Making Music Out of Math
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Was math created or discovered?
Was music created or discovered?
From the bass line of your favorite song to the harmonies of Destiny's Child, math has played a massive yet almost entirely overlooked role in the music we know and love. Recall the age old question: was math created or discovered? The same question can be asked about music. Are the beautiful harmonies and melodies we humans love created from our own ideas, or were they already encoded in the fabric of our universe just waiting to be uncovered? Maybe by exploring the manners in which of math and music intertwine we can get a couple steps closer to answering those questions.
This presentation will explore the connections between math and music and connect those ideas back to the beautiful, intricate, and expansive universe of African American music.
Math in the Beat
WHOLE NOTES
Lasts the 'whole' duration of the measure.
HALF NOTES
Lasts one 'half' of the duration of the measure.
QUARTER NOTES
Lasts one 'quarter' of the duration of the measure
EIGHTH NOTES
Lasts one 'eighth' of the duration of the full measure
…and so on and so on. The same rules apply for rests. Any combination of these note/rest values (and many more) can be used as long as they add up to the total number of beats per measure which is given by the time signature in the beginning of the piece. That is the definition of basic division.
Example: Beats in Hip Hop
Listen closely to be beat in the back! Notice how the 4 beats of the measure are divided up into a series of quarter notes, quarter rests, and eight notes
Introducing Ratios:
Ratio- the relationship in quantity, amount, or size between two or more things
Examples:
  • the ratio for a quarter note is 1:4, meaning for every 1 measure there are 4 beats
  • the ratio for Spelman's student to faculty is 10:1, meaning for every 10 students there is 1 faculty member
  • you could even say Spelman has a 20:2 student to faculty ratio, you just scaled the ratio by a factor of 2
This concept is very important in the relationship between math and music!
Hearing the Math
There is a mathematical correlation behind why we find some combinations of notes so pleasing to our ears.
We can hear because our ears are able to register different frequencies whose values are described by the unit, Herz.
  • Different frequency = different pitch
  • High frequency = high pitch
  • Low frequency = low pitch
(Notice the differences between the movements of the black, blue, and red waves!)
Melodies, Intervals, and Octaves. Oh my!
Connecting Ratios and Frequencies:

31:45

YouTube

The Mathematical Problem with Music, and How to Solve It

There is a serious mathematical problem with the tuning of musical instruments. A problem that even Galileo, Newton, and Euler tried to solve. This video is about this problem and about some of the ways to tackle it. It starts from the basic physics of sound, proves mathematically why some musical instruments can never be perfectly in tune, and then introduces the main solutions that were proposed to solve this problem, along with their upsides and downsides: Pythagorean tuning, Just intonation, the Meantone temperament, and finally – the equal temperament, which is the tuning system almost everybody uses today in the West. === Further Resources === ▶ To learn more about the connections between music and mathematics, I highly recommend the book “Music – A Mathematical Offering” by David Benson. This terrific book is a treasure trove of information, extremely well written, and its thorough discussion of temperaments is just one of the many topics it covers. The book can be downloaded legally and for free from here: https://homepages.abdn.ac.uk/d.j.benson/pages/html/maths-music.html ▶ Sevish is a master of electronic microtonal music. His compositions, despite their ominous genre, sound light and fun. Check him out. https://sevish.com ▶ Paul Davids explaining how and why John Frusciante (of the Red Hot Chili Peppers) “mistuned” his guitar in the song “Scar Tissue”. https://youtu.be/Daw93bRHe4Y ▶ A Madrigal by Nicola Vicentino (1555), played on a 24-tone harpsichord tuned in meantone temperament, by Johannes Keller. https://youtu.be/0akGtDPVRxk ▶ A concise introduction to Arabic music. Pay attention especially to the Albayati, Alsaba, Alsard, and Ahuzaam maqams, with their intense microtonality. https://youtu.be/Rk8mY1cxKNI ▶ The Lumatone Isomorphic Keyboard is a cool interface to microtonal music. https://youtu.be/jyvQnAtlnek === Thanks to === ▶ Yehezkel Raz, the Ableton wizard, for transforming me from a complete Ableton noob to a good-enough user in less than two hours. https://yehezkelraz.com ▶ Dina Lurie, a dear neighbor and a great violinist (in the Irish fiddle tradition!), for contributing two notes and one double glissando. https://www.facebook.com/dina.lurie ▶ Alon Schab, a musicologist, and also a friend, bandmate, and academic colleague, for advice on some musical and historical issues. https://haifa.academia.edu/AlonSchab ------------------ ▶ The intro and outro music is “Snowfall Butterflies” by Asher Fulero (via YouTube Audio Library). https://asherfulero.com ▶ The flute and Qanun sound samples are from FreeSound. https://freesound.org ▶ Photo of the Antegnati Organ in Santa Barbara, Mantua (1565), courtesy of the organist Simon Lloyd. https://simon-lloyd.com === Contents === 00:00 - Intro 00:44 - What is sound? 02:42 - Melodies 04:48 - Intervals 07:00 - Choosing frequencies 11:56 - Pythagorean Tuning 14:33 - Just Intonation 18:36 - Meantone Temperament 24:34 - Equal Temperament 29:50 - Other temperaments 31:09 - Outro ------------------ The sound samples were prepared with Ableton Live 11. To make them piano-like but still as accurate as possible, I used the physical-modelling Pianoteq plugin, with the unison width set to 0, octave stretching ratio set to 1, string length set to its maximum value (to minimize string inharmonicity), hammer noise set to 0.5, pedal noise set to 0, and the velocity-to-dynamics curve considerably lowered (ending at mezzo-piano). ------------------ Created by Yuval Nov for the 2022 "Summer of Math Exposition" (SoME2) competition, hosted by the one and only 3Blue1Brown (Grant Sanderson). ------------------ #math #music #temperament #tuning #Pythagorean #meantone #3b1b #SoME2

Video Excerpt (2:40-4:16) Summary: a melody with 3 notes of 440Hz, 660Hz, and 733.3Hz is played. Then another melody with 3 notes of 550Hz, 825Hz, and 916.6Hz is played. Numerically, the melodies are not the same, but both sound like the song "Twinkle Twinkle Little Star" because the ratios between the 2nd and 1st, and 3rd and 1st notes are the same for both songs (3/2 and 5/3, respectively). Changing all the frequencies by the same factor so the rations between stay the same is called transposition!
MELODIES
Notes may have entirely unrelated Herz frequencies, but as long as they have identical ratios between the notes, the melodies will be the same, just in different keys.
The process of changing all the frequencies by the same factor so the ratios stay the same is called transposition.
INTERVALS
The distance between notes is known as an interval. All intervals correspond to certain ratios. There are some intervals that sound more harmonious and pleasing to the ear than others.
OCTAVES
An octave is an interval that has a ratio of 2:1
i.e. 880Hz : 440Hz or 687Hz : 343.5Hz
The two notes that make up the lower and upper octave have the same note name and sound very harmonious to us.
Other Important Intervals and Ratios
MAJOR 6TH- 5:3
PERFECT 5TH- 3:2
PERFECT 4TH- 4:3
MAJOR 3RD- 5:4
MAJOR 2ND- 9:8
The ChroMATHic Scale
The Chromatic Scale includes 12 equally spaced notes in an octave and is used most commonly in Western music. You are probably familiar with the Major Scale where there are 8 notes in an octave, but the intervals between those 8 notes are not all the same.
In order to determine those identical intervals, the 12th root of 2 needed to be taken to determine the magnitude of the interval between all 12 notes. This wasn't mathematically possible for a long time. But once that was discovered, the 12 corresponding frequencies that were equally spaced where formatted into the Chromatic Scale!
Note: There are several much more complex reasons to why the Chromatic Scales works out the way it does. This is a very simple explanation that does not fully acknowledge the complexities of how music works mathematically/physically.
Applications to African American Music Genres
Let's now explore some of the many ways that these mathematical principles are used in African American Music.
Gospel
Contemporary Gospel groups like Take 6 and Gospel artists like Kirk Franklin that are known for the complex harmonies in their music employ the harmonious intervals that were discussed previously.

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YouTube

Take 6 LIVE - Take 6 Medley

Take6.com With a staggering eighteen Grammy-Award nominations, Take 6 is the most nominated Gospel, Jazz, Pop or R&B artists in Grammy history. In the music industry, Take 6 is so universally recognized as simply being the best, that they have virtually owned Downbeat magazines readers poll having won Favorite Jazz Vocal Group 9 consecutive years. Take 6 has received some of its highest praise from the music industrys icons. Mega producer and longtime collaborator Quincy Jones, has described Take 6 as The baddest vocal cats on the planet. In their stellar career, they have been honored to perform with numerous music legends including Ella Fitzgerald, Miles Davis, David Foster, Al Jarreau, Stevie Wonder, Denyce Graves, The Yellowjackets and Wynton Marsalis among a host of others. The Take 6 style has also reached todays pop culture. Their musical style and tight harmonies have influenced pop groups from Boyz II Men and Backstreet Boys to *NSYNC. antcreations.com

Jubilee Quartets
The typical 4 voice parts of of a Jubilee Quartet also use harmonious intervals. Typically, the ratio between all the voice parts in proportional!
Ragtime
Ragtime is known for its swinging and syncopated rhythms. Think back to the division principles from the beginning that say you can have any combination of note and rest values as long as they add up to the total number of beats per measure. That is what syncopation is as well. Additionally, think of a piano, the main instrument of this genre. Pianos function with a series of vibrating strings that correspond to a note finely tuned to a specific frequency in Herz. The type of frequency wave generated by a vibrating piano string is very similar to the animation of the black line below. Notice that the black line is fixed at both ends, while the red and blue line move freely at the ends.
The same can be said for Hip-Hop music. Recall the example at the beginning of the presentation, or think of some of your favorite rap songs!
Jazz
Think of all the different instruments used to play jazz: pianos, guitars, saxophones, drums, clarinets, trumpets, etc. All of those instruments vibrate out different frequencies in different ways. The different types of sound waves they output is what makes a saxophone and a trumpet sound so different, but still very similar, and what makes those two instruments sound so much different from a drum.
THANK YOU FOR LISTENING!
Music and Math
Evidently, music and math share so many more commonalities than one would ever guess, and there are hundreds of more connections that were not mentioned in this particular presentation. You almost can't have one without the other. At this point there is more research about the similarities between math and music than there is research about how those math principles can be applied and analyzed uniquely through the lens of African American music. With our intersectionalities as Black women along different career paths, it is important that we close that gap in all research fields. Nevertheless, it is clear that you can make math out of music and music out of math!
Sources
E, M. (2022, October 3). How to Read Music - Part 1: Music notation. School of Composition. https://www.schoolofcomposition.com/music-notation/
Lynch, P. (2017, May 25). The Mathematician’s Patterns, Like Those of the Composer, Must be Beautiful. The Irish Times. https://www.irishtimes.com/news/science/the-mathematician-s-patterns-like-those-of-the-composer-must-be-beautiful-1.3096020
Merriam-Webster. (n.d.). Ratio definition & meaning. Merriam-Webster. https://www.merriam-webster.com/dictionary/ratio
Roberts, G. E. (2016). From Music to Mathematics: exploring the connections. JHU Press.
Wright, D. (2009). Mathematics and music (Vol. 28). American Mathematical Soc..
YouTube. (2014, August 18). Music and Mathematics - Mathematician & Concert Pianist Eugenia Cheng. YouTube. https://www.youtube.com/watch?v=6b5MM4x-3aQ
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