Pepsi

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]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)⁹ = 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 3² * ~701.955 ¢ = (3/2)⁹.)

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

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

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

31/25

16/9

21/11

78/55

...