Factor 9 grid: Difference between revisions
Cmloegcmluin (talk | contribs) "optimal GPV sequence" → "optimal ET sequence", per Talk:Optimal_ET_sequence |
m →Theory |
||
| (10 intermediate revisions by 2 users not shown) | |||
| Line 1: | Line 1: | ||
'''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. | '''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 == | == Theory == | ||
The scale is an isoharmonic sequence | 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 [[12edo|12-tone equal temperament]], often accompanied by references to the symbolic or "sacred" 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 == | == Intervals == | ||
| Line 47: | Line 51: | ||
== Attempts at representing the scale through regular temperament theory == | == 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. | |||
This being said, | 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 [[wikipedia:666 (number)|666]], which is a value widely regarded in popular numerological traditions, and Western popular culture as a whole as [[wikipedia:the Number of the Beast|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. | 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. | ||
| Line 64: | Line 74: | ||
{{Optimal ET sequence|legend=1|171f, 495, 666}}. | {{Optimal ET sequence|legend=1|171f, 495, 666}}. | ||
== Mysticism and reality == | |||
The motivation behind the scale, as with many esoteric just intonation proposals, is the claim that the irrational pitch relationships of equal temperament produce acoustically unpleasant effects which, in turn, are said to propagate into subconscious perception and negatively affect human well-being. | |||
However, claims that specific tuning systems (such as just intonation or particular frequency standards like 432 Hz) have direct effects on public health, social cohesion, or global conditions are not supported by empirical evidence. While differences in tuning can influence perceived consonance, timbre, and listener preference, these effects operate at the level of auditory perception and musical aesthetics rather than large-scale societal outcomes. | |||
Furthermore, from a mathematical perspective, it is not possible to simultaneously achieve the exact rational interval relationships of just intonation and the structural evenness of equal temperament. The irrationality inherent to equal divisions of the octave has been recognized since antiquity, most commonly through proofs such as the irrationality of √2. For example, if there were an exact just intonation fraction corresponding to the 600-cent tritone, its numerator and denominator would be required to satisfy mutually incompatible conditions — [[wikipedia:Square root of 2#Proof by infinite descent|being both even and coprime]]. Similarly, if a stack of pure fifths (3/2) were to close exactly at the octave, the resulting comma {{Monzo|-X Y}} would have to equal 1. In this number, numerator X must be a power of 2 and the denominator Y a power of 3, thus implying the existence of an even power of 3, which is not possible. | |||
More broadly, while differences in tuning systems can affect perceived consonance, beating, and timbral character, these effects remain within the domain of auditory perception and musical aesthetics. There is no established physiological or cognitive mechanism by which specific frequency ratios could influence complex outcomes such as public health, social stability, or quality of life at scale. Such phenomena are determined by a wide range of economic, environmental, and social factors, none of which have demonstrated dependence on musical tuning standards. | |||
Accordingly, claims that particular tuning systems produce measurable improvements in human well-being would require empirical verification under controlled conditions. In the absence of such evidence, assertions of this kind remain unsupported, particularly when presented within numerological or occultist frameworks rather than scientific methodology. | |||
== External links == | == External links == | ||
Latest revision as of 22:44, 1 April 2026
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 "sacred" 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
Optimal ET sequence: 171f, 495, 666.
Mysticism and reality
The motivation behind the scale, as with many esoteric just intonation proposals, is the claim that the irrational pitch relationships of equal temperament produce acoustically unpleasant effects which, in turn, are said to propagate into subconscious perception and negatively affect human well-being.
However, claims that specific tuning systems (such as just intonation or particular frequency standards like 432 Hz) have direct effects on public health, social cohesion, or global conditions are not supported by empirical evidence. While differences in tuning can influence perceived consonance, timbre, and listener preference, these effects operate at the level of auditory perception and musical aesthetics rather than large-scale societal outcomes.
Furthermore, from a mathematical perspective, it is not possible to simultaneously achieve the exact rational interval relationships of just intonation and the structural evenness of equal temperament. The irrationality inherent to equal divisions of the octave has been recognized since antiquity, most commonly through proofs such as the irrationality of √2. For example, if there were an exact just intonation fraction corresponding to the 600-cent tritone, its numerator and denominator would be required to satisfy mutually incompatible conditions — being both even and coprime. Similarly, if a stack of pure fifths (3/2) were to close exactly at the octave, the resulting comma [-X Y⟩ would have to equal 1. In this number, numerator X must be a power of 2 and the denominator Y a power of 3, thus implying the existence of an even power of 3, which is not possible.
More broadly, while differences in tuning systems can affect perceived consonance, beating, and timbral character, these effects remain within the domain of auditory perception and musical aesthetics. There is no established physiological or cognitive mechanism by which specific frequency ratios could influence complex outcomes such as public health, social stability, or quality of life at scale. Such phenomena are determined by a wide range of economic, environmental, and social factors, none of which have demonstrated dependence on musical tuning standards.
Accordingly, claims that particular tuning systems produce measurable improvements in human well-being would require empirical verification under controlled conditions. In the absence of such evidence, assertions of this kind remain unsupported, particularly when presented within numerological or occultist frameworks rather than scientific methodology.
External links
- Sonic Geometry: The Language of Frequency and Form - the original video
- Testing 432Hz Frequencies and Temperaments - refuting by Adam Neely