Keep It Greasey: the tempo, measured

1. The lock, and what came in

The previous session closed with three measured blockers rather than an appeal to effort, and a queue item asking which of four tempo numbers is correct: the printed 134, the metronome stem's 136, the folder name's 142, and a local grid sweep peaking at 143.5. This page answers that item.

2. The metronome stem is synthetic

The previous session leaned on a stem named metronome in the 15-stem separation set, reading 136.00 BPM from it and using its first click as the phase anchor for grid tests. Before any tempo arithmetic, that asset was classified.

A source separation cannot produce a stem shorter than its siblings, and it cannot produce 1,171 bit-identical transients at full scale over digital silence.

3. Nine values per interval

1,171 clicks give 1,170 intervals. Every single interval falls within one sample of a multiple of 882 samples, which is 20 ms at 44.1 kHz. The fraction inside two samples of that lattice is 1.0000. The beat times were produced at 50 frames per second and then rendered.

Each single interval can take nine values and no others. A run of alternating steps does encode an intermediate average, so the lattice constrains instantaneous values rather than long-window means. The claim below rests on one interval and on a median of intervals, and both are lattice-bound.

The folder name's 142bpm is the same instrument talking. 142.857 is the 420 ms bin and the mean interval across the file is 428.15 ms, which is 140.14 BPM. The filename and the metronome stem are one estimate, not two.

4. Control: the audio is not speed-shifted

The stem folder is named D minor-142bpm-442hz. A file that had been resampled to a 442 Hz reference would carry every tempo figure with it, so it was tested first, and the result gates every BPM figure below. Tuning was estimated per stem against the A440 equal-tempered grid, with 30-second windows for stability.

5. Method, and the bias it replaces

Onsets come from the summed kit, six stems: kick, snare, hat, cymbals, toms and other_kit. The measurement therefore no longer rests on quiet snare detections at -42 to -50 dBFS, which is the third inherited blocker. Onset detection runs at hop 128, 44.1 kHz.

The statistic is circular concentration over a candidate period T:

R = |mean( exp( 2πi · t_k / T ) )|     E[R] under the null = √π / (2√N)

Phase is optimized implicitly, so no anchor click enters the measurement and the rendered metronome is never touched.

6. The estimator, validated end to end

Signal built to the measured conditions of the 198-238 s window: onset density 7.3 per second, two amplitude populations 22 dB apart with a quiet share of 0.545, kit-like exponentially decaying broadband transients, and a -60 dBFS noise floor.

Clean condition, 12 ms jitter

The detector reports every onset about 5.5 ms late. A constant shift leaves the period estimate untouched, because the statistic is phase-invariant. It matters for anchoring, and it is carried into section 12.

Matched difficulty, route A: raise jitter until R meets the real 0.4377

The method has a cliff and the real data sits well above it. Down to R near 0.27 the estimator holds to 0.06 BPM. Below R near 0.19 it breaks completely, missing by 26 BPM. The observed R of 0.4377 is roughly 2.3 times the breakdown level.

Matched difficulty, route B: model the measured drift instead of raising jitter

The 5-second table in section 9 shows the passage accelerating from 135.72 to 137.87. A linear ramp across exactly that range, with ordinary 12 ms jitter, was synthesised and run end to end.

7. The metrical level, tested

Calling 109.12 ms "the sixteenth" is a claim, so it was scanned rather than assumed. Wide sweep, 40 to 900 ms, over the solo passage.

One fundamental and its half. A previous version of this table labelled 518.32 ms "one 19/16 bar", which was wrong: 518.32 ms is 4.75 sixteenths and the bar is 19 of them, 2073.28 ms. The bar period is now measured and listed, at R = 0.0376. Every longer multiple sits at or below the null, which is what a running sixteenth subdivision with no eighth-level accent looks like. Reading 109.12 ms as the sixteenth gives a 19/16 bar of 2.073 s, against the tab's notated 2.111 s for measure 102. Two independent routes to the same bar length settle the level.

8. The result at 3:18

Solo passage 198 to 238 s. Both Alfred page 64 and the DRUM Groove Analysis page mark this passage from 3:18, and the lead stem enters at 198.275 s against that printed 198.000 s, so the window is tied to the printed marker by audio. Nothing ties it to a measure number by audio. 293 kit onsets. A single best-fit period across the whole 40 seconds gives quarter = 137.46, R = 0.4377 against a null expectation of 0.0518, Rayleigh p = 4.2 × 10^-25, and above all 200 draws of an interval-shuffle null whose max-R averaged 0.226.

9. The drift curve

A single 40-second fit hides tempo drift, so the window was split. Every sub-window is individually strong, R between 0.59 and 0.83 against nulls near 0.15.

Other windows, for completeness. Bootstrap intervals here measure onset sampling noise only. They do not measure tempo drift inside a window, and the table above shows drift is the larger term.

The song is not at one tempo, which agrees with the tab carrying 19 tempo automations between 64 and 139.

10. What this settles

11. The three blockers, before and after

12. The anchor, unmeasured

Queue item q-2026-09-06-acaf55. It asks where measure 102 begins in the audio. It ran twice, and the second run overturned the first.

Run one, and why it does not stand

The first run scanned a global offset across plus or minus 1.5 s and reported a cross-family range of 1,928 ms, with kick at +1.176 and hat at +1.174. The cymbal peak sat at +1.490, ten milliseconds from the search boundary, which is the tell that the window was too narrow.

Run two: plus or minus 3.0 s, 3,001 offsets, with a null that sees the same search

Added: a scan-aware permutation null, shuffling the notated inter-onset intervals and rescanning, so the null faces the same 3,001-point search the observation did. Added: a bootstrap over notated events. Added: a sweep of the peak-over-mean cut, since the guitar vote missed the published 1.35 by 0.02.

Threshold sensitivity, since the guitar vote sat 0.02 from the cut

No cut produces convergence. The picture is not sensitive to the threshold, which was the one reassuring thing the sweep could have shown and did.

Run three: the external route, on the review's own design

Queue item q-2026-09-06-c1d6d3, taken in the same pass. Several distinctive lead and vocal landmarks rather than one entrance, a local map instead of one global offset, monotonic with a slope-change penalty of 0.5 s per two-measure block, and every landmark set split so the fit is scored only on held-out events. No part 8 lane enters, so no ghost label and no drum stem is involved.

Run four: DTW against a score chromagram

q-2026-09-06-a42f3c. It had been queued as blocked on rendering nine parts to audio. That was wrong. FMP chapter 3 alignment builds a score chromagram from notated pitches, so no synthesis is needed, and part 8 carries no pitch so the alignment cannot see a ghost flag.

Attempt two used the sum of the eight pitched stems so drum energy cannot drive the chroma, with the band tightened to plus or minus 3.6 s. Measure 102 lands at 197.880 s, offset +1.025 s, only 28% of the band, path slope 1.0188, and the map drifts smoothly from +1.077 at measure 99 to +1.393 at measure 107.

Nothing about the tempo measurement changes. That result covers the audio between 198 and 238 s and nothing wider. No external evidence on this machine places that window at measures 102 to 104, and the broader external-alignment question stays open.

13. Blocker 3, tested with a second separator

The same mixture, 180 to 300 s, separated again with demucs 4.0.1 htdemucs. Every detection in the original snare stem was then checked for a corroborating onset in that independent drums stem. The item had been queued as needing a second engine. demucs was already installed, so it was never blocked.

Loud snare detections survive an independent separation at 1.53x chance. Quiet ones do not, at 0.48x, which is below what random placement would give. Quiet minus loud is -27.5 percentage points. In 41 regions totalling 10.25 s where the independent drums stem sits at its noise floor, 6 original-snare detections still fire, 0.585 per second against an overall 3.55.

Masking control, because whole-kit drums can hide a quiet stroke

The comparison stem is the whole kit, so a genuine quiet stroke under a loud kick would score as uncorroborated. Masking predicts uncorroborated quiet detections sit under louder kit energy.

No flag moves on this. If anything it argues for leaving all 629 where the transcriber put them, since the audio evidence on both sides is now weaker than a reading made by eye and by ear.

14. The drift simulation, driven by measurement

The review's objection was that the 24 ms jitter condition in section 6 was selected to match observed R, making it a sensitivity calibration, and that a linear ramp reproducing one summary statistic shows compatibility rather than causation. This run is driven by the eight measured local tempi and the observed per-window onset counts, jitter held at an ordinary 12 ms, with nothing tuned to R. 30 seeds, end to end.

A single 40-second fit misses by more than 1 BPM on every one of 30 seeds and lands in one of two basins. The sub-window median recovers the truth to 0.125 BPM. That validates the choice already made on this page: 137.46 was withdrawn and 136.7 from the sub-window curve reported instead. The simulated sub-window-minus-global gap is -1.452 BPM and the real audio's is -0.80, same sign and order.

15. Instrument-specific detector latency

The +5.5 ms lateness subtracted in section 12 came from modelled kit transients. The review asked whether bass, piano and cymbal attacks carry their own. They do.

Maximum deviation from kit is 6.14 ms, so a single flat correction is not adequate. Latency also depends on density: kit reads 3.42 ms on sparse isolated hits against 5.5 ms in the dense end-to-end run, so the flat value used sits between them.

16. The 13 px Atkinson scan, now classified

The record called this scan unclassified rather than unreadable, and said classifying it needs a measured error rate for telling a bracketed notehead from an ordinary one using that same scan. Pages 6 to 9 extracted at native 180 ppi.

391 real ordinary noteheads were taken from the scan itself, copied, given a synthetic bracket at the 2.3 px stroke width the record measured, and the scan's own paper noise re-added at the measured sd of 0.0176. A best-threshold classifier was then scored on real-ordinary against synthetic-bracketed.

What would change it: a publisher-resolution copy of the Modern Drummer Rock Charts pages, still open as q-2026-09-06-e64c44. At the 18 to 19 px of the DRUM Odd Meter Lesson pages brackets read by eye.

17. What stays open

• An external anchor. Section 12 leaves the offsets spanning 4.290 s with five of six sources failing a scan-aware null. Closing it needs several distinctive lead or vocal landmarks rather than one entrance, a monotonic local map with a slope-change penalty, and landmarks held out for verification.
• Blocker 3. Low-level detector reliability on a separated stem, with a second separation method as the named remedy.
• Was 134 ever right? A printed tempo can describe another pressing, another transfer, or the transcriber's rounding. This page measures the recording on disk and says nothing about which artifact the publishers were listening to.
• The publication date of Marc Atkinson's Modern Drummer Rock Charts transcription, 1997 per Paiste's profile against an October 2000 page footer in a third-party scan.
• A publisher-resolution copy of that complete chart, the only complete-song printed source.

18. Provenance

Audio

Scripts

All under the session tool directory for s604777. Hashes identify the exact file that produced each section.

Python 3.11.15, librosa 0.11.0, numpy 1.26.4. Random seeds fixed at 7, 11 and 17.

Printed sources named on this page

• Rich Lackowski, On the Beaten Path: Progressive Rock, Alfred, page 64, Track 21. Read at 600 dpi from the publisher's free sample, content.alfred.com/catpages/00-31503.pdf, md5 116b3ee7df76b9d11a1c3d4d18781d12. Prints 19/16, quarter = 134.
• DRUM! Magazine Groove Analysis page for this song, headed quarter = 134.
• Marc Atkinson, Modern Drummer Rock Charts, the only complete-song chart, held here as a 180 ppi third-party scan with incomplete chain-of-custody.
• Songsterr s604777, live revision r8852151 by Ben Dibden1.