A=444Hz Tuning and the Magic 528Hz Frequency

One of my guitar-playing friends recently posted the following article to Facebook as a joke:

http://themindunleashed.org/2014/03/miracle-528-hz-solfeggio-fibonacci-numbers.html

I know that my friend was just being silly, but the actual content of that piece is more drivel in a long line of mathematical silliness which forces me to heave a deep sigh for the fate of humanity. The article in question reinforces my conviction that some people will believe just about anything: bigfoot, aliens, unicorns, Obamacare, leprechauns, etc. But one of my personal favorites is the assertion that altering the base frequency in a tuning scale will somehow lead to a perfect universe.

What a bunch of hooey.

As I mentioned earlier, I know that my friend was posting the article to be silly, but just for the sake of argument, I can't resist taking a look at the math from the article. At the risk of being overly self-indulgent, I know that I have used my A=432Hz Tuning blog post to refute concepts like this in the past. But that being said, my blog post examines a lot of the actual math behind these sorts of silly ideas, and they just don't stand up to scrutiny. Oh sure, there's a bunch of purported facts in the article that my friend posted, (once you get past the gooey new age crap). But as I said earlier, people will believe just about anything.

Here's a case in point: when I visited Machu Picchu I was assured by my tour guide that one of the stones in one of the walls had been certified by NASA as the harmonic center point of all nature. I didn't believe my guide, but in hindsight her statement seems considerably more plausible than anything that was presented in the "Magic 528Hz" article. (Note: I meant that humorously; you can't trust NASA to find the harmonic center point of anything.)

In any event - let's take a look at some of the math from the 528Hz article, shall we?

If you use A=444Hz as the article suggests, that does NOT make the frequency for C fall on an even interval - it's off by a diminutive fraction:

Note Frequency
A 444.00 Hz
Bb 470.40 Hz
B 498.37 Hz
C 528.01 Hz
C# 559.40 Hz
D 592.67 Hz
Eb 627.91 Hz
E 665.25 Hz
F 704.81 Hz
F# 746.72 Hz
G 791.12 Hz
G# 838.16 Hz
A 888.00 Hz

As you can see, the frequency for C falls pretty close to 528Hz. But as I mentioned in my blog, what your ear actually wants to hear are frequencies which harmonically-derived perfect intervals across the scale. However, the frequencies in the tuning scale that the article's author is using are based on equal-temperament, which is a harmonically imperfect standard. Because of this fact, you would not use equal-tempered tuning if you were actually trying to calculate harmonically-perfect intervals, so the 528Hz article is completely busted right there. (On a side note, even frequencies in a full scale like this do not matter to your ear - because they just don't. Period. You can have uneven decimal points for perfect intervals in a harmonically-derived scale if you do your math correctly; arguing about decimal points is just stupid.)

That being said, the author spends a great deal of time rambling on and on about Fibonacci sequences, (which are really cool by the way). However, the author completely fails to mention (or perhaps to even notice) that 528 doesn't fall in the standard Fibonacci sequence:

1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377,
610, 987, 1597, 2584, 4181, 6765, 10946, etc.

Now, if the number 528 had actually fallen inside the standard Fibonacci sequence, that would have been a pretty cool factoid for the article. But that being said, it still wouldn't mean anything.

Just for the fun of it, let's see how we can manipulate the math a little, shall we?

For example, if you use A=431.33333Hz as your base frequency, then the frequency for Eb will be 610.00Hz, which is actually a valid number in a standard Fibonacci sequence. That's kind of amusing, but it doesn't mean anything useful. All that means is that I spent a lot of time in Excel typing in random base frequencies until I bumped into a number that worked. Likewise, if you use A=443.99Hz as your base frequency, then your C will actually be 528Hz, but that's just as useless. (And good luck trying to find a tuner that will let you use A=443.99Hz as your base frequency.)

In the end, the article which my friend posted to Facebook is an amusing work of fiction, although reading it will waste several minutes of your life which could have been spent doing something considerably more productive.

A=432Hz Tuning versus A=440Hz Tuning

A coworker recently pointed me to the following blog post, and he asked if it had any basis in reality: 432Hz: Crazy Theory Or Crazy Fact. After looking at that blog, I think a better title for it would be "432Hz: Misinterpreted Theory and Misconstrued Facts." I honestly mean no disrespect to the author by my suggestion; but the blog's author clearly does not understand the theory behind what he is discussing. And because he misunderstands some basic concepts, his discussion on this subject offers little by way of practical information. As such, I thought that I would set the record straight on a few things and offer some useful information on the subject.

First of all, the author's suggestion that using A=432Hz for a reference when tuning will put your guitar in Pythagorean Tuning is completely false; all you are doing is changing the base frequency that you are using, but your guitar will still be in Standard Tuning.

Discussing the base frequency is about as effective as discussing the merits of an E-Flat Tuning versus Standard-E Tuning; either one is fine, and it just comes down to user preference as to which one is better. The same thing holds true for choosing A=432Hz over A=440Hz - it's a preference choice. (Unless you have Perfect Pitch, in which case  A=432Hz is probably going to annoy you more than words can say.)

However, there is one major difference: if you choose to record music by using an other-than-normal base frequency, you'll frustrate the heck out of someone who just tuned their guitar with a standard tuner and attempts to sit down and learn your music. ("Hmm... this just doesn't sound right.") And you could retune just to annoy them for fun, of course. ;-]

That being said, any discussion of Pythagorean Tuning and the guitar is utterly useless, because a guitar is not fretted for Pythagorean Tuning. Here is where the real confusion lies, because the author of that blog is confusing changing the base frequency with somehow putting the guitar into a different temperament, which is not possible without re-fretting your instrument. Here's what I mean by that:

The physical interval between the frets on a guitar neck is based on Equal Temperament, which is a constant that is defined as the 12th root of 2. In Microsoft Excel that formula would be 10^(LOG(2)/12), which comes to 1.0594630944. We all know that an octave is double the frequency of the base pitch, so with A=440Hz you would get A=880Hz for the next higher octave. By using the above constant, you can create the following scale from an A to an A in the next higher octave by multiplying each frequency in the scale by the constant in order to derive the resultant frequency for each successive note:

Note Frequency
A = 440.00Hz
Bb = 466.16Hz
B = 493.88Hz
C = 523.25Hz
C# = 554.37Hz
D = 587.33Hz
D# = 622.25Hz
E = 659.26Hz
F = 698.46Hz
F# = 739.99Hz
G = 783.99Hz
Ab = 830.61Hz
A = 880.00Hz

In contrast to the claims that were made by the blog's author, you do not magically get whole-number frequencies (e.g. with no decimal points) if you change the base frequency to A=432Hz; the math just doesn't support that. Here is the list of resulting frequencies for an octave if you start with a base frequency of A=432Hz, and I have included a comparison with a base frequency of A=440Hz:

Note Frequency 1 Frequency 2
A = 432.00Hz <-> 440.00Hz
Bb = 457.69Hz <-> 466.16Hz
B = 484.90Hz <-> 493.88Hz
C = 513.74Hz <-> 523.25Hz
C# = 544.29Hz <-> 554.37Hz
D = 576.65Hz <-> 587.33Hz
D# = 610.94Hz <-> 622.25Hz
E = 647.27Hz <-> 659.26Hz
F = 685.76Hz <-> 698.46Hz
F# = 726.53Hz <-> 739.99Hz
G = 769.74Hz <-> 783.99Hz
Ab = 815.51Hz <-> 830.61Hz
A = 864.00Hz <-> 880.00Hz

When you look at the two sets of frequencies side-by-side, you see that tuning with either base frequency yields only two even frequencies - one for each of the A notes. However, when you use the standard A=440Hz tuning, you have two frequencies (the F# and G) that almost fall on even frequencies (at 739.99Hz and 783.99Hz respectively). Not that this really matters - your ear is not going to care whether a frequency falls on an even number. (Although it might look nice on paper if you have Obsessive Compulsive Disorder and you rounded every frequency to the nearest whole number.)

Since the frets on the guitar are based on this temperament, that's all you get - period. You can fudge your base frequency up or down all you want, but in the end you're still going to be using Equal Temperament, unless you completely re-fret your guitar as I already mentioned. (Note: See the FreeNotes website for guitar necks that are fretted for alternate temperaments.)

If you had a background that included synthesizers, (and as a guitar player I must apologize for my side hobby on keyboards), you might remember that back in the 1980s there was a passing phase with microtonality on keyboards. If you had a keyboard that supported this technology, you were able to play your keyboard by using intonation that was different than the Equal Temperament, which was sometimes pretty cool.

Why would someone want to do this? Because many of the old composers never used Equal Temperament; that's a fairly recent invention. So if you want to hear what a piece of piano music sounded like for the original composer, you might want to set up your keyboard to use the same microtonality temperament that the composer actually used.

But that being said, before the invention of Equal Temperament, there were several competing temperaments, and each was usually based on tuning some interval like the fourth or fifth by ear, and then finding intervals in-between those other intervals that sounded acceptable. What this resulted in, however, were a plethora of tunings/temperaments that sounded great in some keys and terrible in others. More than that, if you continue to work your way up a scale based on intervals based on sound, you will eventually introduce errors. Using the actual Pythagorean Tuning suffers from this problem, so if you put a microtonal keyboard into Pythagorean Tuning and attempted to play a piece of music that extended past a couple of octaves, it sounded terrible. (See Pythagorean Tuning for an explanation.)

But on that note, almost every guitarist suffers from this same problem, but you just don't know it. Have you ever tuned your guitar by using the 5th fret and 7th frets harmonics? Of course you have, and so have I. But here's a side point that most guitarists don't know - when you tune your guitar by using those harmonics, you slowly introduce errors across the guitar, and as a result it will seldom seem completely in tune with itself.

Here's an excerpt from a write-up that I did for the Christian Guitar website a while ago that describes what I mean:

There have been many different temperaments used in the Western Hemisphere, and many of these centered around specific intervals. For example, start with a C note, then find the perfect octave above; you now have the starting and ending points for your scale. Next, find the harmonically perfect 5th of G by tuning and listening to pitches, then use these intervals to find E, which is the major 3rd. Once done, you now have three notes of your scale and the octave. If you jump up to G and use the same process to find the 3rd and 5th, you get the B and D notes. If you keep repeating the process, you eventually derive all of the diatonic notes for your major scale. On a piano that would be just the white keys.

Leaving sharps and flats out of this example, (the piano's black keys), the problem is that if you keep using the perfect 5th for a reference, you gradually find that the notes in your scale are not lining up as you travel around the circle of 5ths. This occurs because using perfect 5ths will eventually introduce slight errors on other intervals, and the result will be that your scale works great in one or two keys, but other keys sound noticeably awful.

Here's why this happens: after having gone around the entire circle using perfect 5ths as a tuning guide, by the time you get to the octave above your starting note, the physical frequency for the octave is not the same as the last pitch that you derived from tuning based on the perfect 5ths. This is especially problematic when you use one particular note/key to tune an instrument, and then try to play in another key. For example, if you tune an instrument using perfect 5ths and start on a C note, the key of C# will sound distinctively out-of-tune.

The only trouble that some people might have with equal-temperament is that the intervals within the octave are not based on perfect intervals, but rather intervals based on the constant. This causes a lot of problems with people who tune by ear using perfect 5ths, which many guitarists do without realizing when they tune their guitars using harmonics over the 7th fret.

For example, if you were to tune an E note using an A note as a reference point, your ear would want to hear the perfect 5th for E which is 660.00Hz, not the equal-tempered E that is 659.26Hz. Although the difference is very small, it is compounded over time as you tune the other notes within the scale. If you continued to tune using 5ths, your next note higher would be the B that is a 5th over E. Your ear would want to hear the perfect 5th again, so you would wind up with 990.00Hz for B instead of the equal-tempered 987.77Hz. Another perfect 5th would be 1485Hz instead of the equal-tempered 1479.98Hz, then 2227.50Hz instead of 2217.46Hz, etc.

I personally find the math part of music fascinating, and I've obviously spent a bunch of time (perhaps too much time  ;-]) studying notes, scales and tunings from a mathematical perspective. Because of that, I view the whole guitar neck as a numerical system and all chords/scales as algorithms. I know that's really geeky, but it's still pretty cool. In the end, I think that math might be my 2nd-favorite part of music. (My favorite part is turning the amps up to 11 and feeling the actual notes as they tangibly pass through my body - it's like a physical feedback loop. Very cool...)

The net result of this discussion is - use a tuner when you are tuning your guitar, not your ear. And it doesn't matter what your base frequency is when you are tuning your guitar - you are still using Equal Temperament because that's the way that your guitar is made. ;-]

Restoring an Old Friend Back to Life

My Love Affair with Explorer-style Guitars

Many years ago - more years than I will care to admit - I saw Cheap Trick in concert. (Okay, just to give you an idea of how long ago this was - Cheap Trick was touring to promote their Cheap Trick at Budokan album; you can do the math from there.) At this point in my life, I hadn't been playing the guitar for very long, and my main guitar at the time was a cheap 3/4-size nylon-string acoustic that my dad had bought for me from a store on a military base. Military bases aren't known for keeping great guitars in stock, so it needs little explanation that I was fascinated by any cool guitar that came along. This made seeing Cheap Trick even more entertaining, because their lead guitar player, Rick Nielson, used something like 1,000 different guitars throughout the show.

But one particular guitar caught my eye - an Explorer; something about it's futuristic shape seemed to me like the coolest guitar ever. Rick Nielson played an Explorer from Hamer Guitars, but I soon learned that Hamer's Explorer was a copy of the Gibson Explorer, and that became the 'Guitar to Have' for me.

Rick Nielson (left) playing a Hamer Explorer onstage with Cheap Trick.
(Note: This image is originally from Wikipedia.)

About this time I was in my first rock band with my good friend Gene Faith. Even though we both actually played the guitar, we liked to create fake instruments for ourselves - I made myself a fake guitar out of scrap wood that looked like an Explorer, even though it was hollow and had strings that were made out of rubber bands. But it was cool - there was no doubt in my mind about that. Once we had some 'instruments' at our disposal, we'd put on a record and pretend to actually play these fake instruments and jump around my dad's living room like we were rock stars. (Hey, don't laugh so hard - I was only 12 or 13 years old.)

My first electric guitar was a cheap copy of a Gibson SG that I purchased at Sears for somewhere around $100. (And believe me - I delivered a lot of newspapers to earn the $100 to buy that guitar.) It was okay as a starter guitar, but I soon found myself wanting a better axe. A year or so later I saved up more of the proceeds from my newspaper route and I bought an Explorer copy from an off-brand company named Seville - it was nowhere near as good as a Gibson, but it was the best that I could do on a paperboy's budget. It had a hideous tobacco sunburst paint job, so I removed the neck and hardware, sanded the body down to the bare wood, stained it with a dark wood color, and then I shellacked the body with a clear finish. When I reassembled the guitar, it looked pretty good. I played that Explorer for a few years, and I eventually sold it to my friend Gene.

That's me on the right
playing my Seville
Explorer back in 1981.
Gene posing with my
Seville Explorer.

Jumping ahead a few decades, another good friend, Harold Perry, was moving from Seattle to San Francisco, so he was parting with a bunch of musical gear. I'm always in the market for seasoned gear that needs a new home, so Harold and I were going through a bunch of his old items while I was deciding what I might want to buy. Harold had bought a 1980 Gibson Explorer II several years earlier as a 'project guitar' - it had been badly treated by a previous owner and needed a lot of repair work. Since Harold was moving, he didn't expect to have time to finish the guitar, and he wanted it to find a good home, so he sold it to me for a great price.

And so my adventure with guitar restoration began as a labor of love.

Restoring My Gibson Explorer II

When I took the guitar home, the first thing that I did was strip all of the remaining hardware off the guitar; thereby leaving nothing but the wood body. I then proceeded to polish every inch of the guitar for a few hours. Whoever had owned the guitar before Harold apparently had some hygiene issues and it seemed like he had never cleaned the guitar despite voluminous amounts of caked sweat that coated much of the surface. What's more, his sweat had corroded all of the stock hardware, so nearly all of the hardware would need to be replaced. With that in mind, I decided that this would be a long-term project and I would take my time with it.

The Explorer with all of
the hardware removed.
Grotesquely-corroded
original hardware.

The next thing that I needed to do was to polish the hardware that I intended to keep - which was just the brass nut and frets, all of which looked pretty hideous. I used Mr. Metal to polish the hardware, which seemed a strangely apropos title for a former heavy metal dude.

Badly-tarnished frets and nut. Dude - it's "Mr. Metal." :-O
The pile of used cotton patches
after I finished polishing.
Shiny frets and brass nut!

Over several months I slowly bought new hardware that I needed. I'll spare you most of the details, but suffice it to say that it took a long time for me to locate and purchase all of the right replacement parts that I wanted. I primarily bought the hardware from Stewart McDonald, Musician's Friend, and Guitar Center, and I had the guys at Parson's Guitars create a new truss rod cover to replace the original that had been lost before the guitar had found its way to me. In the end, I replaced the bridge, tailpiece, volume & tone potentiometers, tuning machines, strap locks, toggle switch, and speed knobs. (The folks at Parson's Guitars thought that replacing the stock Gibson parts was a sacrilege, even though I explained that keeping the stock parts left the guitar unplayable.)

All new hardware. New truss rod cover.

Before I started wiring the guitar, I lined the inside of the routing cavities with copper tape - this is supposed to reduce EMI on the guitar. I've never used it before, so it's something of an experiment. In any event - lining the routing took several hours to complete; time will tell if it was worth it.

Lining the interior routing cavities with copper tape.

The next part of the project was to install the new guitar tuning machines. Oddly enough, Gibson won't sell their inline-6 set of tuners for an Explorer to customers, so I had to buy tuning machines from another company. I eventually decided on tuning machines from Gotoh, which I was able to order through Stewart McDonald. The trouble is, once I mounted them on the headstock, I discovered that the screw holes for the tuning machines were off by a little over a millimeter. (If you look at the image, you can see that the screw holes are angled slightly downward on the right side of the machines, but they needed to be perpendicular to the machine shafts.)

Bad news - these tuning machines don't fit. :-(

After doing some additional research, I discovered that the only Gotoh tuning machines that Stewart McDonald sells are Gotoh's SG381 tuning machines, and I needed their SG360 tuning machines for my Explorer. After a quick call to Stewart McDonald, I verified that they cannot order Gotoh's SG360 tuning machines for me, so I searched the Internet until I found a distributer in Australia who could ship them to me. It took several weeks for the tuners to make the journey to the United States, but when they arrived they were a perfect fit.

Good news - these tuning machines fit. :-)

Once I had the right tuning machines installed, I started the long process of wiring and soldering the electronics.

Installing the pickups and
running the wires.
Soldering the pickup
selector switch.
Soldering a capacitor on
the tone potentiometer.
Installing the pickup selector
switch and running the wires.
Testing some of the
wiring before final soldering.
Soldering completed!

Once I completed the wiring, the last hurdles were to re-string the guitar, tune it up, adjust the string height and intonation, and test it out. (Which is the fun part.)

That about sums it up. The guitar looks great and plays great, although I might drop it by the folks at Parson's Guitars and have them them give it a quick tune-up for good measure.

Special thanks go to Harold for hooking me up with this guitar; and I also owe a big set of thanks to my wife, Kathleen, for humoring me while I took over one of the rooms in our house for the several weeks that I spent working on this project. ;-)

The Days Grow Shorter...

Back in the 1980s I was a big fan of the Canadian Power Trio named "Triumph." As far as arena rock was concerned, few bands could put on a show that was anywhere near as entertaining as a Triumph concert. It wasn't just about being a fan - there are any number of great bands out there who could put on a good show if you already liked them; but Triumph put on a killer show whether you liked them or not.

At the height of their popularity, Triumph recorded what was to become one of their greatest hits, which was a song that was titled "Fight the Good Fight." Many guitar players - myself included - spent a good deal of time learning that song, and I always enjoyed playing it live in the various rock bands that I played in throughout my teenage years.

As the first official day of Autumn is just around the corner here in Seattle, the opening lines to "Fight the Good Fight" seem to take on special meaning:

"The days grow shorter,
And the nights are getting long.
Feels like we're running out of time."

As I look out of my office window, that's exactly what I see:

Our short-lived Pacific Northwest Summer appears to have come to a close, and the clouds seem like they're here for the duration. The sun is setting a little earlier each day, and within a few months the choleric combination of miserable mists and depressing dusk will shorten the average day to six hours or less of daylight. And yet the most discouraging fact that I have to wrestle with today is the knowledge that the weather will be this way for the next nine months.

[I exhale a deep sigh...] Storm cloud

Three months from now is the Winter Solstice, at which time we will confront the shortest day of the year; after that, we will at least have the small consolation that each day will be a little longer than the last, but we still won't see much of the sun until sometime next June or July.

[I heave another deep sigh...] Storm cloud

I wonder how much a plane ticket to Hawaii would cost in January? Island with a palm tree