Factor 9 grid: Difference between revisions

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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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	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.		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.		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.