Mathematics, Music, and Emotion

Copyright © 2010, Glenn Story

I begin with a warning: this is going to be technical, both mathematically and musically. My hope is to explain a complex subject clearly.

I remember once being in a movie theater in Japan and laughing along with the audience even though I hadn’t understood the dialog. Perhaps you’ve experienced something similar. And I’m sure you’ve felt sadness and tears when you see someone else cry. This empathetic stirring of emotions based on the emotional state of others is, I believe, a normal part of human nature.

I’m sure you’ve also experienced having your emotions stirred by a piece of music: felt joy or sorrow from listening to a great piece of music–not from the lyrics, but from the notes themselves.

But how? How is it that a combination of abstract sounds played by man-made instruments can touch our heart and soul in a deep way; can move us to tears or laughter?

A friend loaned me a book about Indian music, which contains the following quote:

“‘Intelligible sound (Naada) is the treasure of happiness for the happy, the distraction of those who suffer, the winner of the hearts of hearers, the first messenger of the God of Love. Easy of access, it is the nimble beloved of passionate women. May it for ever be honoured. It is the fifth approach to the Eternal Wisdom, the Veda.’ ( Sangiita Bhaashya)” quoted in The Raaga-s of Northern Indian Music by Alain Danielou.

Further along in the same book, after a detailed analysis of the harmonic intervals (more on which later) is the following quote:

“The expression conveyed by these different classes of intervals is in no way arbitrary. It is related to psycho-physiological facts upon which all music depends. These intervals are used in all music more or less instinctively, by good singers and players of stringed instruments.”

I was stunned and intrigued by this assertion. It is saying that emotions conveyed by music are directly derived from mathematical relationships in the intervals between the notes.

What does this mean? The key word here is “interval”. An interval is simply the distance between two notes.

Every note has a certain pitch because a vibration is produced at a certain frequency. For example in western music A above middle C has a frequency of 440 cycles per second, or 440 Hertz (abbreviated Hz.) That is to say if a human voice, or violin, or flute or whatever sings or plays that note, it will be vibrating 440 times per second.

If two instruments (or voices) sound the same note we will recognize it as the same pitch because they are both vibrating at the same rate. They may have a different sound quality (or timbre) but their pitch is heard as being the same. The ratio of their frequencies is 440:440 or 1:1. Musicians call this interval a “unison”.

The next most fundamental interval has a ratio of 2:1. It is called the octave. In both western and Indian music (and also Japanese, and probably every other musical tradition as well) two notes that are one (or more) octaves apart have the same name or letter. Thus if we play A above middle C and then A an octave higher the latter note would have a frequency of 880 Hz. The ratio is 880:440 or 2:1. Similarly A an octave lower has a frequency of 220 Hz. 440:220 or 2:1.

The octave gets its name because the notes are eight notes apart in the standard western scale.

The next interval is the fifth with a ration of 3:2. The fifth is five notes away from the starting note in the western scale.

Then we have the fourth which has a ratio of 4:3.

The unison, octave, fifth and fourth are so fundamental that they are part of every musical tradition on earth. (In Indian classical music, it is these intervals that are played on the tampura as a tonal backdrop to the performance.)*

Thus, although there are differences in musical “languages” from one culture to another, there are certain aspects that are universal. And those aspects are based on mathematical relationships–ratios–of the frequencies of the notes.

Next, in western music, comes the major third with a ratio of 5:4 and then the minor third with a ratio of 6:5. Chords and melodies employing the major third sound happy; those employing the minor third sound either sad or angry. Thus we introduce the idea of emotion associated with certain harmonies.

Several years ago Leonard Bernstein, in a series of lectures at Harvard University (the “Norton Lecture Series”) suggested that the emotional quality of certain intervals derives from these ratios: The more complex the ratio the less happy he emotion.

Maybe. However, passion is often expressed in western music by a sixth with a ratio of 5:3. Perhaps too much passion is not a happy thing.

Nevertheless the fundamental idea that Bernstein was expressing is that there is a relationship between the physics of the note intervals and the psychology of the emotions they convey. “It is related to psycho-physiological facts upon which all music depends,” to quote again from the book on Indian music.

Let us now delve deeper into the Indian theory of music. In many aspects, Indian music is more complex than Western European music. The theory of the relationships of pitch intervals to emotions is no exception.

Before I begin this quotation I must explain that Indian notes are known by syllables as follows: Sa, Ri,Ga, Ma, Pa, Dha, Ni, Sa. Note that the first and last note are both “sa”, the latter being one octave higher than the former. There is not a definite pitch associated with each note the way C, D, E, etc. are used in western notation, even though the text I am about to quote implies otherwise.. Rather these syllables represent pitch intervals, similar to the way solfege (Do, Re, Mi, Fa Sol, etc.) does in western music. I’m sure the author realizes this and uses the absolute pitchs e.g. Sa = C for illustrative purposes.

Now the quote:

“Starting from Sa (C) = 1/1 which is neutral, the ascending fifths are all of the form 3″/2″+’, i.e. ratios fromed by 3 and its multiples divided by multiples of 2. Thus: G (Pa) = 3/2, D (Ri) = 9/8 = 32 / 23. A+ [The “+” represents a microtonal interval above the base pitch, in this case A. A+ is a smaller distance above A than is A#.] (Dha+) = 27/16 = 33/42…..

“These intervals represent an “active principle”: they all express sunshine, strength and joy.

“If we now take the descending fifths from Sa (C) these are all of the form 2n+x/3n. Thus F (Ma) = 4/3 = 22/3, B flat (Kakali Ni) = 16/9 = 24 / 32….

“These intervals represent a “passive principle”, they all express moonlight, beauty, peace.

“The expression of these two basic series has an impersonal character. Their ratios never use a prime number higher than three.

“In the next series a new element appears the prime number 5. We shall discover that whenever 5 appears as a constituent of the numerator, the interval whose ratios is so expressed conveys tenderness…. Whenever it appears as a constituent of the denominator, the interval whose ratio is so expressed conveys passion. ”

The author goes on to give ever more numerically complex intervals for various emotional colorations, including such nuances as Mriduh (tender), Karuna (pathetic), and Dipta (fiery).

I am amazed and somewhat overwhelmed by this complexity and detail.

I know of no similarly detailed analysis of emotions mapped to intervals in western music. Bernstein’s comments in the Norton lectures comes closest, but it is far simpler than that from which I have just quoted (which in turn is far simpler still than the text in its entirety.)

It is not clear to me how scientifically accurate all this analysis is. I think it would prove difficult to even measure people’s emotional reactions to music employing these various intervals. It is also not clear how much of this correlation between different types of intervals and specific emotions varies by cultures and how much is universal.

What is clear is that people everywhere emotionally respond to music. I recently heard a talk by Michael Tilson Thomas in which he said something like this (quoting from memory):

“We humans are capable of experiencing a vast range of emotions with many subtle nuances. Words cannot express all those nuances; music can.”

Reference: Danielou, Alain, The Raga-s of Northern Indian Music, 1968, Barrie & Rockliff, London

* A friend of mine disputes this assertion of universality, giving as his counter-example the didgeridoo, an instriument used by Australian aborigines. This instrument has very little pitch range, and relies almost entirely on changes in timbre (sound quality).

2 Replies to “Mathematics, Music, and Emotion”

  1. Music and Emotions

    The most difficult problem in answering the question of how music creates emotions is likely to be the fact that assignments of musical elements and emotions can never be defined clearly. The solution of this problem is the Theory of Musical Equilibration. It says that music can’t convey any emotion at all, but merely volitional processes, the music listener identifies with. Then in the process of identifying the volitional processes are colored with emotions. The same happens when we watch an exciting film and identify with the volitional processes of our favorite figures. Here, too, just the process of identification generates emotions.

    An example: If you perceive a major chord, you normally identify with the will “Yes, I want to…”. If you perceive a minor chord, you identify normally with the will “I don’t want any more…”. If you play the minor chord softly, you connect the will “I don’t want any more…” with a feeling of sadness. If you play the minor chord loudly, you connect the same will with a feeling of rage. You distinguish in the same way as you would distinguish, if someone would say the words “I don’t want anymore…” the first time softly and the second time loudly.
    Because this detour of emotions via volitional processes was not detected, also all music psychological and neurological experiments, to answer the question of the origin of the emotions in the music, failed.

    But how music can convey volitional processes? These volitional processes have something to do with the phenomena which early music theorists called “lead”, “leading tone” or “striving effects”. If we reverse this musical phenomena in imagination into its opposite (not the sound wants to change – but the listener identifies with a will not to change the sound) we have found the contents of will, the music listener identifies with. In practice, everything becomes a bit more complicated, so that even more sophisticated volitional processes can be represented musically.

    Further information is available via the free download of the e-book “Music and Emotion – Research on the Theory of Musical Equilibration:

    or on the online journal EUNOMIOS:

    Enjoy reading

    Bernd Willimek

  2. I have long wanted to respond to the comments by Bernd Willimek but until now have not found the time. I apologize for the delay.

    I have to say that I don’t find his arguments entirely persuasive. And I confess I probably don’t understand it completely. For example, what exactly is a “volitional process?” A google search returns no exact results but does include this quote from Wikipedia as a definition of “volition”: “Volition or will is the cognitive process by which an individual decides on and commits to a particular course of action.” If that definition is correct then “volition” is a “cognitive process” and thus “volitional process” is redundant.

    Nevertheless I get the sense that “volitional process” is a reference to the will, a sense that is reinforced by statements such as “If you perceive a major chord, you normally identify with the will ‘Yes, I want to…'” and a similar statement regarding minor cords and “I don’t want any more”. Thus, I believe the writer is suggesting that emotion (at least in regards to music) arises from will.

    I don’t believe this suggestion is correct, either from my own experience or from my (limited) knowledge of the organization of the brain. In short, I don’t believe emotion arises from volition. We may think so sometimes, as in I’m happy when my will is satisfied and upset when it is not.

    But experientially I have often felt a strong upwelling of emotion when listening to certain pieces of music. I can find no explanation of that upwelling, certainly not from satisfied or thwarted will, as to why that emotion arose. The best I can say is that the music simply invokes the emotion. It seems this is true directly.

    That is my experience. As an experience I claim it is valid. (I claim all experiences are valid as experiences.) As an explanation it is perhaps too simplistic. A deeper explanation involving psychology, neurophysiology, or (as in the case of my original post and the Indian work it quotes) the physics of sound.

    Yes, a deeper explanation is wanted. That doesn’t necessarily mean that the respondent’s explanation is the correct one.

    Let us then turn to the anatomy and physiology of the brain. It is my understanding that emotion comes from the lower more primitive areas of the brain, whereas will emanates from higher regions. So does will lead to emotion or does emotion lead to will? I claim both can happen. I could give examples of both but I’m sure the reader can find them from his/her own experience. In the case of music, again falling back on my own experience, I am conscious of no act of will either as cause or effect of the emotions I feel from music.

    I will also say that the respondent’s reference to “leading tones” etc., relates more to dissonance resolving to consonance. Musically, this is quite different from the concepts of major and minor (and their more complex analogs in Indian music) that are discussed in my post.

    In conclusion, to the extent I understand the respondent’s arguments correctly, I do not think will necessarily leads to emotion, and I have no conscious experience of it ever doing so in me with respect to music.

    Perhaps I should read the thesis to which the respondent refers, in order to better understand what the respondent means, and to see what empirical evidence there is behind it, but frankly I have no will to do so. And I’m not sad about that.

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