EQ (equalization) plugins are volume controls for specific parts of the frequency spectrum. Every DAW, mixing board and guitar amp has EQ controls, and they can radically transform your sounds. But while EQ is an essential part of audio engineering, it is also a source of confusion for beginners. In this post, I lay out some key vocabulary.
Category Archives: Math
Tuning is hard
I am committed to teaching my New School music theory students something about the history of tuning in Western European music. I don’t expect them to retain any details or do any math, I just want them to know that the history exists. In preparation, I continue to refine my explanation of this history to myself.
Before the year 1400 or so, Western Europeans mainly tuned their instruments in three-limit just intonation, which they called Pythagorean tuning. (Don’t be fooled by the name; this system was in use in Mesopotamia centuries before the Greeks described it.) Three-limit just intonation is based on the first three harmonics of a vibrating string. Western Europeans really like the pitch ratios produced by these harmonics, as do people from many other cultures (though not all of them). In this post, I will explain why Europeans liked three-limit just intonation, why they nevertheless eventually abandoned it, and what came after.
The harmonic family tree
My blog stats have made it crystal clear that very few of you want to read about tuning systems. However, a vocal minority of you do love reading about them, and I definitely enjoy writing about them. So, let’s dig in and see how much Western harmony we can derive from the natural overtone series!
Seventh chords in just intonation vs 12-TET
I enjoy listening to Jacob Collier explain his music more than I enjoy the music itself. His arrangement of “Moon River” is mostly exhausting. However, Miles Comiskey pointed me to an interesting moment in this explainer video at the 1:04:22 mark where Jacob talks about how Kontakt enables you to change your instrument tuning on the fly.
Jacob takes a dominant seventh chord and plays it in two different tuning systems: twelve-tone equal temperament, the system we’re all used to, and just intonation, which is a more “pure” harmonics-based system. The chord sounds very different in the two systems. That is a profound musical concept that is not easy to understand! Jacob buries it in his song under five thousand other ideas, but I thought it would be helpful if I built a whole track around it:
The problem with just intonation – a visual guide
Tuning is the final frontier of my musical understanding. I start reading about it, and then I hit a big table of fractions or logarithms and my eyes immediately glaze over. However, tuning is important and interesting! So I continue to struggle on. Fortunately, as with so many music theory concepts, the right computer software can open up lots of new learning avenues. I have been having a great time with MTS-ESP by Oddsound. It was designed to help you hear and play different tuning systems, but it also visualizes them in an attractive circular way. If you read this blog, you know how much I love a good circular music visualization scheme.
So here is the basic problem with tuning. An ideal system (for Western people) would be based on the natural harmonic series, because we love how harmonics sound. This kind of tuning system is called just intonation. It sounds lovely! Unfortunately, just intonation makes it impossible to change keys orĀ tune your guitar. Let’s use MTS-ESP to figure out why that is. Continue reading
The Well-Tempered (and not-so-well-tempered) Clavier
Bach wrote The Well-Tempered Clavier as a showcase for a new tuning system that could play in all twelve major and all twelve minor keys. Up until that point, the various European tuning systems only worked for some keys, not all of them. If you were in or near the key of C, you were usually okay, but as you moved further out on the circle of fifths, things got ugly fast. So this new tuning system that actually sounded good in all the keys was an exciting development.
However… no one knows what tuning system Bach used. All we know is that it wasn’t twelve-tone equal temperament, the one we all use now. There were many systems in circulation at the time that people called “well temperament.” Was Bach using Werckmeister? Kirnberger? Kellner? Some idiosyncratic system of his own invention? No one knows. This video sums up the situation well:
Until this gets resolved, at least technology makes it easy to hear these different systems for yourself. I used Oddsound MTS-ESP to run some of the Well-Tempered Clavier preludes through various historical tuning systems. Here’s what I got:
Just intonation and key changes
Western people like two things in harmony: intervals derived from the natural overtone series, and the ability to play in multiple keys. Unfortunately, it’s not possible to do both of these things within the same tuning system. If you want to use just intonation intervals derived from harmonics, then they will not work in every key. So we as a civilization have decided to use a tuning system that enables you to play in lots of different keys, even though it means that all of the keys are slightly out of tune. Fortunately, the computer makes it easy to explore alternative tuning systems. I have been experimenting with this cool tuning plugin called MTS-ESP.
I have struggled my whole life to understand how tuning works, so I made a track to demonstrate to myself how just intonation sounds when you use it in all twelve keys.
What you are hearing in my track is a tuning system that is “perfect” in C major, but not so perfect in other keys, and very not perfect in a few of them. Let’s figure out why!
The blues and the harmonic series
In this post, I’m going to expand on an idea in my blues tonality treatise: that the distinctive scales and chords of the blues are an approximation of African-descended tuning systems based on the natural overtone series. Gerhard Kubik argues in his book Africa and the Blues that blues tonality comes from the overtone series of I and IV, and can only be approximated using instruments tuned to standard twelve-tone equal temperament (12-TET). Let’s unpack what that means!
Circular chord charts
Being home with my kids all day is not very conducive to dissertation writing, but my fragmented attention is still up to the task of making infographics. I’ve been thinking about ways of visually representing grooves. Since circles work so well for rhythms, maybe they can work for harmonies too. Here’s a circular view of twelve bar blues in C:
Think of this as a chord chart wrapped in a circle rather than written in a line. Each cell is a measure. Start on the C7 at the top and move clockwise.
What does the Well-Tempered Clavier sound like in actual well temperament?
First, some niche Twitter comedy:
Twelve-tone equal temperament is socialism, Make Intonation Just Again
— Dr Ethan Hein (@ethanhein) June 26, 2020
The Well-Tempered Clavier is a book of JS Bach compositions for keyboard instruments in each of the twelve major and twelve minor keys. The name refers to Bach’s preferred tuning system, which made it possible to play (sort of) in tune in every key. This was a big deal, because in the usual tuning systems of Bach’s era, only some of the keys sounded good, while others sounded horrible. The history of tuning in Western music is complicated and abstruse, and I won’t go into detail about it in this post, but you can learn some of how it works here. The key facts:
- Western tuning systems, keys and scales are based on the natural harmonic series.
- Harmonics are based on prime numbers.
- Prime numbers don’t divide into each other evenly.
The practical consequence is that your music can either be in perfect tune, or it can use more than one key, but it can not do both. In Hindustani classical tradition, they opted for being in tune, so everything is in a single “key” defined by the omnipresent drone. Western Europeans wanted to be able to change keys, however, and that required some tuning compromises.