The following is a musical tuning based upon the Golden Ratio of Phi.
The process is to multiply each frequency by 1.61803398875, and then to bring the resulting frequencies down through even division so that all of the notes reside within the same musical octave. This process is necessary as the actual resulting frequencies are limited in their musical application, and go beyond human audible sound.
Following this process for 11 steps fills in a note at each of the 12 chromatic positions on a traditional keyboard, though this process can theoretically be done an infinite number of times. The starting frequency will be the unity measure of one cycle per second, which raised through eight octaves equals the musical pitch of C256.
The following chart details the process step; the resulting frequency as a phi proportion to the one previous to it; the resulting frequency divided down into the fourth audible musical octave; and then the resulting frequency’s equivalent as an Indian svara, as a musical pitch and as a distance in cents to the closest equivalent at A440 equal temperament, which can be input into a compatible synthesizer or sound module.
| Step | Frequency | 4th Octave | Svara | Pitch | Cents |
|---|---|---|---|---|---|
| 0 | 256 | 256 | Sa | C | C -37 |
| 1 | 414.21670112 | 414.217 | Dha-k | Ab | G# -05 |
| 2 | 670.21670112 | 335.108 | Ga | E | E +29 |
| 3 | 1084.43340224 | 271.108 | Re-k | Db | C# -38 |
| 4 | 1754.65010336 | 438.663 | Dha | A | A -05 |
| 5 | 2839.08335056 | 354.885 | Ma | F | F +28 |
| 6 | 4593.73360896 | 287.108 | Re | D | D -39 |
| 7 | 7432.81711456 | 464.551 | Ni-k | Bb | A# -06 |
| 8 | 12026.5507235 | 375.830 | Ma-t | F# | F# +27 |
| 9 | 19459.3678381 | 304.053 | Ga-k | Eb | D# -40 |
| 10 | 31485.9185616 | 491.967 | Ni | B | B -07 |
| 11 | 50945.2863997 | 398.010 | Pa | G | G +26 |
The cent amounts are to the nearest frequency possible considering 1/100th resolution. Some synthesizers are able to accept Scala files for greater tuning precision; and some allow for direct frequency input to the thousandths place, such as Native Instrument’s Absynth.
If one plays the lowest audible C on the keyboard, and then climbs up each step through the chart above, it is possible to play through several of the initial 11 steps according to their true phi intervals.
Progressing beyond the first 11 steps creates variations or alternates of the initial 12 musical notes, which can be used to express different shades or flavors of pitch, similar to the 23 srutis of North Indian music, or to the 65 notes described in the Universal Scale of Sounds as detailed in Alain Danielou’s book “Music and the Power of Sound: The Influence of Tuning and Interval on Consciousness.”
It is interesting to note that the mathematical relationships between the majority of the adjoining intervals in this scale, when considered within the same octave, are very close to 1.0590…, while the relationships between adjoining notes within western equal temperament is 1.05946309436. This means that a phi ratio based scale, following this approach, when converted to and played as a unified octave, creates intervals within +/- 5/10,000ths of western equal temperament.
Beauty is in the ear of the beholder.
