Ddb).

{| class="wikitable center-1 center-3"

|+ style="font-size: 105%;" | Hemififths nomenclature for selected intervals

|+ style="font-size: 105%;" | Hemififths nomenclature for selected intervals

|-

! Ratio

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Below is tabulated how to notate the prime harmonics with an arrow representing a Pythagorean comma (thus ^C = B#).

{| class="wikitable center-1 center-3"

|+ style="font-size: 105%;" | Hemififths nomenclature for selected intervals

|+ style="font-size: 105%;" | Hemififths nomenclature for selected intervals

|-

! Ratio

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== Tunings ==

=== Tuning spectrum ===

{| class="wikitable center-all left-4"

|-

! Edo generator

! Edo generator

! [[Eigenmonzo|Eigenmonzo (unchanged-interval)]]*

! [[Eigenmonzo|Eigenmonzo (unchanged-interval)]]*

! Generator (¢)

! Comments

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|

| 351.220

| Lower bound of 11- to 15-odd-limit and 13-limit 21-odd-limit diamond monotone

| Lower bound of 11- to 15-odd-limit and 13-limit 21-odd-limit diamond monotone

|-

|

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| 25/24

| 351.472

| Very close to [[Logarithmic approximants#Argent temperament|argent ~~temperament~~]] with neutral intervals (351.47186 cents)

| Very close to [[Logarithmic approximants #Argent temperament|argent tuning]] with neutral intervals (351.47186 cents)

|-

|

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|

| 352.941

| Upper bound of 7- to 15-odd-limit and 13-limit 21-odd-limit diamond monotone

| Upper bound of 7- to 15-odd-limit and 13-limit 21-odd-limit diamond monotone

|-

|

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|

|}

<nowiki />* Besides the octave

<nowiki/>* Besides the octave

== References ==

[[Category:Temperaments]]

@@ Line 1: / Line 1: @@
-'''Hemififths''' is the [[temperament]] [[tempering out]] the breedsma, [[2401/2400]], and the hemifamity comma, [[5120/5103]], and as the name suggests, uses a neutral third as a generator. '''Hemif''' is the no-5 subgroup version of hemififths. It is supported by [[41edo|41-]], [[58edo|58-]], and [[99edo|99et]].
+'''Hemififths''' is a [[regular temperament|temperament]] that uses a neutral third as a [[generator]], just as the name suggests. A stack of 13 generators represents [[7/4]] and a stack of 25 generators represents [[5/4]], [[tempering out]] the breedsma, [[2401/2400]], and the hemifamity comma, [[5120/5103]]. It extends fairly naturally to the 11- and 13-limit by treating the generator as [[11/9]][[~]][[16/13]]. The no-5 subgroup [[restriction]], called '''hemif''', is also notable. Possible tunings include [[41edo|41-]], [[58edo|58-]], and [[99edo]].
 Hemififths was named by [[Gene Ward Smith]] in 2004<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_10541.html Yahoo! Tuning Group (Archive) | ''Names for important high-complexity temperaments'']</ref>.
-See [[Breedsmic temperaments #Hemififths]] and [[No-fives subgroup temperaments#Hemif]] for more technical data.
+See [[Breedsmic temperaments #Hemififths]] and [[No-fives subgroup temperaments #Hemif]] for more technical data.
 == Interval chain ==
-In the following table, odd harmonics 1–21 are labeled in '''bold'''.
+In the following table, odd harmonics 1–21 and their inversions are labeled in '''bold'''.
 {| class="wikitable center-1 right-2"
 |-
-! rowspan="2" | &#35;
+! rowspan="2" | #
 ! rowspan="2" | Cents*
-! colspan="2" | Approximate Ratios
+! colspan="2" | Approximate ratios
-! rowspan="2" | [[Ups and downs notation|Ups and downs<br />notation]]**
+! rowspan="2" | [[Ups and downs notation|Ups and downs<br>notation]]**
 |-
 ! 7-limit
-! 13-limit Extension
+! 13-limit extension
 |-
 | 0
@@ Line 197: / Line 197: @@
 | ^A3 = \A4
 |}
-<nowiki />* In 7-limit CTE tuning, {{nowrap|generator {{=}} 351.445¢,|P5 {{=}} 702.89¢|and c {{=}} 2.89¢}}
+<nowiki/>* In 7-limit CTE tuning, {{nowrap|generator {{=}} 351.445¢ }}, {{nowrap|P5 {{=}} 702.89¢}} and {{nowrap|c {{=}} 2.89¢}}
-<nowiki />** Enharmonic equivalences: vvA1 and v\m2. Cents: {{nowrap|^1 {{=}} 50¢ + 3.5c}} and {{nowrap|/1 {{=}} 50¢ &minus; 8.5c}}
+<nowiki/>** Enharmonic equivalences: vvA1 and v\m2. Cents: {{nowrap|^1 {{=}} 50¢ + 3.5c}} and {{nowrap|/1 {{=}} 50¢ − 8.5c}}
 == Notation ==
@@ Line 208: / Line 208: @@
 Below is tabulated how to notate the prime harmonics with an arrow representing a syntonic~septimal comma (thus ^C = Ddb).
 {| class="wikitable center-1 center-3"
-|+ style="font-size: 105%;" | Hemififths nomenclature<br />for selected intervals
+|+ style="font-size: 105%;" | Hemififths nomenclature<br>for selected intervals
 |-
 ! Ratio
@@ Line 237: / Line 237: @@
 Below is tabulated how to notate the prime harmonics with an arrow representing a Pythagorean comma (thus ^C = B#).
 {| class="wikitable center-1 center-3"
-|+ style="font-size: 105%;" | Hemififths nomenclature<br />for selected intervals
+|+ style="font-size: 105%;" | Hemififths nomenclature<br>for selected intervals
 |-
 ! Ratio
@@ Line 274: / Line 274: @@
 == Tunings ==
 === Tuning spectrum ===
 {| class="wikitable center-all left-4"
 |-
-! Edo<br />generator
+! Edo<br>generator
-! [[Eigenmonzo|Eigenmonzo<br />(unchanged-interval)]]*
+! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]*
 ! Generator (¢)
 ! Comments
@@ Line 310: / Line 309: @@
 |
 | 351.220
-| Lower bound of 11- to 15-odd-limit<br />and 13-limit 21-odd-limit diamond monotone
+| Lower bound of 11- to 15-odd-limit<br>and 13-limit 21-odd-limit diamond monotone
 |-
 |
@@ Line 350: / Line 349: @@
 | 25/24
 | 351.472
-| Very close to [[Logarithmic approximants#Argent temperament|argent temperament]] with neutral intervals (351.47186 cents)
+| Very close to [[Logarithmic approximants #Argent temperament|argent tuning]] with neutral intervals (351.47186 cents)
 |-
 |
@@ Line 450: / Line 449: @@
 |
 | 352.941
-| Upper bound of 7- to 15-odd-limit<br />and 13-limit 21-odd-limit diamond monotone
+| Upper bound of 7- to 15-odd-limit<br>and 13-limit 21-odd-limit diamond monotone
 |-
 |
@@ Line 462: / Line 461: @@
 |
 |}
-<nowiki />* Besides the octave
+<nowiki/>* Besides the octave
 == References ==
-<references />
+<references/>
 [[Category:Temperaments]]