User:R-4981/Pepsi: Difference between revisions

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 [[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]]. 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>.)
 {| class="wikitable"
 |-
@@ Line 30: / Line 40: @@
 |-
 | [[3/2]]
-| [[64/63]]
+| [[100/99]]
-| [[27/16]]
+| [[76/45]]
 | [[36/35]]
 | [[6/5]]
-| [[13/12]]
+| [[12/11]]
-| [[19/11]]
+| [[45/26]]
 | [[9/7]]
 | [[13/10]]
 | [[21/20]]
-| [[35/32]]
+| [[10/9]]
 | [[15/13]]
+| [[15/11]]
+| [[49/32]]
+| [[56/45]]
+| [[25/14]]
+| [[25/13]]
+| [[64/45]]
+|-
+| [[3/2]]
+| [[105/104]]
+| [[27/16]]
+| [[39/38]]
+| [[6/5]]
+| [[38/35]]
+| [[26/15]]
+| [[32/25]]
+| [[43/33]]
+| [[43/41]]
+| [[31/28]]
+| [[38/33]]
+| [[19/14]]
+| [[55/36]]
+| [[41/33]]
+| [[57/32]]
+| [[48/25]]
+| [[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]]
-| [[56/45]]
+| [[31/25]]
-| [[9/5]]
+| [[16/9]]
 | [[21/11]]
-| [[45/32]]
+| [[78/55]]
 | ...
 |}
-== See also ==
+[[Category:Tuning]]
-* [[Redbull]]
-* [[Super-pitch]]
-* [[EDSO]]
-[[Category:Temperaments]]
-[[Category:Pages with Scala files]]
-{{Stub}}