φ_audio — perceptual descriptors

Fifteen dimensions, kept compact and put on axes a linear model can express a preference along.

Computed on Hann-windowed frames of the normalized mono render (2048 samples, 50% hop), plus a few time-domain and segment-local measurements. Every field is finite by construction, because vetting ran first.

The coordinates

The :p2 suffix is the stimulus generation tag, and it is the migration mechanism rather than a comment.

Why these axes and not the obvious ones

The model downstream is linear in $\varphi$, so the axis a feature lives on decides what preferences are expressible at all.

Frequency features are logarithmic, not linear in Hz

Brightness and pitch perception are octave-based. On a linear-Hz axis normalized by Nyquist, moving a patch from 200 Hz to 400 Hz (a full octave, an enormous audible change) shifts the coordinate by 0.009, while 8 kHz → 16 kHz shifts it by 0.36.

A linear model in that coordinate cannot represent "I like my basses a shade brighter": the entire usable range is swallowed by the bright tail of the pool. The preference is not hard to learn, it is inexpressible.

So log_axis puts centroid, rolloff and ZCR on a shared octaves-above-20 Hz scale, normalized to $[0,1]$ at Nyquist:

$$\text{log_axis}(f) = \frac{\log_2!\big(\max(f, f_0)/f_0\big)}{\log_2!\big(\max(f_{\text{Nyq}}, 2f_0)/f_0\big)}, \qquad f_0 = 20\ \text{Hz}$$

20 Hz because below it frequency is not audible as pitch and the ratio scale stops meaning anything. Normalizing at Nyquist keeps the vector sample-rate agnostic.

Note that a zero-crossing rate is a frequency (two crossings per cycle), so it goes on the same axis:

$$\text{zcr_mean} = \text{log_axis}!\left(\frac{r \cdot f_s}{2}\right)$$

where $r$ is the crossing fraction. Leaving it as a raw fraction would put a frequency-like quantity on a non-frequency axis beside three that are on one.

Heavy tails are logged

crest spans 1 to 40+; tail_ratio spans three orders of magnitude. Standardizing either raw hands the model a coordinate whose z-score is near-constant for most of the pool and $+4$ for a handful of outliers: a coordinate that separates nothing except the outliers.

$$\text{crest} = \log!\frac{\text{peak}}{\text{RMS} + \epsilon}, \qquad \text{tail_ratio} = \log!\left(\frac{\text{RMS}_{\text{last 300 ms}}}{\text{RMS} + \epsilon} + 10^{-3}\right)$$

The $10^{-3}$ floor inside the tail log matters: a pluck fully decayed by the last 300 ms would otherwise send the log to $-\infty$, and "silent tail" and "very quiet tail" are the same judgement to a listener anyway.

The attack crossing is interpolated, not floored

Quantizing the 90%-of-peak crossing to the analysis-window index makes attack_s exactly zero for every patch whose first window is already at peak (most percussive patches), turning a continuous axis into a zero-inflated spike.

So the envelope uses a fine grid (4 ms window, 1 ms hop) and interpolates linearly between the last sub-threshold hop and the first one over it:

$$h^\star = (i-1) + \frac{0.9,\max(e) - e_{i-1}}{e_i - e_{i-1}}, \qquad \text{attack_s} = \log!\left(\frac{h^\star \cdot \text{hop}}{f_s} + 0.005\right)$$

The measurement window is onset → the second note's onset (2.0 s under the v2 phrase), and the $+5$ ms inside the log keeps the fast end resolved instead of compressing every percussive patch into the same value.

Spectral definitions

Per frame, with magnitudes $m_i$ over $\text{bins} = 1024$ and $\text{power} = \sum m_i^2$:

Centroid. The magnitude-weighted mean frequency, then log-axised:

$$f_c = \frac{\sum_i i \cdot \Delta f \cdot m_i}{\sum_i m_i}$$

Rolloff. The lowest bin at which cumulative power reaches 85% of the total.

Flatness. Geometric over arithmetic mean of the power spectrum, clamped to 1:

$$\text{flatness} = \min!\left(1,\ \frac{\exp!\big(\tfrac{1}{N}\sum_i \log(m_i^2 + \epsilon)\big)}{\tfrac{1}{N}\sum_i m_i^2 + \epsilon}\right)$$

Flux. Normalized by the combined magnitude sum of both frames:

$$\text{flux} = \frac{\lVert m^{(t)} - m^{(t-1)} \rVert_2}{\sum_i m^{(t)}_i + \sum_i m^{(t-1)}_i + \epsilon}$$

Dividing by the current frame alone is the obvious choice and it explodes: a loud frame decaying into near-silence gives an enormous flux for a change that is barely audible. The combined denominator keeps it in roughly $[0,1]$.

Frames whose power is below $10^{-12}$ contribute to none of the spectral means: a silent frame has no centroid, and averaging in a zero would drag brightness down in proportion to how much silence the phrase happens to contain.

Segment-local coordinates

The last three are measured over one note's gate-on span, and they exist because whole-phrase statistics conflate things a listener does not.

Roles are found by property, not position, which is what keeps them meaningful if the phrase changes:

• held: the first note.
• high: the highest note at least half an octave above the held one.
• chord: the first note with chord voices.

A phrase missing a role yields 0.0 for its features, which reads as "no evidence" rather than as a measurement.

held_centroid_std is the important one. centroid_std over the whole phrase conflates note-to-note register jumps with genuine timbral motion: a static patch played across two octaves has a large centroid_std. Restricted to the held note's span the coordinate is register-constant by construction, so it is the axis on which "a filter sweeping at 0.4 Hz" and "a static patch" are different patches at all. It needs at least 3 frames in the span, or it reports 0.0.

high_ratio = $\log$ of the high note's span RMS over the held note's. Does the patch speak in the upper register, or does its filter choke it?

chord_flatness_delta = mean flatness over the chord span minus the held span. Intermodulation and mud when voices stack.

Deliberately compact

Fifteen dimensions is a choice. The model is a mixture of linear experts, and interpretable axes are the point: "bright", "noisy", "slow attack", "long tail" are things the DIRECTIONS tab can name and a person can recognise in their own preferences.

A 128-dimensional MFCC bank would carry more information and would be unreadable, and would make the cold start dramatically worse: every dimension is posterior variance to pay down before the model says anything at all.

Known collinearity

Measured over 1200 prior draws (cargo run -p auracle-features --example pipeline_stats --release -- 1200), the variance inflation factors are mostly comfortable, with one cluster that is not:

That is the brightness cluster — three genuine measurements of one perceptual thing. It is left standing deliberately: dropping any of them discards real signal rather than redundancy, since they disagree in informative ways (a bright noisy patch and a bright tonal patch differ in ZCR-versus-centroid). The right fix is a shared or fused prior over the cluster, which is a modelling change rather than a feature change, and is not done.

For contrast, φ_struct had two exact linear dependencies, which is a different and worse problem and was fixed by dropping columns.