M.I.R Homework 2 -solved

(2) identify every beat/downbeat position of a song, and (3) identify the meters of a song. 
The definitions of beat and downbeat have been mentioned in the course slides. Meter refers to the 
regularity of repeating patterns of music. In a narrow sense, meter here refers to the relationship 
between beats and bars. For example, time signature 3/4 means that each bar contains 3 beats, and each 
beat is a quarter note; therefore, its meter is 3-beats. The commonly seen meters in our everyday music 
are 3-beats, 4-beats or their multiples, while 5-beats and 7-beats are also used sometimes. 
In this assignment, we will run five datasets. They are: 
 
ISMIR2004 (tempo) 
 
Ballroom (tempo, beat, downbeat) 
 
SMC (beat) 
 
JCS (beat, downbeat, meter) 
 
ASAP (beat, downbeat, meter) 
These datasets are available here (our TA have dealt with most of the label files for you): (Ballroom), 
(JCS), (Other three datasets) 
You might also need to use the following functions in librosa: 
 
librosa.feature.fourier_tempogram 
 
librosa.feature.tempogram 
 
librosa.beat.tempo 
 
librosa.beat.beat_track 
 
librosa.tempo_frequencies 
 
librosa.fourier_tempo_frequencies 
And maybe others. Read the documentation of librosa carefully and discover the useful functions 
by yourself. Also, you can choose to implement the tempograms and the related features by yourself. 
Task 1: tempo estimation 
Q1  Design an algorithm that estimate the tempo for the ISMIR2004 and the Ballroom dataset. 
Assume that the tempo of every clip is constant. Note that your algorithm should output two predominant tempi for each clip: 𝑇1 (the slower one) and 𝑇2 (the faster one). For example, you may 
simply try the two largest peak values in the tempogram over the whole clip. Please compare and 
discuss the results computed from the Fourier tempogram and the autocorrelation tempogram. 
The evaluation metrics of tempo estimation is as follows. We need to compute a “relative saliency of 
𝑇1” defined by the strength of 𝑇1 relative to 𝑇2. It is to say, for the tempogram 𝐹(𝑡, 𝑛), we have the 
saliency 𝑆1 = 𝐹(𝑇1, 𝑛)/(𝐹(𝑇1, 𝑛) + 𝐹(𝑇2, 𝑛)) for tempo value 𝑡 at a specific time at n. For an 
excerpt with ground-truth tempo G, the P-score of the excerpt is defined as 
𝑃 = 𝑆1𝑇
𝑡1 
+ 
(
1 
− 
𝑆
1
)𝑇𝑡
2 
𝑇𝑡𝑖 = {
1 if |𝐺 
− 
𝑇
𝑖 
𝐺 
| 
≤ 
0
.
08 
0 
otherwise 
, 
𝑖 
= 1,2 
Another score function is the “at least one tempo correct” (ALOTC) score, defined as 
𝑃 = {
1 if |𝐺 −
𝐺 𝑇1| ≤ 0
.08 or |𝐺 −
𝐺 𝑇2| ≤ 0.08 
0 
otherwise 
Compute the average P-scores and the ALOTC scores of the ISMIR2004 dataset and the eight genres 
(Cha Cha, Jive, Quickstep, Rumba, Samba, Tango, Viennese Waltz and Slow Waltz) in the Ballroom 
dataset using your algorithms. The above process can all be found in the evaluation routine 
mir_eval.tempo.detection. 
Note 1: if you want to use librosa.beat.tempo directly, you have to find some ways to let it 
output two tempi. 
Q2 

: Instead of using your estimated 
[𝑇1, 𝑇2
] in evalu
ation, try to use 
[
𝑇1/2, 𝑇
2
/
2
] , 
[𝑇1
/3
, 𝑇
2/
3
], 
[2𝑇1, 2
𝑇2], and [3
𝑇1, 3𝑇2] for estimation
. What are the resulting P
-score values
? 
Also, 
compare 
and discuss 
the results 
using the Fourier tempogram 
and the autocorrelation tempogram
. 
Q3  The window length is also an important factor in tempo estimation. Try to use 4s, 6s, 8s, 
10s, 12s for both Fourier tempogram and the autocorrelation tempogram and compare the ALOTC of 
the eight genres in the Ballroom dataset and ISMIR2004 dataset. 
Task 2: using dynamic programming for beat tracking 
Q4 Using librosa.beat.beat_track to find the beat positions of a song. Evaluate 
this 
beat tracking algorithm on the Ballroom dataset. 
The 
F-score of beat tracking is de
fi
ned as 𝐹 
≔ 
2𝑃𝑅/(
𝑃 + 
𝑅), with Precision, P, and Recall, 
R
, being 
computed from the number of correctly detected 
onsets 
TP, the number of false 
alarms FP
, and the number of missed onsets 
FN
, where 
𝑃 
≔𝑇𝑃/(𝑇𝑃 + 𝐹𝑃) and 𝑅 ≔ 𝑇𝑃/(𝑇𝑃 
+ 
𝐹𝑁
). Here, a detected 
beat is considered a true positive 
when it 
is located within a tolerance of 
±
70 
ms around the 
ground truth ann
otation. If there are more than 
one detected beat in this tolerance 
window, only one is counted as true positive, the others are counted 
as false alarms. If a detected onset is within the tolerance window of two annotations, then one true 
positive and one false negative will be counted. This process can be done with mir_eval.beat. 
Similarly, please compute the average F-scores of the eight genres in the Ballroom dataset and discuss 
the results. 
Q5  Also use this algorithm on the SMC, JCS, and ASAP datasets. Compare and discuss the 
results together with the results of the Ballroom dataset. Could you explain the difference in 
performance? 
N. 
Task 3: meter recognition 
Q6The meter of a song can be 2-beats, 3-beats, 4-beats, 5-beats, 6-beats, 7-beats, or others. 
There might be multiple meters existing in a song (e.g., a 4-beats section followed by a 3-beats section). 
As a task combining both beat tracking and downbeat tracking, meter recognition is still a challenging 
task. Could you design an algorithm to detect the instantaneous meter of a song? Test the algorithm on 
the clips in the JCS dataset, and report frame-wise accuracy. The 1, 2, 3, 4, 5 after every line in the 
annotation file is the meter annotation. You can simply use madmom.features.beats (the state
of-the-art beat tracker) or combine other functions mentioned above. 
The deadline for this homework is June 7 (Tue).