Bug and beep: Difference between revisions

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'''~~Beep~~''' is a remarkable low-complexity, though high-badness [[7-limit]] [[~~temperament~~]]~~. It is also~~ called '''titanium''' by [[Mason Green]]. ~~It [[tempering out|~~tempers out]] the [[21/20|septimal minor semitone (21/20)]], making it a [[septisemi temperaments|septisemi temperament]], and the [[49/48|slendro diesis (49/48)]], making it part of the [[~~slendro~~ clan]]. As such, ~~[[6/5]],~~ [[7/6]], and [[8/7]] ~~are all represented by the same interval (which, in fact, is the [[generator]])~~. Two of these generators make a very ~~sharp~~ [[4/3|~~fourth~~]] (which is also a very ~~flat~~ [[7/5]]). Since three fifths make a minor ~~(not major) sixth~~, and four make a minor ~~(not major) third~~, it is also similar to the [[~~pelogic~~]] temperament ([[superpelog]] in particular~~). It also tempers out [[27/25]] and is part of the [[bug family]]; because of this, the generator also functions as [[10/9]] (meaning this tuning has a ''negative'' syntonic comma~~). Finally it can be also considered a sort of messed-up variant of [[orwell]] temperament as well, since the generator falls into the same range of sizes.

{{Infobox regtemp

| Title = Bug / Beep

| Subgroups = 2.3.5, 2.3.5.7

| Comma basis = [[27/25]] (2.3.5); <br>[[21/20]], [[27/25]] (2.3.5.7)

| Edo join 1 = 4 | Edo join 2 = 5

| Mapping = 1; 2 3 1

| Generators = 5/3

| Generators tuning = 938

| Optimization method = CWE

| MOS scales = [[1L 3s]], [[4L 1s]], [[5L 4s]]

| Pergen = (P8, P4/2)

| Color name = Guguti

| Odd limit 1 = 7 | Mistuning 1 = 44.4 | Complexity 1 = 4

| Odd limit 2 = 9 | Mistuning 2 = 72.6 | Complexity 2 = 5

}}

'''Bug''' is a remarkable [[complexity|low-complexity]], though [[badness|high-badness]] [[exotemperament]]. It [[tempering out|tempers out]] [[27/25]] and is part of the [[bug family]]; as such, [[5/3]] and [[9/5]] are represented by the same interval which, in fact, is the [[generator]]. With the [[7-limit]] [[extension]] called '''beep'''<ref group="note">Aliased ''titanium'' by [[Mason Green]]. </ref>, it tempers out the [[21/20|septimal minor semitone (21/20)]], making it a [[septisemi temperaments|septisemi temperament]], and the [[49/48|slendro diesis (49/48)]], making it part of the [[semaphoresmic clan]]. As such, the generator also represents [[7/4]] and [[12/7]]. Two of these generators make a very flat [[3/2|fifth]] which is also a very sharp [[10/7]]. Two fifths make a second that is neutral in [[interval quality|quality]], so a good tuning typically has a ''negative'' [[syntonic comma]]. Since three fifths make a sixth that sounds minor, and four make a third that sounds minor, it is also similar to the [[mavila]] temperament ([[superpelog]] in particular). Finally it can be also considered a sort of messed-up variant of [[orwell]] temperament as well, since the generator falls into the same range of sizes.

The edos whose [[patent val]]s support beep are [[4edo]], [[5edo]], [[9edo]], and [[14edo]]. Many other edos can be used as non-patent vals, such as 13.

As a scale archetype, bug represents [[4L 1s]] and [[5L 4s]], and more generally [[Ploidacot/Alpha-dicot|alpha-dicot]].

The name ''beep'' was attested as early as at least January, 2004<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_8809.html Yahoo! Tuning Group | ''114 7-limit temperaments'']</ref>, and is likely a reference to the smallest step of the [[Bohlen–Pierce]] scale.

See [[Bug family #Bug]] and [[Bug family #Beep|#Beep]] for technical data.

== Interval chain ==

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! #

! Cents*

! Approximate ~~Ratios~~

! Approximate ratios

|-

| 0

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| 15/8

|}

<nowiki>*~~</nowiki> octave reduced, in~~ 7-limit CTE tuning

<nowiki/>* In 7-limit CTE tuning, octave reduced

== Chords and harmony ==

In beep, the 7-limit tetrad has very low [[Graham complexity]] (only 3). The fact that [[7/5]] is also [[4/3]] allows a type of "tritone substitution" distinct from that which appears in [[jubilismic]] temperaments; namely, one in which the 4/3 of one chord becomes the 7/5 of the next or vice versa. This is equivalent to modulating upward or downward by a generator. The more usual type of modulation (upward or downward by fifths) is also easy since two generators make a fifth. These tetrads, despite being relatively inaccurate, are still easily recognizable and not necessarily unpleasant-sounding (though of course it depends on the timbre of the instrument they are played on). Another thing to watch out for is that due to tempering, the tetrads are of the form Lsss, which means they are their own inverses (i.e. the utonal and otonal tetrads are the same). The 4:5:6:7 and 5:6:7:9 tetrads are also the same. This gives them a chameleonic quality, akin to power chords; they can sound either major or minor, which strongly depends on the voicing used. The terms major and minor can still be used to refer to ''inversions'' of the basic tetrad (the major inversion being Lsss, and the minor inversion being sLss).

In beep, the 7-limit tetrad has very low [[Graham complexity]] (only 3). The fact that [[7/5]] is also [[4/3]] allows a type of "tritone substitution" distinct from that which appears in [[Jubilismic_clan|jubilismic]] temperaments; namely, one in which the 4/3 of one chord becomes the 7/5 of the next or vice versa. This is equivalent to modulating upward or downward by a generator. The more usual type of modulation (upward or downward by fifths) is also easy since two generators make a fifth. These tetrads, despite being relatively inaccurate, are still easily recognizable and not necessarily unpleasant-sounding (though of course it depends on the timbre of the instrument they are played on). Another thing to watch out for is that due to tempering, the tetrads are of the form Lsss, which means they are their own inverses (i.e. the utonal and otonal tetrads are the same). The 4:5:6:7 and 5:6:7:9 tetrads are also the same. This gives them a chameleonic quality, akin to power chords; they can sound either major or minor, which strongly depends on the voicing used. The terms major and minor can still be used to refer to ''inversions'' of the basic tetrad (the major inversion being Lsss, and the minor inversion being sLss).

Beep forms enneatonic scales which may be either of the form LLsLsLsLs or ssLsLsLsL; where both step sizes are equal this simply gives [[9edo]]. An enneatonic scale has six tetrads, and just like the diatonic scale it allows a variant of the familiar "I-IV-V" type of chord progression (which, using enneatonic notation, would be I-V-VI). Beep's enneatonic scales provide an alternative to the Erlich decatonic; they are lower in accuracy, but are also simpler and offer more freedom of modulation, and thus for some listeners they might actually be less xenharmonic.

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|-

! Cents

! Generator ~~Steps~~

! Generator steps

|-

| 0

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|-

| 116

| -4

| −4

|-

| 271

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|-

| 387

| -3

| −3

|-

| 542

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|-

| 658

| -2

| −2

|-

| 813 (or 774)

| 3 (or -6)

| 3 (or −6)

|-

| 929

| -1

| −1

|-

| 1045

| -5

| −5

|}

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=== Suggested timbre ===

If using brittle ~~titanium~~ (23cd, 14edo, etc.), one might want to consider using this as a guideline. With this spectrum, no partials are more than 25 cents above or below their perfectly harmonic values, and when using 14edo, no intervals will be more than 26 cents out of tune. This is only a guideline, and only with synthesized tones would it be possible to achieve this perfectly. With physical idiophones (celestas, gamelans, etc.) it should still be possible to get a great approximation using CAD (or trial and error).

If using brittle beep (23cd, 14edo, etc.), one might want to consider using this as a guideline. With this spectrum, no partials are more than 25 cents above or below their perfectly harmonic values, and when using 14edo, no intervals will be more than 26 cents out of tune. This is only a guideline, and only with synthesized tones would it be possible to achieve this perfectly. With physical idiophones (celestas, gamelans, etc.) it should still be possible to get a great approximation using CAD (or trial and error).

Fundamental (1st): just

* Fundamental (1st): just

* Octave (2nd): just

* 3rd: −12.5

* 4th: just

* 5th: +25

* 6th: −12.5 cents

* 7th: −25 cents

* 8th: just

* 9th: −25 cents

* 10th: +25 cents

* 11th: −25 cents

* 12th: −12.5 cents

* 13th: +15 cents

* 14th: −25 cents

* 15th: +12.5 cents

* 16th: just

~~Octave~~ (~~2nd~~)~~: just~~

== Tunings ==

=== Tuning spectrum ===

~~3rd:~~ -12.5

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

|-

~~4th: just~~

! Edo<br>generator !! Eigenmonzo<br>(unchanged-interval) !! Generator (¢) !! Comments

|-

~~5th: +25~~

| || 5/3 || 884.3 ||

|-

~~6th: -12~~.5 ~~cents~~

| 3\4 || || 900.0 || Lower bound of 7-odd-limit diamond monotone

|-

~~7th:~~ -~~25 cents~~

| || 7/5 || 908.7 ||

|-

~~8th: just~~

| 10\13 || || 923.1 ||

|-

~~9th:~~ -~~25 cents~~

| || 5/4 || 928.8 || 5- and 7-odd-limit minimax

|-

~~10th: +25 cents~~

| || 7/6 || 933.1 ||

|-

~~11th:~~ -~~25 cents~~

| 7\9 || || 933.3 ||

|-

~~12th:~~ -12.~~5 cents~~

| 11\14 || || 942.9 ||

|-

~~13th: +15 cents~~

| || 9/7 || 945.0 || 9-odd-limit minimax

|-

~~14th: -25 cents~~

| || 3/2 || 951.0 ||

|-

| 4\5 || || 960.0 || Upper bound of 7-odd-limit diamond monotone<br>9-odd-limit diamond monotone (singleton)

|-

| || 7/4 || 968.8 ||

|-

| || 9/5 || 1017.6 ||

|}

~~15th: +12.5 cents~~

== Notes ==

~~16th: just~~

== References ==

[[Category:~~Temperaments]]~~

[[Category:Bug| ]]

~~[[Category:Beep~~| ]]

[[Category:Rank-2 temperaments]]

[[Category:Exotemperaments]]

[[Category:Bug family]]

[[Category:~~Slendro~~ clan]]

[[Category:Semaphoresmic clan]]

[[Category:Septisemi temperaments]]

@@ Line 1: / Line 1: @@
-'''Beep''' is a remarkable low-complexity, though high-badness [[7-limit]] [[temperament]]. It is also called '''titanium''' by [[Mason Green]]. It [[tempering out|tempers out]] the [[21/20|septimal minor semitone (21/20)]], making it a [[septisemi temperaments|septisemi temperament]], and the [[49/48|slendro diesis (49/48)]], making it part of the [[slendro clan]]. As such, [[6/5]], [[7/6]], and [[8/7]] are all represented by the same interval (which, in fact, is the [[generator]]). Two of these generators make a very sharp [[4/3|fourth]] (which is also a very flat [[7/5]]). Since three fifths make a minor (not major) sixth, and four make a minor (not major) third, it is also similar to the [[pelogic]] temperament ([[superpelog]] in particular). It also tempers out [[27/25]] and is part of the [[bug family]]; because of this, the generator also functions as [[10/9]] (meaning this tuning has a ''negative'' syntonic comma). Finally it can be also considered a sort of messed-up variant of [[orwell]] temperament as well, since the generator falls into the same range of sizes.
+{{Infobox regtemp
+| Title = Bug / Beep
+| Subgroups = 2.3.5, 2.3.5.7
+| Comma basis = [[27/25]] (2.3.5); <br>[[21/20]], [[27/25]] (2.3.5.7)
+| Edo join 1 = 4 | Edo join 2 = 5
+| Mapping = 1; 2 3 1
+| Generators = 5/3
+| Generators tuning = 938
+| Optimization method = CWE
+| MOS scales = [[1L&nbsp;3s]], [[4L&nbsp;1s]], [[5L&nbsp;4s]]
+| Pergen = (P8, P4/2)
+| Color name = Guguti
+| Odd limit 1 = 7 | Mistuning 1 = 44.4 | Complexity 1 = 4
+| Odd limit 2 = 9 | Mistuning 2 = 72.6 | Complexity 2 = 5
+}}
+'''Bug''' is a remarkable [[complexity|low-complexity]], though [[badness|high-badness]] [[exotemperament]]. It [[tempering out|tempers out]] [[27/25]] and is part of the [[bug family]]; as such, [[5/3]] and [[9/5]] are represented by the same interval which, in fact, is the [[generator]]. With the [[7-limit]] [[extension]] called '''beep'''<ref group="note">Aliased ''titanium'' by [[Mason Green]]. </ref>, it tempers out the [[21/20|septimal minor semitone (21/20)]], making it a [[septisemi temperaments|septisemi temperament]], and the [[49/48|slendro diesis (49/48)]], making it part of the [[semaphoresmic clan]]. As such, the generator also represents [[7/4]] and [[12/7]]. Two of these generators make a very flat [[3/2|fifth]] which is also a very sharp [[10/7]]. Two fifths make a second that is neutral in [[interval quality|quality]], so a good tuning typically has a ''negative'' [[syntonic comma]]. Since three fifths make a sixth that sounds minor, and four make a third that sounds minor, it is also similar to the [[mavila]] temperament ([[superpelog]] in particular). Finally it can be also considered a sort of messed-up variant of [[orwell]] temperament as well, since the generator falls into the same range of sizes.
 The edos whose [[patent val]]s support beep are [[4edo]], [[5edo]], [[9edo]], and [[14edo]]. Many other edos can be used as non-patent vals, such as 13.
+As a scale archetype, bug represents [[4L 1s]] and [[5L 4s]], and more generally [[Ploidacot/Alpha-dicot|alpha-dicot]].
+The name ''beep'' was attested as early as at least January, 2004<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_8809.html Yahoo! Tuning Group | ''114 7-limit temperaments'']</ref>, and is likely a reference to the smallest step of the [[Bohlen–Pierce]] scale.
+See [[Bug family #Bug]] and [[Bug family #Beep|#Beep]] for technical data.
 == Interval chain ==
@@ Line 10: / Line 31: @@
 ! #
 ! Cents*
-! Approximate Ratios
+! Approximate ratios
 |-
 | 0
@@ Line 36: / Line 57: @@
 | 15/8
 |}
-<nowiki>*</nowiki> octave reduced, in 7-limit CTE tuning
+<nowiki/>* In 7-limit CTE tuning, octave reduced
 == Chords and harmony ==
-In beep, the 7-limit tetrad has very low [[Graham complexity]] (only 3). The fact that [[7/5]] is also [[4/3]] allows a type of "tritone substitution" distinct from that which appears in [[jubilismic]] temperaments; namely, one in which the 4/3 of one chord becomes the 7/5 of the next or vice versa. This is equivalent to modulating upward or downward by a generator. The more usual type of modulation (upward or downward by fifths) is also easy since two generators make a fifth. These tetrads, despite being relatively inaccurate, are still easily recognizable and not necessarily unpleasant-sounding (though of course it depends on the timbre of the instrument they are played on). Another thing to watch out for is that due to tempering, the tetrads are of the form Lsss, which means they are their own inverses (i.e. the utonal and otonal tetrads are the same). The 4:5:6:7 and 5:6:7:9 tetrads are also the same. This gives them a chameleonic quality, akin to power chords; they can sound either major or minor, which strongly depends on the voicing used. The terms major and minor can still be used to refer to ''inversions'' of the basic tetrad (the major inversion being Lsss, and the minor inversion being sLss).
+In beep, the 7-limit tetrad has very low [[Graham complexity]] (only 3). The fact that [[7/5]] is also [[4/3]] allows a type of "tritone substitution" distinct from that which appears in [[Jubilismic_clan|jubilismic]] temperaments; namely, one in which the 4/3 of one chord becomes the 7/5 of the next or vice versa. This is equivalent to modulating upward or downward by a generator. The more usual type of modulation (upward or downward by fifths) is also easy since two generators make a fifth. These tetrads, despite being relatively inaccurate, are still easily recognizable and not necessarily unpleasant-sounding (though of course it depends on the timbre of the instrument they are played on). Another thing to watch out for is that due to tempering, the tetrads are of the form Lsss, which means they are their own inverses (i.e. the utonal and otonal tetrads are the same). The 4:5:6:7 and 5:6:7:9 tetrads are also the same. This gives them a chameleonic quality, akin to power chords; they can sound either major or minor, which strongly depends on the voicing used. The terms major and minor can still be used to refer to ''inversions'' of the basic tetrad (the major inversion being Lsss, and the minor inversion being sLss).
 Beep forms enneatonic scales which may be either of the form LLsLsLsLs or ssLsLsLsL; where both step sizes are equal this simply gives [[9edo]]. An enneatonic scale has six tetrads, and just like the diatonic scale it allows a variant of the familiar "I-IV-V" type of chord progression (which, using enneatonic notation, would be I-V-VI). Beep's enneatonic scales provide an alternative to the Erlich decatonic; they are lower in accuracy, but are also simpler and offer more freedom of modulation, and thus for some listeners they might actually be less xenharmonic.
@@ Line 48: / Line 69: @@
 |-
 ! Cents
-! Generator Steps
+! Generator steps
 |-
 | 0
@@ Line 54: / Line 75: @@
 |-
 | 116
-| -4
+| −4
 |-
 | 271
@@ Line 60: / Line 81: @@
 |-
 | 387
-| -3
+| −3
 |-
 | 542
@@ Line 66: / Line 87: @@
 |-
 | 658
-| -2
+| −2
 |-
 | 813 (or 774)
-| 3 (or -6)
+| 3 (or −6)
 |-
 | 929
-| -1
+| −1
 |-
 | 1045
-| -5
+| −5
 |}
@@ Line 87: / Line 108: @@
 === Suggested timbre ===
-If using brittle titanium (23cd, 14edo, etc.), one might want to consider using this as a guideline. With this spectrum, no partials are more than 25 cents above or below their perfectly harmonic values, and when using 14edo, no intervals will be more than 26 cents out of tune. This is only a guideline, and only with synthesized tones would it be possible to achieve this perfectly. With physical idiophones (celestas, gamelans, etc.) it should still be possible to get a great approximation using CAD (or trial and error).
+If using brittle beep (23cd, 14edo, etc.), one might want to consider using this as a guideline. With this spectrum, no partials are more than 25 cents above or below their perfectly harmonic values, and when using 14edo, no intervals will be more than 26 cents out of tune. This is only a guideline, and only with synthesized tones would it be possible to achieve this perfectly. With physical idiophones (celestas, gamelans, etc.) it should still be possible to get a great approximation using CAD (or trial and error).
-Fundamental (1st): just
+* Fundamental (1st): just
+* Octave (2nd): just
+* 3rd: −12.5
+* 4th: just
+* 5th: +25
+* 6th: −12.5 cents
+* 7th: −25 cents
+* 8th: just
+* 9th: −25 cents
+* 10th: +25 cents
+* 11th: −25 cents
+* 12th: −12.5 cents
+* 13th: +15 cents
+* 14th: −25 cents
+* 15th: +12.5 cents
+* 16th: just
-Octave (2nd): just
+== Tunings ==
+=== Tuning spectrum ===
-rd: -12.5
+{| class="wikitable center-all left-4"
+|-
-th: just
+! Edo<br>generator !! Eigenmonzo<br>(unchanged-interval) !! Generator (¢) !! Comments
+|-
-th: +25
+| || 5/3 || 884.3 ||
+|-
-th: -12.5 cents
+| 3\4 ||  || 900.0 || Lower bound of 7-odd-limit diamond monotone
+|-
-th: -25 cents
+|  || 7/5 || 908.7 ||
+|-
-th: just
+| 10\13 ||  || 923.1 ||
+|-
-th: -25 cents
+|  || 5/4 || 928.8 || 5- and 7-odd-limit minimax
+|-
-th: +25 cents
+|  || 7/6 || 933.1 ||
+|-
-th: -25 cents
+| 7\9 ||  || 933.3 ||
+|-
-th: -12.5 cents
+| 11\14 ||  || 942.9 ||
+|-
-th: +15 cents
+| || 9/7 || 945.0 || 9-odd-limit minimax
+|-
-th: -25 cents
+| || 3/2 || 951.0 ||
+|-
+| 4\5 ||  || 960.0 || Upper bound of 7-odd-limit diamond monotone<br>9-odd-limit diamond monotone (singleton)
+|-
+| || 7/4 || 968.8 ||
+|-
+| || 9/5 || 1017.6 ||
+|}
-th: +12.5 cents
+== Notes ==
+<references group="note"/>
-th: just
+== References ==
-[[Category:Temperaments]]
+[[Category:Bug| ]] <!-- main article -->
-[[Category:Beep| ]] <!-- main article -->
+[[Category:Rank-2 temperaments]]
+[[Category:Exotemperaments]]
 [[Category:Bug family]]
-[[Category:Slendro clan]]
+[[Category:Semaphoresmic clan]]
 [[Category:Septisemi temperaments]]