## The debut of The MIT Centrifugues

April 15, 2013

For my last semester as an MIT undergraduate, I started a new bastard pop

*a cappella*group at MIT called The Centrifugues. After a semester of hard work, we had our debut performance at the annual Spring Greater Boston Invitational Songfest (SGBIS) concert last Friday. We also have a joint concert with another MIT*a cappella*group called the Asymptones on April 27th, and a couple of other gigs in the works for the end of the semester.Two of the Centrifugues are also in the Asymptones, and we were serendipitously placed back-to-back in last week’s concert, so this lucky duo got to remain on stage between the performances. Coincidentally, these two members are myself (President of the Centrifugues) and David Xiao (President of the Asymptones):

Our soloists are incredibly talented vocalists. Pictured below is Rachel Nations, Class of 2016, taking the lead on a mashup of Born this Way (Lady Gaga) and When Love Takes Over (David Guetta):

Following our first song, Sumin Kim, Class of 2014, performed a bootleg of Talking to the Moon (Bruno Mars) and The One that Got Away (Katy Perry):

We’ve spent countless hours rehearsing, arranging, publicizing, perfecting our pitches, rhythms, and appearance, applying for recognition and funding from MIT, recruiting, scheduling, holding auditions, and probably several more things I can’t think of. It’s been an incredibly productive first semester, and I can’t wait to see what the future holds.

Photo credit: Joseph Lee.

## Piano dueling with Sumin

April 14, 2013

Sumin and I had a “call and response” piano improv battle today.

## Highest note on a guitar

December 29, 2012

I just bought a new guitar, and I asked myself:

*what is the highest note you can play in standard tuning?*When plucked, the high E string vibrates at \(440 \times 2^{-\frac{5}{12}}\) Hz, or approximately 329.628 Hz.

If we pluck this string while fingering the innermost fret (assuming there are 24 frets), we hit the E a major 18th above middle C, with frequency \(440 \times 2^{\frac{19}{12}}\) Hz, or approximately 1318.510 Hz. We can also place our finger just past this fret and hit the next F or maybe even the F# before we run out of fretboard.

However, we can also pluck the string on the other side of the fret to get even higher notes. The highest note we can produce this way happens when we pluck the high E string on the far side of the first fret, with the finger pressing the string against this fret from the inward side.

This note will be higher than any audible harmonics you will be able to produce. Let’s calculate its pitch. To do this, we need to know the width of the space between the nut and the first fret (the string is effectively this long when we press it against the first fret from the other side).

Halving the length of a string raises its pitch by an octave. This generalizes, of course, to arbitrary intervals and length ratios. If a string is length \(x\), we can raise its pitch by \(n\) semitones by reducing its length to \(\left( \frac{1}{2} \right)^{\frac{n}{12}} x\).

Using this we can calculate the width of the first segment of the fretboard. Each segment of the fretboard constitutes a semitone in pitch. If we want to raise the high E string (of length \(x\)) by a semitone, we neet to shorten it to \(\left( \frac{1}{2} \right)^{\frac{1}{12}} x\), which is about \(0.944x\). This means the piece we cut off, which is the part we’re will be playing, has length \(\left( 1 - \left( \frac{1}{2} \right)^{\frac{1}{2}} \right) x\), or about \(0.056x\).

This means we changed the length of the string by a factor of about 0.056. We can use the formula above to calculate how many semitones by which we have increased. Solving \(1 - \left( \frac{1}{2} \right)^{\frac{1}{2}} = \left( \frac{1}{2} \right)^{\frac{n}{12}}\) for \(n\), we find that we have gone up about \(n = 49.862\) semitones (about 4 octaves and a major 2nd) from the E, which brings us approximately to a really high F#, a tritone above the highest note on a piano. The frequency of this note is exactly

\[ 440 \times 2 ^ { \left( 12 \log \left( 1 - \left( \frac{1}{2} \right) ^ { \frac{1}{12} } \right) / \log \left( \frac{1}{2} \right) - 5 \right) / 12 } \text{ Hz,} \]

which is roughly 5.873 kHz. Some adults will not be able to hear this note.

As my collegue Ben Sena points out, you can hit even higher notes by pressing the string against two frets and plucking in between. You can pluck the highest note then by plucking between the innermost two frets on the high E string. Let’s calculate the frequency of this note, assuming the guitar has 24 frets. If the string has length \(x\), then the length of this piece is \( \left( \left( \frac{1}{2} \right) ^ { \frac{23}{12} } - \left( \frac{1}{2} \right) ^ { \frac{24}{12} } \right) x \), which is about \( \left( 1.487 \times 10^{-2} \right) x\). Using the method we used above, we solve for \(n\) in the equation \( \left( \frac{1}{2} \right) ^ { \frac{23}{12} } - \left( \frac{1}{2} \right) ^ { \frac{24}{12} } = \left( \frac{1}{2} \right) ^ { \frac{n}{12} } \). We find that \( n = 12 \log \left( \left( \frac{1}{2} \right) ^ { \frac{23}{12} } - \left( \frac{1}{2} \right) ^ { \frac{24}{12} } \right) / \log \frac{1}{2} \), which is about 72.862 semitones, raising the high E string by 6 octaves and almost a minor 2nd. This means that, assuming a 24 fret quitar, the highest note you can produce is approximately an F. The frequency of this note is exactly

\[ 440 \times 2 ^ { \left( 7 + 12 \log \left( \left( \frac{1}{2} \right) ^ { \frac{23}{12} } - \left( 1 / 2 \right) ^ { \frac{24}{12} } \right) / \log \left( \frac{1}{2} \right) \right) / 12 } \text{ Hz,} \]

which is about

**44.347 kHz**. You will not be able to hear this note!This analysis uses a very simple model of guitar. We assume that the strings are perfectly elastic and ignore strain forces, air resistance, and other factors that are likely to affect the sound in these extreme conditions.

**TLDR:**F, with some waving of hands.