Factor 9 grid: Difference between revisions

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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.

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.

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 "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.

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.

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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.

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 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.
+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.
-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 "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.
+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.
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 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.