User:R-4981/Pepsi: Difference between revisions
Relate tuning with existing concepts, cleanup |
→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 |
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== Interval chain == | == Interval chain == | ||
In this table, the intervals are [[octave-reduced]]. | 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/ | | ~[[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]] | ||
| [[ | | ~[[39/38]] | ||
| [[6/5]] | | ~[[6/5]] | ||
| [[13/12]] | | ~[[38/35]] | ||
| [[19/11]] | | ~[[26/15]] | ||
| [[ | | ~[[32/25]] | ||
| [[ | | ~[[43/33]] | ||
| [[21 | | ~[[43/41]] | ||
| [[ | | ~[[31/28]] | ||
| [[ | | ~[[38/33]] | ||
| [[35/26]] | | ~[[19/14]] | ||
| [[50/33]] | | ~[[55/36]] | ||
| [[ | | ~[[41/33]] | ||
| [[9 | | ~[[57/32]] | ||
| [[21/11]] | | ~[[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]] | |||
| ... | | ... | ||
|} | |} | ||
[[Category:Tuning]] | [[Category:Tuning]] | ||
Revision as of 03:02, 4 June 2024
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. 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.
| 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 | ... |