User:R-4981/Pepsi - Revision history

Lériendil: Lériendil moved page Pepsi to User:R-4981/Pepsi: seems niche enough

2025-03-31T23:13:32Z

Lériendil moved page Pepsi to User:R-4981/Pepsi: seems niche enough

← Older revision	Revision as of 23:13, 31 March 2025
(No difference)

Godtone: /* Interval chain */ missing plural

2024-06-04T15:09:44Z

Interval chain: missing plural

← Older revision		Revision as of 15:09, 4 June 2024
Line 14:		Line 14:

	== Interval chain ==		== Interval chain ==
	In this table, the intervals are [[octave-reduced]]. It is up to the user whether they want to use this reduced version of the scale as an octave-repeating scale, or whether they want to use the non-octave version of this scale (in which case one must keep in mind that the octave-reductions shown are only to help simplify analysis). ~~Ratio~~ given in the below rows are approximated by the corresponding pitch of the tuning (or are exact in the few cases without a tilde ('''~''')). The ratios are shown in order of size, so that the most plausible interpretations tend to be near the middle, while alternative interpretations that may harmonize better in various contexts are shown above and below.		In this table, the intervals are [[octave-reduced]]. It is up to the user whether they want to use this reduced version of the scale as an octave-repeating scale, or whether they want to use the non-octave version of this scale (in which case one must keep in mind that the octave-reductions shown are only to help simplify analysis). Ratios given in the below rows are approximated by the corresponding pitch of the tuning (or are exact in the few cases without a tilde ('''~''')). The ratios are shown in order of size, so that the most plausible interpretations tend to be near the middle, while alternative interpretations that may harmonize better in various contexts are shown above and below.

	Note that [[3/2]], [[27/8]] and (3/2)<sup>9</sup> = 19683/512 (which when octave-reduced is [[19683/16384]]) are tuned perfectly, hence the otherwise-surprising inclusion of the last ratio (although for practically all purposes it is more useful to think of it as an approximation of 6/5). (This is because 3<sup>2</sup> * ~701.955{{cent}} = (3/2)<sup>9</sup>.)		Note that [[3/2]], [[27/8]] and (3/2)<sup>9</sup> = 19683/512 (which when octave-reduced is [[19683/16384]]) are tuned perfectly, hence the otherwise-surprising inclusion of the last ratio (although for practically all purposes it is more useful to think of it as an approximation of 6/5). (This is because 3<sup>2</sup> * ~701.955{{cent}} = (3/2)<sup>9</sup>.)

Godtone: /* Interval chain */ correct & clarify

2024-06-04T03:08:30Z

Interval chain: correct & clarify

← Older revision		Revision as of 03:08, 4 June 2024
Line 16:		Line 16:
	In this table, the intervals are [[octave-reduced]]. It is up to the user whether they want to use this reduced version of the scale as an octave-repeating scale, or whether they want to use the non-octave version of this scale (in which case one must keep in mind that the octave-reductions shown are only to help simplify analysis). Ratio given in the below rows are approximated by the corresponding pitch of the tuning (or are exact in the few cases without a tilde ('''~''')). The ratios are shown in order of size, so that the most plausible interpretations tend to be near the middle, while alternative interpretations that may harmonize better in various contexts are shown above and below.		In this table, the intervals are [[octave-reduced]]. It is up to the user whether they want to use this reduced version of the scale as an octave-repeating scale, or whether they want to use the non-octave version of this scale (in which case one must keep in mind that the octave-reductions shown are only to help simplify analysis). Ratio given in the below rows are approximated by the corresponding pitch of the tuning (or are exact in the few cases without a tilde ('''~''')). The ratios are shown in order of size, so that the most plausible interpretations tend to be near the middle, while alternative interpretations that may harmonize better in various contexts are shown above and below.

	Note that 3/2, 27/8 and (3/2)<sup>9</sup> = 19683/16384 are tuned perfectly, hence the otherwise-surprising inclusion of the last ratio (although for practically all purposes it is more useful to think of it as an approximation of 6/5).		Note that [[3/2]], [[27/8]] and (3/2)<sup>9</sup> = 19683/512 (which when octave-reduced is [[19683/16384]]) are tuned perfectly, hence the otherwise-surprising inclusion of the last ratio (although for practically all purposes it is more useful to think of it as an approximation of 6/5). (This is because 3<sup>2</sup> * ~701.955{{cent}} = (3/2)<sup>9</sup>.)
	{\| class="wikitable"		{\| class="wikitable"
	\|-		\|-

Godtone: /* Interval chain */ remove tildes and instead convey the info of three intervals being exact just before the table, as was before and to make table less wide

2024-06-04T03:05:22Z

Interval chain: remove tildes and instead convey the info of three intervals being exact just before the table, as was before and to make table less wide

← Older revision		Revision as of 03:05, 4 June 2024
Line 15:		Line 15:
	== Interval chain ==		== Interval chain ==
	In this table, the intervals are [[octave-reduced]]. It is up to the user whether they want to use this reduced version of the scale as an octave-repeating scale, or whether they want to use the non-octave version of this scale (in which case one must keep in mind that the octave-reductions shown are only to help simplify analysis). Ratio given in the below rows are approximated by the corresponding pitch of the tuning (or are exact in the few cases without a tilde ('''~''')). The ratios are shown in order of size, so that the most plausible interpretations tend to be near the middle, while alternative interpretations that may harmonize better in various contexts are shown above and below.		In this table, the intervals are [[octave-reduced]]. It is up to the user whether they want to use this reduced version of the scale as an octave-repeating scale, or whether they want to use the non-octave version of this scale (in which case one must keep in mind that the octave-reductions shown are only to help simplify analysis). Ratio given in the below rows are approximated by the corresponding pitch of the tuning (or are exact in the few cases without a tilde ('''~''')). The ratios are shown in order of size, so that the most plausible interpretations tend to be near the middle, while alternative interpretations that may harmonize better in various contexts are shown above and below.

			Note that 3/2, 27/8 and (3/2)<sup>9</sup> = 19683/16384 are tuned perfectly, hence the otherwise-surprising inclusion of the last ratio (although for practically all purposes it is more useful to think of it as an approximation of 6/5).
	{\| class="wikitable"		{\| class="wikitable"
	\|-		\|-
Line 38:		Line 40:
	\|-		\|-
	\| [[3/2]]		\| [[3/2]]
	\| ~[[100/99]]		\| [[100/99]]
	\| ~[[76/45]]		\| [[76/45]]
	\| ~[[36/35]]		\| [[36/35]]
	\| ~[[6/5]]		\| [[6/5]]
	\| ~[[12/11]]		\| [[12/11]]
	\| ~[[45/26]]		\| [[45/26]]
	\| ~[[9/7]]		\| [[9/7]]
	\| ~[[13/10]]		\| [[13/10]]
	\| ~[[21/20]]		\| [[21/20]]
	\| ~[[10/9]]		\| [[10/9]]
	\| ~[[15/13]]		\| [[15/13]]
	\| ~[[15/11]]		\| [[15/11]]
	\| ~[[49/32]]		\| [[49/32]]
	\| ~[[56/45]]		\| [[56/45]]
	\| ~[[25/14]]		\| [[25/14]]
	\| ~[[25/13]]		\| [[25/13]]
	\| ~[[64/45]]		\| [[64/45]]
	\|-		\|-
	\| [[3/2]]		\| [[3/2]]
	\| ~[[105/104]]		\| [[105/104]]
	\| [[27/16]]		\| [[27/16]]
	\| ~[[39/38]]		\| [[39/38]]
	\| ~[[6/5]]		\| [[6/5]]
	\| ~[[38/35]]		\| [[38/35]]
	\| ~[[26/15]]		\| [[26/15]]
	\| ~[[32/25]]		\| [[32/25]]
	\| ~[[43/33]]		\| [[43/33]]
	\| ~[[43/41]]		\| [[43/41]]
	\| ~[[31/28]]		\| [[31/28]]
	\| ~[[38/33]]		\| [[38/33]]
	\| ~[[19/14]]		\| [[19/14]]
	\| ~[[55/36]]		\| [[55/36]]
	\| ~[[41/33]]		\| [[41/33]]
	\| ~[[57/32]]		\| [[57/32]]
	\| ~[[48/25]]		\| [[48/25]]
	\| ~[[27/19]]		\| [[27/19]]
	\| ...		\| ...
	\|-		\|-
	\| [[3/2]]		\| [[3/2]]
	\| ~[[121/120]]		\| [[121/120]]
	\| ~[[32/19]]		\| [[32/19]]
	\| ~[[40/39]]		\| [[40/39]]
	\| [[19683/16384]]		\| [[19683/16384]]
	\| ~[[13/12]]		\| [[13/12]]
	\| ~[[19/11]]		\| [[19/11]]
	\| ~[[14/11]]		\| [[14/11]]
	\| ~[[56/43]]		\| [[56/43]]
	\| ~[[22/21]]		\| [[22/21]]
	\| ~[[21/19]]		\| [[21/19]]
	\| ~[[38/33]]		\| [[38/33]]
	\| ~[[35/26]]		\| [[35/26]]
	\| ~[[50/33]]		\| [[50/33]]
	\| ~[[31/25]]		\| [[31/25]]
	\| ~[[16/9]]		\| [[16/9]]
	\| ~[[21/11]]		\| [[21/11]]
	\| ~[[78/55]]		\| [[78/55]]
	\| ...		\| ...
	\|}		\|}

	[[Category:Tuning]]		[[Category:Tuning]]

Godtone: /* Interval chain */ per request of the designer of the scale, to add more interpretations and to fix implausible ones, and to fix the page generally

2024-06-04T03:02:57Z

Interval chain: per request of the designer of the scale, to add more interpretations and to fix implausible ones, and to fix the page generally

← Older revision		Revision as of 03:02, 4 June 2024
Line 14:		Line 14:

	== Interval chain ==		== Interval chain ==
	In this table, the intervals are [[octave-reduced]]. ~~Except 3/2 which~~ is ~~just by construction~~, ~~every ratio~~ given in the ~~second row is~~ approximated by the corresponding pitch of the tuning.		In this table, the intervals are [[octave-reduced]]. It is up to the user whether they want to use this reduced version of the scale as an octave-repeating scale, or whether they want to use the non-octave version of this scale (in which case one must keep in mind that the octave-reductions shown are only to help simplify analysis). Ratio given in the below rows are approximated by the corresponding pitch of the tuning (or are exact in the few cases without a tilde ('''~''')). The ratios are shown in order of size, so that the most plausible interpretations tend to be near the middle, while alternative interpretations that may harmonize better in various contexts are shown above and below.
	{\| class="wikitable"		{\| class="wikitable"
	\|-		\|-
Line 38:		Line 38:
	\|-		\|-
	\| [[3/2]]		\| [[3/2]]
	\| [[64/63]]		\| ~[[100/99]]
			\| ~[[76/45]]
			\| ~[[36/35]]
			\| ~[[6/5]]
			\| ~[[12/11]]
			\| ~[[45/26]]
			\| ~[[9/7]]
			\| ~[[13/10]]
			\| ~[[21/20]]
			\| ~[[10/9]]
			\| ~[[15/13]]
			\| ~[[15/11]]
			\| ~[[49/32]]
			\| ~[[56/45]]
			\| ~[[25/14]]
			\| ~[[25/13]]
			\| ~[[64/45]]
			\|-
			\| [[3/2]]
			\| ~[[105/104]]
	\| [[27/16]]		\| [[27/16]]
	\| [[36/35]]		\| ~[[39/38]]
	\| [[6/5]]		\| ~[[6/5]]
	\| [[13/12]]		\| ~[[38/35]]
	\| [[19/11]]		\| ~[[26/15]]
	\| [[9/7]]		\| ~[[32/25]]
	\| [[13/10]]		\| ~[[43/33]]
	\| [[21~~/20~~]]		\| ~[[43/41]]
	\| [[35/32]]		\| ~[[31/28]]
	\| [[15/13]]		\| ~[[38/33]]
	\| [[35/26]]		\| ~[[19/14]]
	\| [[50/33]]		\| ~[[55/36]]
	\| [[56/45]]		\| ~[[41/33]]
	\| [[9/5]]		\| ~[[57/32]]
	\| [[21/11]]		\| ~[[48/25]]
	\| [[45/32]]		\| ~[[27/19]]
			\| ...
			\|-
			\| [[3/2]]
			\| ~[[121/120]]
			\| ~[[32/19]]
			\| ~[[40/39]]
			\| [[19683/16384]]
			\| ~[[13/12]]
			\| ~[[19/11]]
			\| ~[[14/11]]
			\| ~[[56/43]]
			\| ~[[22/21]]
			\| ~[[21/19]]
			\| ~[[38/33]]
			\| ~[[35/26]]
			\| ~[[50/33]]
			\| ~[[31/25]]
			\| ~[[16/9]]
			\| ~[[21/11]]
			\| ~[[78/55]]
	\| ...		\| ...
	\|}		\|}

	[[Category:Tuning]]		[[Category:Tuning]]

Fredg999: Relate tuning with existing concepts, cleanup

2024-01-05T05:55:42Z

Relate tuning with existing concepts, cleanup

← Older revision		Revision as of 05:55, 5 January 2024
Line 1:		Line 1:
	[[File:Pepsi Plot.png\|thumb\|A graphical expression of the interval chain of the Pepsi.]]		[[File:Pepsi Plot.png\|thumb\|A graphical expression of the interval chain of the Pepsi.]]

	The '''Pepsi'''{{idiosyncratic}}~~, one of "irregular" temperament,~~ is ~~constructed by~~ the ~~repetition of power of √3 starting from~~ [[3/2]] (701.955¢)~~. It can also be regarded as a~~ [[~~Regular temperament~~]] ~~on the logarithmic axis, i.e., a sequence of intervals with double exponentiational increment (<math>702\cdot3^{n/2}</math>¢)~~.		The '''Pepsi'''{{idiosyncratic}} (name proposed by [[User:R-4981\|R-4981]]) [[tuning system]] is the [[geometric pitch sequence]] (GPS) where the initial interval is [[3/2]] (701.955¢) and each subsequent interval has its pitch [[Interval size measure\|measure]] multiplied by √3.

	~~at first glance this temperament exhibits characteristics similar to [[Redbull]],~~ the ~~direction~~ of ~~its potential use value is fundamentally different because of the different construction methods. Also~~, ~~since the double index is not a tetration~~ (~~obvious~~)~~, this scale cannot be expressed~~ in [[~~EDSO~~]] ~~or [[Super-pitch]]ies.~~		The formula for the ''n''th interval of this tuning, where ''p''(''n'') is pitch in [[cent]]s, is:

	~~== Interval chains ==~~		:<math>p(n) \approx 701.955~¢ \cdot 3^{n/2}</math>,

			or in its exact form:

			:<math>p(n) = \left(1200 \log_2 \left(\frac{3}{2}\right) \right)¢ \cdot 3^{n/2}</math>.

			At first glance, this tuning exhibits characteristics similar to [[Redbull]], namely its common use of √3, but the direction of its potential use value is fundamentally different because of the different construction methods. Also, since the double index is not a tetration (obvious), this scale cannot be expressed in [[EDSO]] or [[super-pitch]].

			== Interval chain ==
			In this table, the intervals are [[octave-reduced]]. Except 3/2 which is just by construction, every ratio given in the second row is approximated by the corresponding pitch of the tuning.
	{\| class="wikitable"		{\| class="wikitable"
	\|-		\|-
Line 50:		Line 58:
	\|}		\|}

	~~== See also ==~~		[[Category:Tuning]]
	* [[Redbull]]
	* [[Super-pitch]]
	* [[EDSO]]

	~~[[Category:Temperaments]]~~
	[[Category:~~Pages with Scala files~~]]
	~~{{Stub}}~~

R-4981 at 05:05, 5 January 2024

2024-01-05T05:05:13Z

← Older revision		Revision as of 05:05, 5 January 2024
Line 1:		Line 1:
	[[File:Pepsi Plot.png\|thumb\|A graphical expression of the interval chain of the Pepsi.]]		[[File:Pepsi Plot.png\|thumb\|A graphical expression of the interval chain of the Pepsi.]]

	The '''Pepsi'''{{idiosyncratic}}, one of "irregular" temperament, is constructed by the repetition of power of √3 starting from [[3/2]] (701.955¢). It can also be regarded as a [[Regular temperament]] on the logarithmic axis, i.e., a sequence of intervals with double exponentiational increment (<math>702\~~cdot2^{3~~^{n/2}}</math>¢).		The '''Pepsi'''{{idiosyncratic}}, one of "irregular" temperament, is constructed by the repetition of power of √3 starting from [[3/2]] (701.955¢). It can also be regarded as a [[Regular temperament]] on the logarithmic axis, i.e., a sequence of intervals with double exponentiational increment (<math>702\cdot3^{n/2}</math>¢).

	at first glance this temperament exhibits characteristics similar to [[Redbull]], the direction of its potential use value is fundamentally different because of the different construction methods. Also, since the double index is not a tetration (obvious), this scale cannot be expressed in [[EDSO]] or [[Super-pitch]]ies.		at first glance this temperament exhibits characteristics similar to [[Redbull]], the direction of its potential use value is fundamentally different because of the different construction methods. Also, since the double index is not a tetration (obvious), this scale cannot be expressed in [[EDSO]] or [[Super-pitch]]ies.

R-4981: Do you want to sell sugared water for the rest of your life, or do you want to come with me and change the music theory?

2024-01-05T05:04:13Z

Do you want to sell sugared water for the rest of your life, or do you want to come with me and change the music theory?

New page

[[File:Pepsi Plot.png|thumb|A graphical expression of the interval chain of the Pepsi.]]

The '''Pepsi'''{{idiosyncratic}}, one of "irregular" temperament, is constructed by the repetition of power of √3 starting from [[3/2]] (701.955¢). It can also be regarded as a [[Regular temperament]] on the logarithmic axis, i.e., a sequence of intervals with double exponentiational increment (<math>702\cdot2^{3^{n/2}}</math>¢).

at first glance this temperament exhibits characteristics similar to [[Redbull]], the direction of its potential use value is fundamentally different because of the different construction methods. Also, since the double index is not a tetration (obvious), this scale cannot be expressed in [[EDSO]] or [[Super-pitch]]ies.

== Interval chains ==

{| class="wikitable"
|-
| 701.955
| 15.822
| 905.865
| 47.465
| 317.595
| 142.396
| 952.785
| 427.187
| 458.355
| 81.56
| 175.065
| 244.679
| 525.196
| 734.038
| 375.587
| 1002.115
| 1126.761
| 606.345
| ...
|-
| [[3/2]]
| [[64/63]]
| [[27/16]]
| [[36/35]]
| [[6/5]]
| [[13/12]]
| [[19/11]]
| [[9/7]]
| [[13/10]]
| [[21/20]]
| [[35/32]]
| [[15/13]]
| [[35/26]]
| [[50/33]]
| [[56/45]]
| [[9/5]]
| [[21/11]]
| [[45/32]]
| ...
|}

== See also ==
* [[Redbull]]
* [[Super-pitch]]
* [[EDSO]]

[[Category:Temperaments]]
[[Category:Pages with Scala files]]
{{Stub}}