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Well Temperaments (Part 1)
The tuning system called ‘quarter-comma meantone’ dominated western keyboard music from about 1550 to roughly 1690. The reason: it has very nice thirds and fifths in many different keys! But as I keep...
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Giovanni Battista Doni
You know “do re mi fa sol la ti”? This system for naming notes was invented by a guy named Guido of Arezzo around 1000 AD. But he only named the first 6 notes, because the whole idea of a 7-note scale...
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Well Temperaments (Part 2)
Last time I ended with a question: why are certain numbers close to 1 so important in tuning systems? It helps to understand a bit about this before we plunge into the study of well temperaments. It...
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Thaats
Recently I learned a little about ragas in Carnatic music, which is prevalent in the south of India. I discussed them here: • Melakarta ragas. Now let me give an equally superficial introduction to...
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Well Temperaments (Part 3)
Last time we saw the importance of some tiny musical intervals: irritating but inevitable glitches in our search for perfectly beautiful harmonies. Today I want to talk about a truly microscopic...
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Well Temperaments (Part 4)
Now I want to start talking about some important well-tempered tuning systems invented by Johann Philipp Kirnberger. But first: who was this guy? As I tried to answer this question for myself I...
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Pied Butcherbird
As my friends are learning about my current obsession with tuning systems, they’re starting to ask interesting questions I don’t know the answers to. For example, Michael Fourman asked me: if...
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Well Temperaments (Part 5)
Okay, let’s study Kirnberger’s three well-tempered tuning systems! I introduced them last time, but now I’ve developed a new method for drawing tuning systems, which should help us understand them...
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Well Temperaments (Part 6)
Andreas Werckmeister (1645–1706) was a musician and expert on the organ. Compared to Kirnberger, his life seems outwardly dull. He got his musical training from his uncles, and from the age of 19 to...
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Nicholas Ludford
At first glance it’s amazing that one of the great British composers of the 1400s largely sank from view until his works were rediscovered in 1850. But the reason is not hard to find. When the...
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Quarter-Comma Meantone (Part 1)
I’ve spent the last few weeks drawing pictures of tuning systems, since I realized this is a good way to show off their mathematical beauty. Now I’ll start deploying them! I’ve already written about...
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Quarter-Comma Meantone (Part 2)
Last time I introduced ‘quarter-comma meantone’, the tuning system shown above. Today let’s see if we can understand all the numbers in this picture. You see 13 circles with letters in them standing...
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Quarter-Comma Meantone (Part 3)
Last time I explained the tuning system shown above. But I didn’t say why it’s called ‘quarter-comma meantone’. Today I’ll tell you what a ‘quarter comma’ is. Not only will this shed new light on this...
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Quarter-Comma Meantone (Part 4)
I’ve been talking about the marvelous tuning system that dominated western music from about 1550 to 1690: quarter-comma meantone. Above I’ve drawn it in its most mathematically beautiful form. For...
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Melakarta Ragas
It’s sort of ridiculous that I’ve never studied even the very basics of Indian music, like what are the ragas. Thanks to my friend Todd Trimble I’m dipping my toe into it. And it turns out that in...
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Quarter-Comma Meantone (Part 5)
I’ve been talking about the quarter-comma meantone tuning system, which dominated western music from about 1550 to 1690. So far I’ve been drawing it using a circle of fifths, as above. This is a...
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Quarter-Comma Meantone (Part 6)
So far I’ve focused on the quarter-comma meantone system in its mathematically beautiful, symmetrical form above. Today I’ll say more about the scale as actually played: how we trim it down from a...
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Quarter-Comma Meantone (Part 7)
I’ve said a lot about quarter-comma meantone and its great properties. It’s almost time to start exploring the vast realm of ‘well-tempered’ tuning systems that flourished starting around 1690. But...
View ArticleThe Harmonic Seventh Chord
Traditional western music theory embraces chords with frequency ratios that are products of powers of 2, 3, and 5… but not higher primes. Barbershop quartets go further! These quartets love four-part...
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Tritone Substitutions
A ‘tritone substitution’ is a popular trick for making your music sound more sophisticated. I’ll show you a couple of videos with lots of examples. But since I’m mathematician, let me start with the...
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