Music of the spheres

The idea of “music of the spheres” dates back to the Pythagoreans. They saw an analogy between orbital frequency ratios and musical frequency ratios.

HD 110067 is a star 105 light years away that has six known planets in orbital resonance. The orbital frequencies of the planets are related to each other by small integer ratios.

The planets, starting from the star, are labeled b, c, d, e, f, and g. In 9 “years”, from the perspective of g, the planets complete 54, 36, 24, 16, 12, and 9 orbits respectively. So the ratio of orbital frequencies between each pair of consecutive planets are either 3:2 or 4:3. In musical terms, these ratios are fifths and fourths.

In the chord below, the musical frequency ratios are the same as the orbital frequency rations in the HD 110067 system.

Here’s what the chord sounds like on a piano:


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The Real Book

I listened to the 99% Invisible podcast about The Real Book this morning and thought back to my first copy.

My first year in college I had a jazz class, and I needed to get a copy of The Real Book, a book of sheet music for jazz standards. The book that was illegal at the time, but there was no legal alternative, and I had no scruples about copyright back then.

When a legal version came out later I replaced my original book with the one in the photo below.

The New Real Book Legal

The podcast refers to “When Hal Leonard finally published the legal version of the Real Book in 2004 …” but my book says “Copyright 1988 Sher Music Co.” Maybe Hal Leonard published a version in 2004, but there was a version that came out years earlier.

The podcast also says “Hal Leonard actually hired a copyist to mimic the old Real Book’s iconic script and turn it into a digital font.” But my 1988 version looks not unlike the original. Maybe my version used a kind of typesetting common in jazz, but the Hal Leonard version looks even more like the original handwritten sheet music.


A few weeks ago I wrote about how the dissonance of a musical interval is related to the complexity of the frequency ratio as a fraction, where complexity is measured by the sum of the numerator and denominator. Consonant intervals have simple frequency ratios and dissonant intervals have complex frequency ratios.

By this measure, the most consonant interval, other than an octave, is a perfect fifth. And the most dissonant interval is a tritone, otherwise known as the diminished fifth or augmented fourth. So in some sense perfect fifths and tritones are opposites, but they are both ways of splitting an octave in half, just on different scales.

Linear scale versus log scale

When we say simple frequency ratios are consonant and complex frequency ratios are dissonant, we are speaking about ratios on a linear scale. But we often think of musical notes on a logarithmic scale. For example, we think of the notes in a chromatic scale as being evenly spaced, and they are evenly spaced, but on a log scale.

If we divide an octave in half on a linear scale, we get a perfect fifth. For example, if we take an A 440 and an A 880 an octave higher, the arithmetic mean, the midpoint on a linear scale, we get E 660.

But if we divide an octave in half on a log scale, we get a tritone, three whole steps or six half steps out of 12 half steps in a chromatic scale. The midpoint on a log scale is the geometric mean. The geometric mean of 440 and 880 is 440 √2 = 622, which is D#.

So we take the midpoint of an octave on a linear scale we get the most consonant interval, a perfect fifth, but if we take the midpoint of an octave on a log scale we get the most dissonant interval, a tritone.

Tritone substitution

Intervals of a fifth are so consonant that they don’t contribute much to the character of a chord. It is common to leave out the fifth.

Tritones, however, are essential to the sound of a chord. In fact, it is common to replace a chord with a different chord that maintains the same tritone. For example, in the key of C, the G7 chord contains B and F, a tritone. The chord C#7 contains the same two notes (though the F would be written as E#), and you’ll often see a C#7 chord substituted for a G7 chord. So a song that had a Dm–G7–C progression might be rewritten as Dm–C#7–C, creating a downward chromatic motion in the base line.

This is called a tritone substitution. You could think of the name two ways. In the discussion above we talked about preserving the tritone in a chord. But notice we also changed the root of the chord by a tritone, replacing G with C#. More generally, replacing any chord with a chord whose root is a tritone away is called a tritone substitution or simply tritone sub. For example, a D minor chord does not contain a tritone, but we could still do a tritone sub, replacing Dm with G#m because D and G# are a tritone apart.

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Jaccard index and jazz albums

Miles Davis Kind of Blue album cover

Jaccard index is a way of measuring the similarity of sets. The Jaccard index, or Jaccard similarity coefficient, of two sets A and B is the number of elements in their intersection, AB, divided by the number of elements in their union, AB.

J(A, B) = \frac{|A \cap B|}{|A \cup B|}

Jaccard similarity is a robust way to compare things in machine learning, say in clustering algorithms, less sensitive to outliers than other similarity measures such as cosine similarity.

Miles Davis Albums

Here we’ll illustrate Jaccard similarity by looking at the personnel on albums by Miles Davis. Specifically, which pair of albums had more similar personnel: Kind of Blue and Round About Midnight, or Bitches Brew and In a Silent Way?

There were four musicians who played on both Kind of Blue and Round About Midnight: Miles Davis, Cannonball Adderly, John Coltrane, and Paul Chambers.

There were six musicians who played on both Bitches Brew and In a Silent Way: Miles Davis, Wayne Shorter, Chick Corea, Dave Holland, and John McLaughlin, Joe Zawinul.

The latter pair of albums had more personnel in common, but they also had more personnel in total.

There were 9 musicians who performed on either Kind of Blue or Round About Midnight. Since 4 played on both albums, the Jaccard index comparing the personnel on the two albums is 4/9.

In a Silent Way and especially Bitches Brew used more musicians. A total of 17 musicians performed on one of these albums, including 6 who were on both. So the Jaccard index is 6/17.

Jaccard distance

Jaccard distance is the complement of Jaccard similarity, i.e.

d_J(A, B) = 1 - J(A,B)

In our example, the Jaccard distance between Kind of Blue and Round About Midnight is 1 − 4/9 = 0.555. The Jaccard distance between Bitches Brew and In a Silent Way is 1 − 6/17 = 0.647.

Jaccard distance really is a distance. It is clearly a symmetric function of its arguments, unlike Kulback-Liebler divergence, which is not.

The difficulty in establishing that Jaccard distance is a distance function, i.e. a metric, is the triangle inequality. The triangle inequality does hold, though this is not simple to prove.

F# and G

I was looking at frequencies of pitches and saw something I hadn’t noticed before: F# and G have very nearly integer frequencies.

To back up a bit, we’re assuming the A above middle C has frequency 440 Hz. This is the most common convention now, but conventions have varied over time and place.

We’re assuming 12-tone equal temperament (12-TET), and so each semitone is a ratio of 21/12. So the nth note in the chromatic scale from A below middle C to A above middle C has frequency

220 × 2n/12.

I expected the pitch with frequency closest to integer would be an E because a perfect fifth above 220 Hz would be exactly 330 Hz. In equal temperament the frequency of the E above middle C is 329.6 Hz.

The frequency of F# is

220 × 29/12 Hz = 369.9944 Hz.

The difference between this frequency and 370 Hz is much less than the difference between equal temperament and other tuning systems.

The frequency of G is

220 × 25/6 Hz = 391.9954 Hz

which is even closer to being an integer.

In more mathematical terms, stripped of musical significance, we’ve discovered that

23/4 ≈ 37/22


25/6 ≈ 98/55.


Humming St. Christopher

The other day I woke up with a song in my head I hadn’t heard in a long time, the hymn Beneath the Cross of Jesus. The name of the tune is St. Christopher.

When I thought about the tune, I realized it has some fairly sophisticated harmony. My memory of the hymns I grew up with was that they were harmonically simple, mostly built around three chords: I, IV, V. But this hymn has a lot going on.

I imagine a lot of things that I remember as being simple weren’t. I was simple, and my world was richer than I realized.


You can find the sheet music for the hymn here. I’ll write out the chord progressions for the first two lines.

    I     idim | I            | V7   ii7 V7  | I   III | 
    vi   iidim | vi VI7 ii vi | II   VIII7♭5 | III     |

If you’re not familiar with music theory, just appreciate that there are a lot more symbols up there than I, IV, and V.

The second line effectively modulates into a new key, the relative minor of the original key, and I’m not sure how to describe what’s going on at the end of the second line.

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Updated pitch calculator

I’ve made a couple minor changes to my page that converts between frequency and pitch. (The page also includes Barks, a psychoacoustic unit of measure.)

If you convert a frequency in Hertz to musical notation, the page used to simply round to the nearest note in the chromatic scale. Now the page will also tell you how sharp or flat the pitch is if it’s not exact.

For example, if you enter 1100 Hz, the page used to report simply “C#6” and now it reports “C#6 – 14 cents” meaning the closest note is C#6, but it’s a little flat, 14/100 of a semitone flat. If you enter 1120 Hz it will report “C#6 + 18 cents” meaning that the note is 18/100 of a semitone sharp.

Octave numbers, such as the 6 in C#6 are explained here.

The other change I made to the page was to add a little eighth note favicon that might show up in a browser tab.

pitch converter favicon

I’ve written several online converters like this: LaTeX to Unicode, wavelength to RGB, etc. See a full list here.



Saxophone with short bell

Paul A. sent me a photo of his alto sax in response to my previous post on a saxophone with two octave keys. His saxophone also has two octave keys, and it has a short bell. Contemporary saxophones have a longer bell, go down to B flat, and have two large pads on the bell. Paul’s saxophone has a shorter bell, only goes down to B, and only has one pad on the bell.

Alto sax with short bell

Here’s a closeup of the octave keys.

Two octave keys

Paul says he found his instrument in an antique shop. It has no serial number or manufacturer information. If you know anything about this model, please leave a comment below.

Saxophone blog posts

All possible scales

Pete White contacted me in response to a blog post I wrote enumerating musical scales. He has written a book on the subject, with audio, that he is giving away. He asked if I would host the content, and I am hosting it here.

Here are a couple screen shots from the book to give you an idea what it contains.

Here’s an example scale, number 277 out of 344.

scale 277

And here’s an example of the notes for the accompanying audio files.

sheet music example, track 22


The acoustics of Hagia Sophia

Hagia Sophia

The Hagia Sophia (Greek for “Holy Wisdom”) was a Greek Orthodox cathedral from 537 to 1453. When the Ottoman Empire conquered Constantinople the church was converted into a mosque. Then in 1935 it was converted into a museum.

No musical performances are allowed in the Hagia Sophia. However, researchers from Stanford have modeled the acoustics of the space in order to simulate what worship would have sounded like when it was a medieval cathedral. The researchers recorded a virtual performance by synthesizing the acoustics of the building. Not only did they post-process the sound to give the singers the sound of being in the Hagia Sophia, they first gave the singers real-time feedback so they would sing as if they were there.

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