User:R-4981/Pepsi: Difference between revisions

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Relate tuning with existing concepts, cleanup
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[[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.
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== See also ==
[[Category:Tuning]]
* [[Redbull]]
* [[Super-pitch]]
* [[EDSO]]
 
[[Category:Temperaments]]
[[Category:Pages with Scala files]]
{{Stub}}

Revision as of 05:55, 5 January 2024

A graphical expression of the interval chain of the Pepsi.

The Pepsi[idiosyncratic term] (name proposed by 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 measure multiplied by √3.

The formula for the nth interval of this tuning, where p(n) is pitch in cents, is:

[math]\displaystyle{ p(n) \approx 701.955~¢ \cdot 3^{n/2} }[/math],

or in its exact form:

[math]\displaystyle{ 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.

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 ...
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