Factor 9 grid
Factor 9 grid is a type of musical scale which was first proposed for esoteric reasons as a supposed replacement to 12edo. The scale also became notable in music theory when esoteric properties of the scale were subsequently refuted by the famous YouTuber, bassist, and composer Adam Neely.
Theory
The scale is an isoharmonic sequence that consists of the following notes, in Hertz: 126, 135, 144, 153, 162, 171, 180, 189, 198, 207, 216, 225, 234, 243, (octave), and their octave displacements. Therefore this is equivalent to 14ado scale, running from 14th to 28th harmonic. This is an arithmetic progression with the difference of 9, hence why this scale was named "Factor 9 grid". Since the largest prime present in the intervals is 23, this makes Factor 9 grid a part of 23-limit just intonation.
The scale, as proposed in the video, contains a major inconsistency - it has 14 notes, while it claims to be supposedly a better replacement towards the current 12-note system and makes references towards the sacrality of the number 12. In addition, the table presented features a 13-note scale where G# and Ab are not enharmonic. Also, the scale in the video skips number 243 for an unknown reason, given that 126 = 9 × 14, and you need 14 notes to count to the octave displacement, if you are using an arithmetic difference of 9. This page will consider the full 14:15:…:28 otonal sequence.
Since the esoteric affliction towards the note A being 432 Hz is usually considered, the scale is supposed to be played starting with A, which leaves a wide semitone inside the octave.
Intervals
The 9/8 interval was skipped by the original video for unknown reasons.
| Frequency (Hz) | Note | Interval |
|---|---|---|
| 432 | A | 1/1 |
| 450 | A#/Bb | 25/24 |
| 468 | B | 13/12 |
| 486 | - | 9/8 |
| 504 | C | 7/6 |
| 540 | C#/Db | 5/4 |
| 576 | D | 4/3 |
| 612 | D#/Eb | 17/12 |
| 648 | E | 3/2 |
| 684 | F | 19/12 |
| 720 | F#/Gb | 5/3 |
| 756 | G | 7/4 |
| 792 | G# | 11/6 |
| 828 | Ab | 23/12 |
| 864 | A | 2/1 |
Attempts at representing the scale through regular temperament theory
In an extremely ironic twist, this allegedly divine and enlightening scale is well-approximated by an evil satanic temperament of 666 equal divisions of the 15/14 interval. However, Eliora when trying to represent the Factor 9 grid via a rank-2 temperament did not realize that x31eq.com resource takes the first interval in the number series as the equivalence interval, and assumed that it is represented by 666edo instead. "q666" in the temperament finder, when 14:15:16:... typed out, stands for 666ed15/14 as opposed to 666ed2.
This being said, this temperament preserves more properties of the "Factor 9 Grid" than the corrected version, since it's period-9 and its period minus reduced generator interval also maps to the smallest interval in the Factor 9 Grid system, 28/27. So in the end, 666edo does ironically represent the Factor 9 grid well. The corrected version would be with 495ed15/14 & 666ed15/14, which corresponds to a 4973 & 6691 temperament in the 23-limit, that has no structure resembling the initial factor-9 grid other than being made of two good edos in the 23-limit.
The 495 & 666 temperament is given the name enneasoteric by Eliora, since music was already composed in the 666edo scale approximating Factor 9 grid, while the name "factor 9 grid" to avoid ambiguity will be retained with the scale only.
Enneasoteric (495 & 666)
Subgroup: 2.3.5.7.11.13.17.19.23
Comma list: 442/441, 715/714, 2300/2299, 3060/3059, 3179/3174, 9025/9009, 57375/57344
Mapping generators: ~250/231, ~336/323
Optimal tuning (CTE): ~336/323 = 70.270
Optimal ET sequence: 171f, 495, 666.
Mysticism and reality
The motivation behind the scale, or any kind of esoteric just intonation proposal, is that the irrational pitch standard of equal temperament has an acoustically unpleasant effect, that subsequently allegedly spreads to the subconscious levels of the mind and results in deterioration of the quality of human life.
However, it is not possible to have both the rationality of just intonation and evenness of equal temperament at the same time. The irrationality of equal temperament has been known since ancient times by various names, the most common being the proof by infinite descent of the square root of two. If there was an exact JI interval fraction which corresponds to the 600 cent tritone, it's numerator and denominator would have to be both even and coprime, which cannot happen. Another example is that if a stack of fifths (3/2) were to close at the octave, the resulting comma [-X Y⟩ would be equal to 1, while having X be a power of 2, and Y be a power of 3. This would imply the existence of an even power of 3, which is nonsense.
If there is credible scientific evidence that playing just intonation music through public loudspeakers, in soundtracks, or other public venues will increase peace and quality of human life, it must be presented properly, without occultist context.
External links
- Sonic Geometry: The Language of Frequency and Form - the original video
- Testing 432Hz Frequencies and Temperaments - refuting by Adam Neely