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 consisting of the following frequencies (in Hz): 126, 135, 144, 153, 162, 171, 180, 189, 198, 207, 216, 225, 234, 243, and their octave equivalents. This sequence forms an arithmetic progression with a constant difference of 9, which gives rise to the name "Factor 9 grid". It hence is identical to the 14ado scale spanning the 14th through 28th harmonics, and since 14ado is a 23-limit just intonation system, Factor 9 grid correspondingly is a part of of 23-limit just intonation.
More precisely, the "Factor 9 grid" refers to a specific mode of 14ado whose tonic is placed on the step corresponding to 432 Hz and its octave equivalents, such as 216 Hz or 864 Hz, which is the 11th step of 14ado itself. It is this particular modal alignment that is commonly associated with "A = 432 Hz" conspiracy theories, where the emphasis is placed on organizing the scale around 432 Hz as a tonic.
Proponents of the "Factor 9 grid" manly present it as the more consonant or acoustically "healthier" alternative to the prevailing 12-tone equal temperament, often accompanied by references to the symbolic or "sacred1 significance of the number 12. However, descriptions of the scale, as they are presented in the video, contain several internal inconsistencies. The underlying structure of the grid, as mentioned above, corresponds to 14ado, which by definition contains 14 distinct steps per octave rather than 12, thus conflicting with the initial claims that the scale is 12-note or is a replacement for 12edo.
Furthermore, in the cited material, the sequence appears to omit the frequency 243 Hz (and its octave equivalents), despite it being a member of arithmetic progression that constitutes the scale. This omission appears to be motivated by an attempt to align the number of pitches with the 12-note framework of standard Western notation, as the presentation maps the resulting tones onto conventional note names. However, the rationale for excluding specifically 243 Hz and its octave displacement, as opposed to any other member of the sequence, is not explicitly addressed. No criteria are provided for why this particular step is removed while the remaining tones are retained, leaving the adjustment unexplained within the context of the scale’s stated arithmetic construction.
Additionally, the accompanying table distinguishes between G♯ and A♭ as separate pitches, resulting in a 13-note scale rather than either 12 or 14. This is not explained in the video as well, and further complicates the stated aim of aligning the system with a 12-note framework, particularly in light of the emphasis placed on the number 12 in the associated commentary.
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
Within regular temperament theory, the Factor 9 grid admits close approximations through equal divisions of just intervals, and is well-supported by any tuning system that either represents the 23-limit intervals well, or the direct intervals mentioned in the scale.
Though, in particular, it is worth noting that it can be represented by 666 equal divisions of the 15/14 interval (666ed15/14), which directly models its harmonic structure, as well as by 666edo, which supports a related regular temperament with similar properties, which is a very ironic aspect of the scale.
The scale, often justified by proponents through numerological arguments emphasizing purity and the special status of certain numbers, is located in close correspondence with equal temperament systems involving the number 666, which is a value widely regarded in popular numerological traditions, and Western popular culture as a whole as the Number of the Beast or satanic. Thus, within the proponents’ own framework, where numerical symbolism is treated as musically or metaphysically significant, this association with 666 directly contradicts the stated rationale for privileging the scale over conventional tuning systems.
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, 666edo, more specifically, the 495 & 666 23-limit temperament, regular 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 scale, 28/27. So in a limited way, 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, however that temperament has no structure resembling the initial factor-9 grid other than closely approximating its constituent intervals.
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
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