Just Intonation, part II

In the last section, we noticed that the Just Major Third, 5:4, is one syntonic comma (81:80) flatter than the Pythagorean Major Third (81:64).  Observe that the 3 notes that differ between the two scales are the major third, the Major Sixth (5:3 for Just vs. 27:16 for Pythagorean) and the Major seventh (15:8 for Just and 243:128 for Pythagorean).  If we perform the division, we notice that each of the Just Intervals is exactly one syntonic comma (81:80) flatter than its Pythagorean counterpart.  This gives rise to some new issues.

MXL 990 Mod

I got a good deal on a cheap Chinese mic and upgraded using parts and a new capsule from Microphone Parts.  The capsule is an RK-47.  The mod went super quickly and I'm really happy with the results.  Matt at Microphone Parts is super easy to work with and I received the parts 2 days afer I placed my order. 

First some photos:

Front view

 

Back view

 I am eventually going to do a head-to-head with a stock MXL 990, but for now, here is a clip of me playing my bass recorded by the modded mic followed by Paul Chambers playing through what I assume is a tube U-47, just for comparison.  By the way, the drummer on my chorus is Billy Jones and on PC's it's Roy Haynes.

Saturday Transcription

Mingus Ah Um is one of those records to recommend to people that want to get into jazz and don't know what to listen to.  I remember when I first discovered this record when I was in High School.  It blew me away.  Everything about it is off the hook.

Anyway, one of the things on this record that knocked me out was the tenor break which starts the song Open Letter to Duke.  Once I found out who this sax player was (Booker Ervin) I went out and bought everything I could find from him.  To me, his best albums as a leader feature the rhythm section of Jaki Byard, Richard Davis and Alan Dawson (The Freedom Book and The Space Book, but The Song Book, which replaced Byard with Tommy Flanagan is pretty killing as well.)  Also check out Tex Book Tenor, which features Kenny Barron, Woody Shaw, Billy Higgins and a bass player I never heard of named Jan Arnet.

Just Intonation, part I

The next evolution after Pythagorean Tuning came into common practice sometime around the fifteenth century.  Before the Renaissance, chords other than fifths weren't played much and intervals were thought of as melodic (played in succession) rather than harmonic (played simultaneously).

One of the major artistic changes which occurred during the Renaissance is that music evolved to be polyphonic and harmonic intervals such major and minor thirds and sixths became common.  To understand how the scale evolved, let's go back to the overtone series.  Recall, the first overtone or harmonic is given by a frequency of 2:1 with the fundamental and produces and octave.  The second is 3:1 and produces an octave plus a fifth, which we reduced to 3:2 to keep it within an octave of the fundamental.  These are the only ratios required to complete the Pythagorean Scale.  The next harmonic is 4:1.  Any power of 2 (2, 4, 8, 16, 32, etc.) will produce an octave, so no new information here.

Problems with the Pythagoren Tuning

Let's say we wanted to change keys using the Pythagorean Tuning based on a fundamental pitch of C.  First we'll transpose up a fifth to move to the key of G.  We expect all of the pitches to remain the same, but the seventh degree f needs to move up to f#, so we maintain a leading tone between the 7th degree and octave.  Remember, to change pitches we multiply (to go up) or divide (to go down) by the appropriate ratio.  In this case we multiply by 3/2 to go up a perfect fifth.

In the figure below, the down arrows indicate multiplying by 3/2.  In some cases, the notes leave the octave, so the next round of arrows divides by 2 (where necessary), to keep all of the pitches in one octave.

Pythagorean Tuning Part 2

We left off with the scale consisting of {C, G, C'} and containing frequencies with ratios of {1:1, 3/2:1, 2:1} with the fundamental pitch.  Notice that we get a second interval for free: the prefect fourth between G and the octave C.  To determine the ratio of these pitches (Pitches be crazy), we divide the higher ratio by the lower ratio.  If you remember back to grade school when you learned how to divide fractions, we simply flip the bottom and multiply straight across:

the upshot being that the ratio of a perfect fourth is 4:3.  

Pythagorean Tuning part 1

I am teaching a class at Cal State Monterey (C-scum) on the Mathematics of Music.  We are currently studying the evolution of our modern tuning, starting with Pythagoras.  (The same guy with the right triangles.)  Pythagorean tuning is based solely on perfect fifths and octaves.  Given a starting pitch, all of the remaining scale degrees are arrived at by going up or down by a fifth (which corresponds to multiplying or dividing the fundamental frequency by 3/2) or going up or down by an octave (which corresponds to multiplying or dividing by 2.)  This came about seemingly because the fifth is the first overtone which differs from the fundamental pitch.

Coltrane Transcription

A great solo from one of Coltrane's earlier records, Lush Life, from 1958.  Like all of my transcriptions,

• It's in bass clef (if you want a copy in treble or transposed, drop me a line)
• If you find a mistake, let me know

DIY Mic Cables

I made these xlr cables.

Same quality as Monster: nuetrik connectors and quad star cable, but at a fraction of the cost and in whatever colors I like.  You should make your own cables too.  This is where I got the materials.  You need to do a few at a time to make the shipping charges worthwhile.  Also, you need a soldering iron, some solder and a little bit of time (about 10 or 15 minutes per cable).