Math and Music History

Math and Music History Photo Credit: Clipart.com

Compositions from different periods in music history have different mathematical properties.


Music history by the numbers. I’m Bob Hirshon and this is Science Update.

The history of Western music can be described mathematically. This according to University of Buenos Aires computer scientist Pablo Zivic, who worked with computational biologist Guillermo Cecci of IBM Research.

Using a massive database of music written between 1730 and 1930, they found that the major stylistic periods were marked by distinctive patterns of intervals, or the distance in pitch from one note to the next. For instance, Baroque melodies tend to take small steps of one or two semitones.

That is a contrastive difference from, for example, the Romantic period, which has a lot of intervals that rely in the three or four semitones.

Which reflects a stronger focus on musical chords. Zivic says their computations support the general consensus of music theorists and historians. I’m Bob Hirshon for AAAS, the Science Society.

Making Sense of the Research

We tend to think of art and science as being opposites: art is subjective, while science is objective; art is about beauty, science is about facts; and so on. However, there is science relevant to all kinds of art, and art can be described scientifically. This study shows how broad, seemingly abstract differences in how music sounds can relate to a relatively simple mathematical property.

Zivic's team used a database called the Peachnote Corpus, a massive collection of digitized sheet music. They analyzed music written between 1730 and 1930, which historians generally divide into four periods: the Baroque (in this data set, before 1750; famous composers include Bach, Handel, and Vivaldi), the Classical (1750-1820; Mozart, Haydn), the Romantic (1820-1900; Beethoven, Chopin, Verdi), and the post-Romantic period (1900-1930; Rachmaninoff, Bartok, Puccini). 

They found that when you lump all the compositions from these respective time periods together, distinct mathematical patterns emerge. All of the patterns hinge on the distance, in semitones, from one note to the next in a composition. A semitone is also known as a “half-step;” it's the smallest interval in Western music. You can hear a semitone interval if you play a white key and then an adjacent black key on a piano, or pluck a guitar string while moving your finger up and down one fret. 

The actual patterns are complex, but very broadly speaking, compositions from earlier periods tended to have shorter intervals than compositions from later ones. The way this translates into sound is that the melodies in earlier compositions tend to creep up and down in small steps, while those in later compositions take bigger leaps from high notes to low ones and vice versa. Those are widely recognized differences in the way music from each period sounds; what's new here is that it's verified and described using just numbers. 

It's important to note that the patterns don't apply strictly to any one composition from a given period, just like weather patterns from one season to the next don't apply to every single day. For instance, there are occasionally warm days in winter and cool days in summer. But stepping back and looking at the periods overall, the differences are clear. And this isn't the first time music and math have been related; music actually has a strong mathematical foundation, and the structure and pattern of notes in a piece is what separates not only the Baroque from the Classical, but also hip-hop from emo from reggae.

Now try and answer these questions:

  1. What was the purpose of this study?
  2. What mathematical relationship defined the different eras in music history that the researchers studied?
  3. How did the mathematical structure of typical music compositions change over time?
  4. Musicians: What other relationships between music and math can you think of?

You may want to check out these related podcasts:

The Science Update Browsing Music, from 2006, describes research that paved the way for sites like Pandora and Spotify to suggest new music based on what you listen to.

The Science Update Computer Composer describes an experimental computer program that writes its own original music.

Going Further

For Educators

For more on the science of sound, explore the lessons Properties of Sound Waves and Making Sound Waves Visible: Exploring Chladni Plates.

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