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. ;-]