Phi (φ) is defined as the arithmetic mean of 1 and √5, similar to how the fifth (3/2) is the arithmetic mean of the octave (2/1). This system maps powers of φ into √5 space, making it the period. As an interval ,√5 , represents a traditionally consonant (though irrational) ninth, situated between 16/7 and 15/7.
By stacking and folding (rotations on the circle in log-coordinates) four powers of φ into √5 space ({ (√5)^n * (√Φ)^m } ∈ [1,√5)), the first five notes (zero index) include unison/identity and four effective powers, implying five quasi-equal divisions of √5.
Similarly, using nine powers of √φ approximates a 10-ed√5, which is also close to a 6-edΦ.(see note)
This introduces a smaller step of about 139¢, forming a suitably sized leading tone, slightly smaller than that in the Bohlen-Pierce scale and 8-edo.
The small difference provides a good initial fifth, approximately 1.489... = (√Φ)^5 * (√5)^-1 It is locally tonic, allowing for 3 or 4 consecutive notes within the "same scale." The next notes, guided by "consonance," align with a different tonal center scale out of phase with the previous one(sparse duodecimability). Additional "belonging" notes appear farther up, as this is an infinite chroma system that avoids near octaves in most practical ranges (nearest at 1253 cents, next at 2368, etc.), while other notes can create a distinct mode/s, as traditionally understood.
The tuning also provides a spanned but functional V7 - I chord progression (1:10).
"Acoustic phi" is dissonant with harmonic timbres, so I avoided it in this guitar solo. It acts as a pivot for the separated tonics, creating a centerless sound.
Why φ Acts Like a Generator of √5-space
The connection comes from expressing φ in terms of equal divisions of √5.
Already at the first step, φ itself is an excellent approximation to a rational power of √5:
𝜙=(1+√5)/2 ≈ √5^{3/5}.
Numerically,
𝜙≈1.6180, √5^{3/5}≈1.6206.
The error here is tiny, small enough that, for all perceptual and practical purposes, φ can be treated as though it were exactly a fractional division of √5.
This means that stacking powers of φ within √5-space is essentially the same as running an equal-step generator chain (just like stacking fifths in 12-EDO approximates octaves).
From this perspective: Four powers of φ fold neatly into √5, yielding an effective 5-ED(√5) division. Using ≈√φ instead, you get 10-ED(√5) (≈ 139¢ steps), which is the scale i actually play with. (here we get the inexact acoustic phi at √5^{6/10}).
edit: i just noticed some type errors in the table graphics,,, and on notation:
In the video, the tuning is described with generators in the form
(√5)^n * (√Φ)^{m mod 10}, with n, m ∈ Z
For m, it doesn’t matter whether m ∈ Z or m ∈ N_0, as modular reduction absorbs negative values.
For example,
(−7 mod 12)=5.
To avoid confusion, we can write m ∈ {0,1,2,…,k−1}, so that m explicitly ranges within this set.
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group theory notation https://xcjb.blogspot.com/2024/08/pythagorean-scale-z12z-z.html
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