Another Aural Temperament or Golden Harmonics

Phi (\(\phi\)) is defined as the arithmetic mean of \(1\) and \(\sqrt{5}\), similar to how the fifth \((3/2)\) is the arithmetic mean of the octave \((2/1)\). This system maps powers of \(\phi\) into \(\sqrt{5}\) space, making it the period. As an interval ,\(\sqrt{5}\) , 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 \(\phi\) into \(\sqrt{5}\) space ( \(\{ (\sqrt{5})^n \times \phi^m \} \in [1,\sqrt{5})\) ), with \(n\) and \(m\) integers, the first five notes (zero index) include unison/identity and four effective powers, implying five quasi-equal divisions of \(\sqrt{5}\).

Similarly, using nine powers of \(\sqrt{\phi}\) 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 \(1.489... = (\sqrt{\phi})^5 \times (\sqrt{5})^{-1}\) 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 \(V_7 \rightarrow I\) chord progression (video/audio 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 \(\phi\) Acts Like a Generator of \(\sqrt{5}\)-space

The connection comes from expressing \(\phi\) in terms of equal divisions of \(\sqrt{5}\). 

Already at the first step, \(\phi\) itself is an excellent approximation to a rational power of \(\sqrt{5}\):

\(\phi = (1+\sqrt{5})/2 ​\approx (\sqrt{5})^{3/5}\).

Numerically,

\(\phi \approx 1.6180\), (\(\sqrt{5})^{3/5} \approx 1.6206\).

The error here is tiny (~ 833 vs 835 cents), small enough that, for all perceptual and practical purposes, \(\phi\) can be treated as though it were exactly a fractional division of \(\sqrt{5}\).

This means that stacking powers of \(\phi\) within \(\sqrt{5}\)-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 \(\sqrt{5}\), yielding an effective 5-ED(√5) division. Using \(\sqrt{\phi}\) instead, you get 10-ED(√5) (≈ 139¢ steps), which is the scale i actually play with. (here we get the inexact acoustic phi at \((\sqrt{5})^{6/10}\) ).

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

\( (\sqrt{5})^n \times (\sqrt{\phi})^{m \bmod 10}\), with \(n, m \in \mathbb{Z}\)

 For \(m\), it doesn’t matter whether \(m \in \mathbb{Z}\) or \(m \in \mathbb{N_0}\), as modular reduction absorbs negative values.

For example,

\((−4 \bmod 10)=6\).

To avoid confusion, we can write \(m \in \{0,1,2,…,k−1\}\), so that m explicitly ranges within this set.

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The Phi “Intervalizations”

In Western music theory, the constant phi has been interpreted as a musical interval in a couple of ways, with the most common being “acoustic phi” and “logarithmic phi.” Acoustic phi takes the normalized value of phi (about 1.618) and treats it as a frequency ratio, similar to how intervals like a fifth are 3/2 or 1.5. An interesting feature of this approach is that the resulting combination tones are also in golden proportion. Logarithmic phi, on the other hand, represents the golden ratio of the octave, calculated directly in cents as 1200 × 0.618 approx 741 cents.

Formulas and calculations vary among authors and theorists, and there are many different ways to interpret or "hear" the golden ratio.

Scala file:

! rphi9-rfive.scl

!
9 powers of square root of phi, mapped into square root of five space. rational approximations with gcd tolerance .000001
10
!
2983/2768
1597/1364
491/386
451/329
1741/1169
987/610
1109/636
646/341
743/361
2207/987

group theory notation https://xcjb.blogspot.com/2024/08/pythagorean-scale-z12z-z.html