Series: Robot Music I, Robot Music II: Modes, Robot Music III: The Circle of Fifths, Robot Music IV: Scales of the World

Robot Music Logo - Circle of Fifths

New logo baby.

Robot Music I proved that a pleasing song can be randomly generated under the right guidelines. Those guidelines stuck to a 4/4 rhythm and the C Major Pentatonic Scale.

Robot Music II proved that the C key could be replaced with any other (changing the root). It also showed that the Major Pentatonic Scale (5 notes) could be expanded to the Major Scale (7 notes) without harming our method. And most importantly, this section covered modes and demonstrated that shifting a song’s mode can significantly change its tone. Knowing that the tone can be changed, how can we control it?

Robot Music III introduces the Circle of Fifths. It is a basic musical tool that will help us predict the impact of a mode of the Major Scale. With it, our Robot can place notes along 2 dimensions (Consonance vs. Dissonance, Dominance vs. Sub-Dominance) and estimate the emotional impact of a note according to an arbitrary algorithm based on personal tastes.

This installment is the first to go into the perception of music. To begin, we must discuss aural perception and some musical fundamentals. That technical information will be related to our previous musical concepts by the use of the Circle of Fifths. And finally, a demonstration of mode relationships and some discussion of their popular uses.

You probably already know some of this, so I’ll try to move fast.

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How Sound is Felt

The sensation of sound is created by waves of varying pressure in the air. When a guitar string is plucked, it smacks against air molecules, smashing them together on one side while pulling them apart on the other. Then the string swings back the other way, pulling apart the air on the first side and smashing it on the second. That vibration continues, creating a pattern of high and low pressure ripples in the air that flow out from the string, just like if you splash in water, you see ripples spreading out from your hand. The more quickly the string vibrates, or the hand splashes, the more ripples are created in a short time. That raises the frequency of the ripples. The ripples in the air hit your ear and vibrate your ear drum, just like the ribbon in a microphone. Your inner-ear converts that to an electrical signal (just like a microphone does) and transmits it to the brain. The brain then interprets that data, filtering it and deciphering its meaning.

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Technical Terms

A sound is characterized by the shape of its wave. To better understand waves and a couple other musical concepts, you’ll have to understand these terms:

Pitch and Frequency: The frequency of a wave is interpreted as pitch, or how “high” or “low” a note is. Higher frequencies create a higher pitched sound and lower frequencies, a lower one.

Wavelength: The length of a ripple (1 cycle) is the wavelength. Waves with high frequencies are made of ripples that are very close together so they have shorter wavelengths.

Period: The period is the time it takes for 1 cycle to pass. In regards to sound waves, a shorter period is related to a shorter wavelength.

Amplitude: The height of a wave is its amplitude. It is the “loudness.”

Interval: 2 notes played at the same time.

Chord: 3 or more notes played at the same time.

Beat: When an interval is played, the two waves combine, and they interfere with each other. Sometimes the waves amplify each other (become louder), sometimes they cancel out and become quieter. Most of the time, the waves go back and forth between loud and quiet. The “going back and forth” is a beat. When you hear a guitar or piano chord ring out, listen for the beat going “woOowoOowoOowoOowoOow.”

Consonance and Dissonance: Wave combinations that create an extremely tiny or very long beat are considered consonant while waves that create a short beat are called dissonant.* Consonance is generally considered beautiful, if not boring, and dissonance can be interesting, but very harsh.

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Creating Consonance or Dissonance

To be consonant, waves must be able to “fit together” neatly. This means, that if you tried to fill the space of one wave with another wave, you could do it with as few remainders as possible. Yes, I mean remainders like in math; division. If you have a remainder, the beat will be equal to the number of cycles necessary for the remainder to add up to 1 (more or less).

Waves and Beats

Illustration of Waves A, B, and C

For example, if Wave B has 3/2 the frequency of Wave A, they share a lot of qualities. Waves A and B will both loop at the same points. Wave B will loop 3 times for every 2 loops of A, but every time that A’s cycle ends and begins again, so does B’s. These two waves are extremely consonant and will amplify each other. Wave C, however, has 11/10 the frequency of Wave A. If both waves start at the same time, the ending/beginning point of their cycles won’t meet again until Wave A has looped 10 times and Wave C has looped 11 times.

The time it takes for the two waves to begin and meet again is the length of their beat. The beat created by Waves A and B is so short (equal to twice the period of Wave A) it cannot be heard.** The beat created by Waves A and C would be much more distinct, it would sound very rough.

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Note Ratios

These are the ratios of the popular 12 notes (multiply a note’s ratio by the frequency of C to discover the frequency of that note–Or look at this chart):

Note Ratios

Note: These are the ratios according to Just Intonation. In fact, modern instruments are generally tuned to Equal Temperament. Equal Temperament involves a different set of ratios that were developed to allow musicians to easily change keys, but to come as close to Just Intonation as possible.

You can probably tell that some of these ratios will fit together better than others (3/2 creates a beat of 2 cycles; 16/15 is 15 cycles). In Robot Music I, we used the half-step as a basic unit of measure for notes. It places them in order of frequency from lowest to highest. Now, by comparing the consonance and dissonance of different intervals, we can create a new sequence of notes; the Circle of Fifths.

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The Circle of Fifths

This is the circle of fifths:

Circle of Fifths - NotesCircle of Fifths - DegreesCircle of Fifths - Ratios

Here it is written in notes, then again in degree numbers and in ratios.***

Note: I’ve begun using “b” to mean flat instead of “#” to mean sharp. You should know that an A# is equal to a Bb; they are the same note. # means “+1 half-step” and b means “-1 half-step.” In the case of degrees using b is more common.

Each note is the 5th degree (the dominant) of the note before it and the 4th degree (the sub-dominant) of the note after it in the sequence. In other words, G is the 5th of C and F is the 4th of C. Or if you look at the E note, you know that B is the 5th of E and A is the 4th of E.

C – D – _E _FG – A – _B

E – F# – G# AB – C# – D#

1 – 2 – _3 _45 – 6 – _7

The greater the distance between any two notes in the sequence; the more dissonant they are as an interval. Therefore 2 intervals of equal distance from the root are equally consonant or dissonant. Furthermore, 2 notes of nearly equal distance in the sequence have a nearly equal consonance or dissonance.

In addition to consonance and dissonance, the circle of fifths illustrates dominance and sub-dominance.

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Dominance and Sub-Dominance

Disclaimer: I am making these comments on dominance and sub-dominance not because I have learned of it elsewhere, but because I have felt them to be accurate and useful. Elsewhere, the term dominant is used synonymously with the 5th degree and the sub-dominant with the 4th degree. This concept will be readdressed later.

Dominance and Sub-Dominance involve which note is established as the root in a chord or riff. If you don’t know which note is the root, you don’t know what perspective your scale is being played from so you can’t control its effects. Here the circle is divided into dominant and sub-dominant halves.

Circle of Fifths - Dominance

You can consider everything on the left side to be dominant, everything on the right to be sub-dominant, and the blue 5b to be completely neutral. I find that playing the intervals on the dominant side clearly establishes the interval’s 1st degree as the root of your riff or chord. The sub-dominant intervals may be less clear or even establish themselves as the root.

For example, in rock songs you often hear a 5th interval (1 and 5) like the C and G notes being strummed at the same time. The 4th interval is equally popular (1 and 4), like C and F strummed at the same time. The difference between the two is that between C and G, C will be established as the root note, but between C and F, F will take control.**** Often sub-dominant notes can obscure your root. That’s not necessarily bad, but it must be known. The closer the interval gets to the neutral 5b, the less dominant or sub-dominant effects they have and the more neutral they become.

The first note played and the lowest note played also are more likely to be established as the root, but the dominance/sub-dominance effects won’t really come in until chords and inverses are being used. We won’t get into that yet.

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A Spectrum

As you can see, the general effects or what you might call the “meanings” of notes are accurately mapped out by the Circle of Fifths. It presents notes on a continuous spectrum according to their similarity as the ear and brain perceive them.

Circle of Fifths - Degrees - Hues

The Circle of Fifths is often compared to the color spectrum and remarkably similar to it. Coincidence? I don’t know, but I doubt it (there are theories). The important thing is that you recognize the sound of the 1st degree as very similar to the 4 and 5. Somewhat less similar to 2 and 7b. Less similar to 6 and 3b. And so on down the list.

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Modes Over the Circle

You should remember modes from Robot Music II. Now we can marry those ideas to the Circle of Fifths.

A mode of the Major Scale is defined by its root degree. Each one has a unique tone about it, but we haven’t yet discussed the relationships of the different tones. In fact, just as the Circle of Fifths maps out perceived similarities in notes, it also maps out similarities in the modes of those notes in the Major Scale.

For example, the 1st Degree’s mode (Unison) is almost identical to the mode of the 5th degree (Myxolydian):

1st Mode/Unison:
x – x – x x – x – x x

5th Mode/Myxolydian:
x – x – x x – x – x x

The only difference is the placement of a single note. The same goes when you compare the Unison to the mode of the 4th Degree (Lydian):

1st Mode/Unison:
x – x – x x x – x – x

4th Mode/Lydian:
x – x – x x x – x – x

However, the Unison is very different from the mode of the 7th Degree (Locrian):

1st Mode/Unison:
x x x x x x x

7th Mode/Locrian:
x x x x x x x

The Unison and the Locrian only share 2 notes. That’s still 1 more note than is shared between the Lydian and Locrian which are complete opposites (as they are on the circle):

4th Mode/Lydian:
x x x x x x x

7th Mode/Locrian:
x x x x x x x

The relationship between the mode of each degree is analogous to the relationship between the placement of each degree on the Circle of Fifths. Since the modes themselves are related in this way, the tones they generate are related as well. This is how the Circle of Fifths can allow our robot to control mood changes by shifting modes.

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A Tone Spectrum

Often the words used to describe the tones of the opposing ends of the modal spectrum are “Light” and “Dark.” You can also say “Consonant” and “Dissonant” which would be somewhat more accurate. Or you can use the chromatic analogy of “Warm” and “Cool.” Regardless of semantics, the effects are the same. There is a general feeling produced by modes in the direction of the Lydian, Unison, and Myxolydian that is significantly different from that of the Locrian, Phrygian, and Aeolian. Your job is to listen to each mode and learn those differences, then you can provide your Robot with an algorithm that selects and changes modes according to your tastes.

Ideally you would listen to each mode in a variety of uses, but it would be a bit too much work for me to record several songs for each mode. If you have access to an instrument, try improvising in the different modes to get a better feel for them. Here are the modes themselves, played in C (try to play different modes back to back):

4th Mode/Lydian:
x – x – x – x x – x – x

I always found it odd that this mode was considered “Bright” because it’s only 1 of 2 modes with the highly dissonant Devil’s Interval (1 + 5b). Still, it shows up often in upbeat music.

1st Mode/Unison:
x – x – x x – x – x – x

Your standard Major Scale. This winds up in all kinds of music these days. You can bet that anything poppy is in the Unison Mode.

5th Mode/Myxolydian:
x – x – x x – x – x x –

The Myxolydian is popular because it’s the same as the Unison, but with a distinctive flat 7 interval (1 – 7b). The flat seven is big in Blues, Jazz and much of Rock.

2nd Mode/Dorian:
x – x x – x – x – x x –

The Dorian seems to be the least popular of modes, though it contains an interesting mixture of the Major and Minor sounds. I’ve heard it played by some recent Rock bands (maybe Post-Rock).

6th Mode/Aeolian:
x – x x – x – x x – x –

The Aeolian is best known as the famous Minor Scale. You’ll find it in any song that means to take a somber tone.

3rd Mode/Phrygian:
x x – x – x – x x – x –

I’ve always liked the Phrygian. It’s rarely used, but it has an interesting sound.

7th Mode/Locrian:
x x – x – x x – x – x –

The Locrian shows up in a lot of what you would expect to be “darker” music: Death Metal, Hardcore, etc.

Jazz musicians after Bebop explored all of these. Of particular note is Miles Davis’ experimentation with Modal Jazz.

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That sums up the 3rd session of Robot Music. Next we’ll go over brand new scales, some exotic, and completely different from the Major Scale. Our Robot will have much to choose from.

-Christopher J. Rock

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*Waves that are exactly the same have an infinitely long beat, they are perfectly consonant and create perfect amplification. Waves that are extremely close in frequency have a beat so small it cannot be heard, but they also have a very long beat that can be heard. Waves with a small difference in frequency have an audible and short beat; a lot of dissonance. Waves with more distant frequencies can have varying beats or none at all.

**If Waves A and B were slightly out of tune, they might generate a very long beat that would also qualify as consonant.

***Another term to know is Inverse. As you move from the 1st degree left or right on the circle of fifths, you’ll find the numbers to the left to be twice the inverse of the numbers to the right (except for the 5b in the middle with a ratio of 7/5). That is because the intervals to the left are the inverse of the intervals to the right. Playing a 4th interval is the same as playing a 5th interval, but from the perspective of the opposite note (a different root). Besides that, the intervals are equal.

****Resulting in an inversed 5th interval.
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Notes

Relativity and Proportion in Perception

Most of our judgment is based on relativity. It’s difficult to judge a person’s height, but you can always tell how tall they were compared to you. It’s bright outside because you’ve been in the dark all day. I can’t see you because the sun is in my eyes. How old is she? I don’t know, but she’s older than I am.

Part of using relativity in our judgment is recognizing proportions. I’m just as big to my little brother as he is to our little sister. How fast was the red car going? I don’t know, about twice as fast as the blue car. A mouse is to a cat as a cat is to a dog. The Moon has 1/3 the gravity of the Earth.

How do these apply to music? When a tenor sings a song and a baritone sings the same song, the difference in pitches is obvious, but it’s also obvious that the songs are identical. Your mind recognizes the relationships of the notes sung by the tenor as proportionate to the relationships of the notes sung by the baritone, even when the notes themselves are completely different. Perception of notes is then contextual (dependent on surrounding notes).

Context is the reason we pay attention to scales. The scale is the general context of a note. The mode (defined by the starting point; the root) modifies the meaning of that context.

About the author:
Christopher J. Rock (http://)
Film student at California State, Long Beach. I want to make the gaming world a better place.
  • Mark Sandborn

    I am a researcher, educator, music theorist and
    technology developer regarding color, sound, and language. For the past 18
    years I have worked professionally in this domain. A significant part of my
    research is predicated upon the pitch helix regarding Shepard’s 1964 seminal
    work, along with the continued advancements made by Deutsch and Hajdu. I have
    advanced the logarithmic model substantially including a

    highly
    detailed mapping of the categorical perception of the hitherto undefined
    microtonal space. S

    pecifically I
    have been focused on the harmonic science of sound (which involves both the
    study of music structure and linguistic structure) and the harmonic structure of
    the photon, the harmonic symmetry of the electromagnetic wave, and the harmonic
    science of tristimulus color. As I have found through my many years of
    investigating the topic of color and sound, virtually all research has been
    specifically focused on the relationship of color to pitch which is flawed in
    that it does not provide a color or sound symmetry solution, i.e., duration must
    be part of the symmetry solution. Most correlations between color and sound
    were a one to one non-scientific mapping of the perceived order of the
    incomplete visible spectrum color to the linear frequency ordering of the
    Chromatic circle of pitch (the first such instance being the color organ of
    Castel in 1740). Prior to Castel, obviously Newton proposed the 7 color rainbow
    ROYGBIV correlation. Upon the introduction of the opponency theory of color
    developed by Ewald Hering in 1892, as well as the tristimulus perception of
    color, these color schemes were proven invalid. In the last century,
    individuals such as Scriabin, Hauer, Otterstrom, and Malinowski began looking
    for other means to correlate color to sound such as the perception of harmonic
    key (correlated to the Circle of Fifths) potentially defining a cumulative
    stimulus similar to color tristimulus. However, none of them, or others could
    define the harmonic science correlating these modalities which most
    significantly resulted in skewed color palettes that could not provide any
    results defining properties of pitch by literal known principles of color
    theory.

    Most scholars from either the fields of music or color are unaware that there
    exists a wide range of research on this topic within the field of linguistics
    dating back to the 1960’s. Specifically this area of research was initiated by
    Yilmaz (1967,1968) regarding the direct correlation between tristimulus color
    and the tristimulus formant structure of phonemes (vowels and consonants).
    Shepard also investigated this in his 1974 paper. My research has not only
    further validated Yilmaz’s research, as well as the work of others (such as
    Liljencrants, 1972), but has evolved the science considerably. The harmonic
    model which I have established is actually defining every attribute of music
    structure by color theory in detail and is a 1:1 correspondence at multiple
    levels of hierarchy. For example, I can define why a particular type of chord
    has a particular perception by it’s color properties. I can show why musical
    key exists and how scales are defined by their color properties, including
    demonstrating a previously unidentified logarithmic scale wave structure that is
    directly tied to the work of Hajdu and Shepard. Furthermore, I have taken the
    work of Yilmaz and shown direct correlation to pitch and duration which he
    strongly hypothesized based on the evidence. To be more specific, I can define
    that not only can pitch be correlated directly with color, but duration (rhythm)
    can be as well. My research is directly following on a substantial amount of
    other research demonstrating that vowels are correlated with pitch and that
    consonants are correlated with duration. My model also demonstrates that the
    duration component of sound defines 12 macro categories of perception (inferred
    by Risset) identical to the 12 macro categories of pitch perception defined by
    Hajdu. From these overarching 12 fundamental categories of perception, I have
    elucidated a hierarchy of both macro and micro harmonic structures, i.e., that I
    have a harmonic mapping of micro pitch and micro duration as it’s counterpart
    for which concepts such as micro scales, micro chords, and micro keys can be
    discerned and manipulated intelligibly through color properties. Two very
    important discoveries which evolved from this research were an understanding of
    a previously unobserved key structure for 24 tone temperament and the discovery
    that 7 Tone-Equal-Temperament defines a monochromatic color key. The totality
    of this science is a fusing of the properties of color, the harmonic structure
    of music, and the harmonic structure of language. This results in the ability
    to implement all of these modalities for research, analysis, music reading,
    music education, and to explore new frontiers within the visual, musical, or
    linguistic domains, or a combination thereof.

    Beyond my theoretical research and writing, I have been intimately
    involved with how these findings can be implemented into music theory, analysis,
    composition, and music reading as an educational system. Since 2001, I have
    developed an entirely new music education methodology which implements these
    theoretical concepts. I mention this because the implementation and teaching of
    this system for nearly a decade has provided results which dramatically defy all
    historical expectations of music cognition and learning curves, lending credence
    to its correctness. Western music notation is still employed, but the
    organizational structure of the educational approach is dramatically altered.
    The use of properly correlated color which contains meaning is a significant
    factor for these results. However, just as significant is the ability to employ
    my advanced logarithmic model for both pitch and duration and easily convey
    these advanced and complex harmonic concepts to young students as if they were
    at the collegiate level. There is no doubt that the ability to define
    properties of color as they would exist for the visual artist, including
    concepts of color mixing and structural organization (i.e, color contrast,
    afterimage, complementary structure, primary structures, analogous
    relationships, etc…), are a significant factor in the linguistic communicating
    of all harmonic concepts.

    The following links provide significantly more information regarding
    this science.

    https://www.facebook.com/Virtuosoism?ref=hl

    Link to a PDF of the mapping of helical harmonic space: http://www.virtuosoism.com/Symmetry%20Field%20Circle%20Reference.pdf

    Link to the book A Rosetta Stone at GoogleBooks: http://books.google.com/books/about/A_Rosetta_Stone.html?id=Qx1MBAAAQBAJ

    YouTube link to Debussy’s Arabesque in Spectrum Color: http://youtu.be/PqItTVJSHnU

    YouTube link to Coldplay’s Viva La Vida within the Spectrum Color format:
    http://youtu.be/PqItTVJSHnU

    • ChrisJRock

      Awesome. I’d love to hear more. I’ll be in touch!