A Short Lesson in Music Theory
The Pythagorean tuning system produces perfectly consonant fifths, but presents several issues for musicians. For example, enharmonic notes such as A♭ and G♯ are considered to be different notes with distinct frequencies. In the equal temper tuning we use today, enharmonic pairs are treated as the same note because they have the same interval ratios and frequencies.
The real issue with this system is that the circle of fifths does not close. That means that as you progress through the notes of the scale, you won't end up on the same note an octave above where you started. Some adjustment needed to be made in order to close the circle, and that adjustment was to lop off the last interval and make it smaller then the rest. Thus, the Pythagorean tuning system consists of 11 intervals of perfectly just fifths, and 1 slightly shorter, badly-tuned interval known as the wolf interval. This wolf interval is the direct result of forcing the circle of fifths, which is actually spiral in nature, to be a circle.
- 3-limit has consonant fourths and fifths, but dissonant major and minor thirds
- 5-limit has consonant major and minor thirds, but some dissonant fifths
- 3-limit tuning lives on today mostly in the form of 2-dimensional isomorphic keyboards. More will be said about these later, but for now it will be noted that the way to eliminate the Wolf interval is to add a spatial dimension to our musical instruments.
- 5-limit tuning became popular in the late Middle Ages because chords and triads are based on three notes, which means major and minor thirds need to be consonant in order to achieve a certain complexity in the music. Chords are the basis for most of the music we are familiar with today.
5-limit tuning also has its own issues, and its own wolf intervals. As shown in the table above, the notes F# and G♭ are extremely close to each other, but not equal in ratio/pitch. By seeking to use just intonation and produce pure harmonic ratios, equal interval sizes are sacrificed. This means that each of the interval types, except for the octaves, has three or even four different sizes. 5-limit tuning uses a combination of justly tuned fifths and non-just fifths of several sizes to close the circle of fifths. It's a complex and imperfect system.
Musical Ratios in 2-Dimensions - Circles
It turns out there are a lot of ways to represent music using 2-Dimensional figures, but circles are the most natural element for this visual representation of sound. Because circles are defined by one number (radius), they can be thought of as a 2-D embodiment of number itself. For more information, see my previous blog post Circles as Numerical Sequences.
The chromatic scale represented by the circle radii above shows all the notes stacked on top of each other. Each of the 7 major diatonic notes are shown individually in the table below. It demonstrates the relative sizes of the intervals.
By using circles to represent musical ratios, we can:
- Express ratios in terms of linear measurements like radius, diameter or circumference
- Express ratios in terms of square measurements like area
- Change the positions of the circles to create an interplay of ratios using linear and square measurements
The red circles are in a perfect fourth ratio (4/3) and the blue circles are in a perfect fifth (3/2) ratio. There are also three pairs of circles in octave ratios (2/1, 4/2, 6/3) and three other perfect fifth pairs (3/1, 6/1, 6/4), some spanning more than one octave.
I've seen designs like this appear in crop fields around the world, most notably in the UK. It appears in many forms, but has the same underlying structure. For example, the following design appeared near Liddington Castle in the UK in June of 2010.