1. the processing lyrics should be

**complete,**which means there is no abbreviation like "repeat 2 times chorus" etc.

2. the lyrics should be

**pre-processed**by segmenting sections with blank line.

3. the chorus parts should be generally the

**same**.

There are

**three steps**in the algorithm:

1. calculate phrase-level similarity matrix, and it's lag-time form,

2. find repetitive sections which are the consecutive horizontal lines in the matrix,

3. group these repetitive sections.

Let's start!

**1. Phrase-level similarity matrix**

The similarity metric is defined as:$$metric=\frac{intersection\,words\,number}{largest\,word\,number\,of\,two\,phrases}$$And the similarity is:$$similarity=\left\{\begin{matrix} 0 & \mathrm{if} & metric < 0.8 \\ 1 & \mathrm{if} & metric \ge 0.8. \end{matrix}\right.$$Applying this similarity with each phrase pair in lyrics, we get the similarity matrix:

Fig. 1 phrase-level similarity matrix |

Fig. 2 lag-time form similarity matrix |

**2. Find the repetitive sections**

Finding the repetitive sections is simple. At this stage, we only consider the consecutive horizontal lines. The number of the beginning and the ending phrase of repetitive sections are stored in the below matrix. Mind that we unwrap each horizontal line by adding the lag-time to its $x$ coordinates. The (gene) means generative section which can be considered as the parent section generating the horizontal lines in Fig. 2.

begin | end | begin (gene) | end (gene) | |
---|---|---|---|---|

repetition 1 | 43 | 50 | 34 | 41 |

repetition 2 | 34 | 41 | 17 | 24 |

repetition 3 | 43 | 50 | 17 | 24 |

By observing this table, we found that these three repetitions belong to the same group because they share either the repetitive section or the generative section. So in next step, we mange to group these repetitions and re-search missing segment.

**3. Group repetitive sections**

Alter grouping the repetitive sections and re-searching the missing sections, we end up with four group:

begin | end | |
---|---|---|

group 1 | 17 | 24 |

34 | 41 | |

43 | 50 | |

group 2 | 1 | 15 |

group 3 | 26 | 32 |

group 4 | 52 | 64 |

Until now, it seems our job is done. But remember we said that we were going to deal with the discrete points in Fig. 2. Now we can be sure that these points are in our group 2, 3 or 4 and that also means one or several these groups belong to group 1 (remember we said these discrete points stand for the variation of repetitive section?). So we can set up a rule to re-group further the groups in Table. 2. For example, we could do the pair comparison of the first three words in the first lines of two groups.

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