Hello again! Our previous topic was Temperaments, a collection of tuning systems for keyboards and other instruments. Today we will cover Scale and Modes, and perhaps some other interesting tidbits.
CLICK HERE to avoid confusion
So what is a scale? A scale is an arrangement of notes in a certain pattern that repeats at the octave. We've already discussed the Solfege "Do-Re-Mi-Fa-Sol-La-Ti-Do" which is the Major scale. The Major scale can be seen in some other ways. For example in C it would be "C-D-E-F-G-A-B-C." But to narrow down exactly what a major scale is, regardless of what pitch you start on, you use a certain order of whole steps and half steps to construct the scale. For a Major scale, you will always use this interval pattern:
W-W-H-W-W-W-H
W being a whole step and H being a half step. It may help you to look at a piano keyboard to visualize this concept. Again, this pattern holds true for any starting pitch. So if I started a major scale on Eb instead of C, my pattern of steps would be the same so my scale would come out to "Eb-F-G-Ab-Bb-C-D-Eb." Easy enough, right?
The Natural Minor scale uses just the same notes as the major scale, but the interval pattern starts in a different place. Instead of WWHWWWH, it uses W-H-W-W-H-W-W. Interestingly, if you don't count the step that brings you back to your starting pitch, Natural Minor is a symmetrical scale, meaning the interval content is the same if the order is reversed. We'll see other scales that share this property.
The Major scale is typically associated with happiness and the Minor scale with sadness. But why? I have no clue, and it honestly makes no sense to me. You can have uplifting songs in a minor key (E.S. Posthumus' Pompeii) and heartbreakingly sad songs in a major key (Married Life, the opening sequence to Pixar's UP, for example). The associations people make about major and minor key probably come from years of social conditioning.
Now, there are other types of Minor scales, called Harmonic and Melodic. They exist in Music Theory to provide for more harmonic complexity when composing or playing in minor keys. Harmonic Minor is:
W-H-W-W-H-m3-H................where there's actually a jump of a minor third in the scale. Harmonic Minor is a way of providing a leading tone to the minor scale. Leading tones are things like B to C, a half-step motion that's a major part of a chord resolution through what's called Voice Leading. Anyway, if you had Harmonic Minor spelled on A you would have:
"A-B-C-D-E-F-G#-A"................so really it's Natural Minor, but with the 7th note raised by a half step.
Melodic Minor is not complicated, it's Natural Minor with a raised 6th and 7th.
W-H-W-W-W-W-H or "A-B-C-D-E-F#-G#-A"
(or you could call it Major with a flat 3, it's your call)
Then so far we've seen the following scales:
Major, Natural Minor, Harmonic Minor, Melodic Minor.
Are these the only scales? Of course not! There are such things as the Gypsy scale, Hungarian scale, Arabic, Persian, and more--but those are kind of advanced so I'll just touch them and leave them here for now.
There are also scales that use more, or less than seven notes. For example the pentatonic scale, which uses 5 notes. It has no half steps between pitches, so dissonance is minimal or nonexistent depending on the ear of the listener. The clearest example of the pentatonic scale is simply the black keys of a piano:
"F#-G#-A#-C#-D#-F#" being "Major Pentatonic" because the notes are in an order that promotes a major sound. Indeed, it's a major scale missing the 4th and 6th.
"D#-F#-G#-A#-C#-D#" being "Minor Pentatonic" for an analogous reason.
The Pentatonic Scales are associated with Asia, Asians, China, Japan, etc. If you plunk around on only the black keys of a piano, you'll recognize what I mean. This association comes from the fact that many Asian culture do utilize some kind of 5-note scalar tuning on their instruments. It's not exactly like our equal-tempered pentatonic scale, but it's close enough that the association stands.
There are also Hexatonic scales, like the Whole Tone Scale or the Prometheus Scale.
Whole tone scale: "C-D-E-F#-G#-A#-C" : uses only whole steps, is made of three tritones, and when this scale is played the sound often seems devoid of a tonal center. If you've watched cartoons/sitcoms at all you'll recognize the whole tone scale; it is often used when characters are dreaming.
The Prometheus Scale is similar to, but /dramatically/ different from the whole tone scale. Only one pitch is different, but that pitch changes the harmonic ballgame tremendously.
Prometheus Scale: "C-D-E-F#-A-Bb-C"
Alexander Scriabin, a composer of the 20th century, utilized something in his work which analysts came to call the "Mystic Chord." The Mystic chord uses the very specific tones of the Prometheus Scale, C-D-E-F#-A-Bb-C , and spells them as what's called a Quartal Hexachord. In layman's terms, that means a chord with six notes (hexa) spelled out vertically in some type of 4ths (quart). The intervals of the Mystic Chord are an augmented fourth, a diminished fourth, another augmented fourth, a perfect fourth, and another perfect fourth. You could more specifically say that is uses that interval pattern with these notes:
"C-F#-Bb, E, A, D" , each higher in pitch than the last. The resulting vertical sonority (read: chord) is Scriabin's Mystic Chord in all its glory. I encourage you to listen to his work "Prometheus: Poem of Fire" for a clear understanding of how awesome the chord is (for whatever reason; the Mystic Chord is rooted not in consonant harmony but in Scriabin's personal spiritual beliefs).
Also there are 8-note scales, Whole-Half Diminished and Half-Whole Diminished. These are obtained by alternating whole and half steps, creating a scale rife with minor seconds, tritones, and diminished chords. Compared to many of the other scales, these could commonly be perceived as ugly. To clarify, the interval patterns are W-H-W-H-W-H-W-H, OR, H-W-H-W-H-W-H-W. It's not complicated stuff.
So that's it for scales that use more or less than seven notes (I'm not gonna talk about "nonatonic" or "decatonic" or "quartatonic" -- especially quartatonic, because that'd mean that 7th chord are scales!). But what about interesting scales that DO use seven notes? A prime example is the Lydian Dominant Scale, which approximates the upper partials of the harmonic series into a musical scale; it has a very cool and unique sound due to this. Its interval pattern is "W-W-W-H-W-H-W." Lydian Dominant is also a Mode of Melodic Minor.
Whoa now. WHOA. NOW. "MODE"? Vat eez deez "Maude" vous zpeek off?
Modes are a delightful consequence of the invention of scales.When you have an interval pattern that repeats, you could start on any pitch of the pattern and treat is an your Tonic. Let me explain what this means briefly.
There are all kinds of modes out there, but first I'll discuss those derived from the Major Scale. To begin with, the Major Scale is itself a Mode. It's called "Ionian" Mode, and for a very neat reason. Back when good ol' Pythagoras was doing his thing, establishing founding principles of music, different cultures in the western world favored some interval patterns over others. The most popular wound up being Ionian (major scale) and Aeolian (minor scale), and these have endured even to this day.
But here are all the possible Modes of Major:
1. Ionian: W-W-H-W-W-W-H
2. Dorian: W-H-WWW-H-W (symmetrical/palindromic; pattern is same if read from end to beginning)
3. Phrygian: H-W-W-W-H-W-W
4. Lydian: W-W-W-H-W-W-H
5. Mixo-Lydian: W-W-H-W-W-H-W
6. Aeolian: W-H-WW-H-W-W (also palindromic, if you don't count the repeated note at the octave)
7. Locrian: H-W-W-H-WWW
They have certain qualities and properties that make them special:
Ionian: Is the Major Scale. Is associated with positive emotions. No more to be said.
CDEFGABC on the piano.
Dorian: Minor scale with a sharped 6th. Its symmetry makes it perfect for chant, as in a cathedral.
DEFGABCD on the piano.
Phrygian: Minor scale with a flattened 2nd. It has a very dark feel to it, also good for ominous chanting.
EFGABCDE on the piano.
Lydian: Major scale with a sharped 4th. Can be peaceful OR jarring depending on usage (read: Saria's Song in Zelda vs Floating Continent in Final Fantasy 6).
FGABCDEF on the piano.
Mixo-Lydian: Major with a flattened 7th. Similar to Lydian Dominant. To me it seems subdued.
GABCDEFG on the piano.
Aeolian: The Minor Scale. Is palindromic, is associated with negative emotions.
ABCDEFGA on the piano.
Locrian: Minor with a flat 2nd and flat 5th. I personally find this mode useless. It's very unsettling, so perhaps it could work thematically in a game or film score.
BCDEFGAB on the piano.
And those are only the Modes of the Major scale. I think the Modes of Melodic Minor are even cooler.
1. Melodic Minor: W-H-W-W-W-W-H
2. Dorian b2: H-W-W-W-W-H-W (Palindromic if you don't count the octave)
3. Lydian Augmented: W-W-W-W-H-W-H
4. Lydian Dominant: W-W-W-H-W-H-W (Also made up of two dominant seventh chords)
5. Mixo-Lydian b6: WW-H-W-H-WW (Palindromic from root to octave)
6. Semilocrian: W-H-W-H-W-W-W (One of my favorites: Made up of two half-diminished seventh chords)
7. Superlocrian: H-W-H-W-W-W-W (more like SUPER-USELESS)
I won't discuss the modes of Harmonic Minor today. Probably because I don't like them as much. ;)
I think I'll devote a second treatise later on to the truly obscure scales and modes. For now, I'll leave it at that! Hope that was helpful. If you have questions, e-mail me at dmguillot@gmail.com, or shoot me a message on Facebook. I'll do my best to elaborate.
I plan on sharing something very special with you all on my next post. It's something I've been researching for over a year, and I think you'll be surprised by how elaborate it all is. Seeya next time! :)
Friday, September 21, 2012
Saturday, August 4, 2012
Temperament, Cents, and Hertz
Hello again, all. Thanks to those of you who decided to read the first installment in this musical series of mine. Last time we discussed the Circle/Spiral of 5ths, some things about Pythagoras, and the basics of the Harmonic Series. Today the focus of our discussion is the concept of Temperament, but we will also cover the idea of Cents and the measurement known as Hertz.
I'll pick up where we left off in the Circle of 5ths lecture. People were stumped by the Pythagorean Comma and wished to find a tuning system that circumvented this problem. Attempts to do so resulted in what are called Temperaments, different tuning systems devised to capture certain natural elements of music; no matter what elements of nature's harmony a temperament replicates, it always falls short in other areas.
I will be discussing three types of temperament: Just, Meantone, and Equal Temperament. I mentioned the Pythagorean musical scale last time, but this is not a tempered scale; instead, it derives its pitches directly from the Spiral of 5ths, and contains the very issues that Temperaments were designed to snuff out.
Just Intonation is designed to tune chords with absolute accuracy. In J.I., the Major chord is 4-5-6 exactly, the natural harmonics in series. The octaves, fifths, major thirds, harmonic sevenths, ninths -- indeed every interval of the diatonic scale (Do Re Mi Fa Sol La Ti Do) is pure. There is no harmonic compromise; this is exact harmony as nature intended. Sounds spiffy, doesn't it?
It certainly does, but there's a significant problem involved. You can tune an instrument--say, a keyboard--to play in Just intonation with a diatonic (=major) scale. The scale of C-D-E-F-G-A-B-C, for example. So you have this beautiful, pure C Major scale on your instrument--BUT. Because you've tuned every interval in relationship to the note C, only the diatonic scale on C (or the relative A Minor scale) can be played in tune. Every other interval against C will sound out of tune and strikingly dissonant, ruining the wonderful harmony.
The basic point here is that when you're using J.I., your instruments cannot transpose. You can't play a piece and C and immediately afterward play a piece in F#. For this reason, if you listen to music of the Baroque era you'll find that almost every song remains in a single key, and uses either a Major or Minor scale (probably both) exclusively. They are beautiful compositions but restricted to a certain number of notes, namely the diatonic seven as opposed to all 12 Chromatic pitches. That is the downfall of Just Intonation: crisp, pure intervals, but no (or very limited) transposition to other keys.
Meantone Temperament was an attempt to reconcile accurate harmonicity with the ability to transpose. There were all kinds of meantone temperaments, each with a different focus. Some aimed to preserve pure Thirds, others pure Fifths, others to approximate all the intervals while allowing for freedom of transposition to other keys. There are enough types of meantone tunings that I don't want to get into them right now, but if you're curious there's a fine Wiki article that can tell you things I wouldn't even know about.
An honorable mention goes to J.S. Bach, a Baroque composer who invented a mythical tuning called Well Temperament. There is no consensus on what exactly Well Temperament sounded like, but it was said to sound in tune in every chromatic key signature while allowing each key signature to have its own unique sound. So the exact intervals used in C Major and F# Major were somewhat different, but the harmony in each of them would be essentially smooth and pleasing to the ear. Some people point to a certain Baroque tuning as likely being Well Temperament, but after listening to example of the tuning I'm not sure I believe them. Others say that Well Temperament was only lost in history for so long because another clever temperament popped up at roughly the same time.
Equal Temperament. 12-tone, equal, temperament. All rise for the pledge of allegiance! *hand over heart*
So 12-tone Equal Temperament, it's the guy we in the Western World look to for our musical tuning system. It caught on like wildfire when the concept appeared toward the end of the Baroque era. So what's so fab about it? First off, it approximates Perfect 5ths almost perfectly. In this case, "almost" is a VERY good thing. You recall last time I said people were looking for a way to close the Spiral of 5ths? Yeah, 12-TET did that. Like a boss. To close the Spiral into a Circle of 5ths, each 5th used in the circle was made ever-so-slightly flat. That way, when you start on A and go up 12 Equal Tempered 5ths, you land on the same A as if you'd gone up 7 Octaves. This notion of a perfect acoustic circle was too good to be true, and it quickly began to dominate the world of instrument tuning /even though/ some people thought it sounded weird at the time.
Alright, we make the Perfect 5ths a little flat. But how much flatter than a real one? The answer: almost exactly 2 Cents flatter.
Wait WAT? What the devil are "Cents"? Well Cents are a notion that only came into being once 12-TET took over the world. There are 12 chromatic pitches in an octave, right? Well, each one of those pitches in 12-TET is equidistant from its neighbors. That is to say, the distance between C and C# or Gb and G is 100 cents. In this way, 12 notes having 100 cents distance between each other make an octave comprising 1200 cents, and 12 notes. Not difficult to process, is it? This also gave us a new way of quantifying intervals from a Root, other than talking about pure harmonic relationships.
Here are the different Chromatic intervals, showing their Cent values:
0 c. = Root
100 c. = half step / semitone
200 c. = whole step / whole tone
300 c. = minor third
400 c. = Major third
500 c. = Perfect Fourth
600 c. = Tritone (I'll talk about this guy)
700 c. = Perfect Fifth
800 c. = minor 6th
900 c. = Major 6th
1000 c. = minor 7th
1100 c. = Major 7th
1200 c. = Perfect Octave (fun fact: in some Meantone temperaments the octaves were NOT perfect!)
These cent values are convenient and easy to remember, perhaps easier to remember than the specific numbered harmonic that is ascribed to each note. Is it easier to remember a Major 6th as 900 cents or the 27th harmonic? Maybe that depends on the person; I know not. In any case, those intervals are the ones we (read: the entire Western world and all "modernized" countries) use on our instruments, our synthesizers, our pitch pipes and all the rest. 12-EQT is actually a bold compromise, as the only TRULY perfect interval is the octave while the 4th and 5th are almost exact but not quite. The Major 3rd we use is 14 cents sharp; our Major 7th is 12 cents sharp; our "harmonic seventh" is 31 cents sharp of the true one, and that's a lot.
Then we have the lovely little lady called the Tritone.
Throughout history the interval of the Tritone was called by many unpleasant names. The most famous of these is "diabolus in musica" -- which means "THE DEVIL IN MUSIC." Why did it get such a bad rap? Quite simply, it was almost impossible to sing in tune. This is partly because there are many different "tritones" in the Harmonic series, rather than a single example to emulate.
The Equal Tempered Tritone, whose ratio is the square root of 2 (1.414...), became the quintessential version of the interval. Because it's an equal division of the octave, it has a strange kind of "stability" while being nonetheless dissonant to our ears (dissonant =/= ugly!). Our tritones have become on a wide scale much easier to sing after years and years of acoustic conditioning. I am personally so comfortable with tritones now that they are no longer dissonant to my ears. Not everyone has this luxury; some still find them hideous or unpleasant.
I suppose now's as good a time as any to begin on Hertz. Hertz are especially easy to understand if we're talking about Just Intonation. The word Hertz refers to a unit of measurement, and what the unit measures is Cycles-Per-Second. This basically means, in acoustic terms, the frequency of a wave. A sound wave.
Every note has a value in Hz, and the best example to begin with is A. In our tuning, we tune the note A to 440 Hz. Thus, every other note A is an octave up or down from 440, which means dividing or multiplying by 2. There's A 440, A 880, A 1760...and on the other hand, A 220, A 110, A 55. This direct, perfect relationship of doubling or halving a frequency is what Octaves are all about.
In J.I. you can take the hertz values even further in their relationship. Let's consider A 110. Recall the harmonic series: 1, 2, 3, 4, 5 etc. If we treat A 110 as the Root, or first harmonic, this is what follows:
110, 220, 330, 440, 550
Which from the Root spells an Octave, Perfect 5th, Double Octave, and Major Third.
Isn't Math fun? :)
Hz and Cents can both be used to compare the similarities and differences of different types of tuning. They can show numerically how the 12-EQT Major Third is considerably sharper than the acoustic/numeric reality, and the slight, nigh-imperceptible difference between an acoustic P5 and a tempered one. I often check the cent values of my tempered notes against the true harmonic series' pitches when I am crafting new chords for my compositions. I also use a waveform analyzer to prove to me visually whether a chord is consonant or dissonant, and in what way.
I took a course on the Physics of Music, and I took from it a strong understanding about Cents and Hertz, on a level that I can't fully communicate here. If any of you want to ask me about it in person, I can show you materials I still have from that class which would back up everything I've said here and reinforce it in new ways. Otherwise, if you're curious enough you can take the mindspark I've shared here and ride it across the web searching for answers I haven't provided.
That wraps up my rant on Temperament, Cents, and Hertz. I hope you found it on some level informative--and once again, if I'm missing any crucial information or didn't cover something you've wondered about just let me know and I'd gladly make an addendum to this post.
Next time: ??? - I dunno, maybe Combination Tones and their relevance to harmony. We'll see. :)
I'll pick up where we left off in the Circle of 5ths lecture. People were stumped by the Pythagorean Comma and wished to find a tuning system that circumvented this problem. Attempts to do so resulted in what are called Temperaments, different tuning systems devised to capture certain natural elements of music; no matter what elements of nature's harmony a temperament replicates, it always falls short in other areas.
I will be discussing three types of temperament: Just, Meantone, and Equal Temperament. I mentioned the Pythagorean musical scale last time, but this is not a tempered scale; instead, it derives its pitches directly from the Spiral of 5ths, and contains the very issues that Temperaments were designed to snuff out.
Just Intonation is designed to tune chords with absolute accuracy. In J.I., the Major chord is 4-5-6 exactly, the natural harmonics in series. The octaves, fifths, major thirds, harmonic sevenths, ninths -- indeed every interval of the diatonic scale (Do Re Mi Fa Sol La Ti Do) is pure. There is no harmonic compromise; this is exact harmony as nature intended. Sounds spiffy, doesn't it?
It certainly does, but there's a significant problem involved. You can tune an instrument--say, a keyboard--to play in Just intonation with a diatonic (=major) scale. The scale of C-D-E-F-G-A-B-C, for example. So you have this beautiful, pure C Major scale on your instrument--BUT. Because you've tuned every interval in relationship to the note C, only the diatonic scale on C (or the relative A Minor scale) can be played in tune. Every other interval against C will sound out of tune and strikingly dissonant, ruining the wonderful harmony.
The basic point here is that when you're using J.I., your instruments cannot transpose. You can't play a piece and C and immediately afterward play a piece in F#. For this reason, if you listen to music of the Baroque era you'll find that almost every song remains in a single key, and uses either a Major or Minor scale (probably both) exclusively. They are beautiful compositions but restricted to a certain number of notes, namely the diatonic seven as opposed to all 12 Chromatic pitches. That is the downfall of Just Intonation: crisp, pure intervals, but no (or very limited) transposition to other keys.
Meantone Temperament was an attempt to reconcile accurate harmonicity with the ability to transpose. There were all kinds of meantone temperaments, each with a different focus. Some aimed to preserve pure Thirds, others pure Fifths, others to approximate all the intervals while allowing for freedom of transposition to other keys. There are enough types of meantone tunings that I don't want to get into them right now, but if you're curious there's a fine Wiki article that can tell you things I wouldn't even know about.
An honorable mention goes to J.S. Bach, a Baroque composer who invented a mythical tuning called Well Temperament. There is no consensus on what exactly Well Temperament sounded like, but it was said to sound in tune in every chromatic key signature while allowing each key signature to have its own unique sound. So the exact intervals used in C Major and F# Major were somewhat different, but the harmony in each of them would be essentially smooth and pleasing to the ear. Some people point to a certain Baroque tuning as likely being Well Temperament, but after listening to example of the tuning I'm not sure I believe them. Others say that Well Temperament was only lost in history for so long because another clever temperament popped up at roughly the same time.
Equal Temperament. 12-tone, equal, temperament. All rise for the pledge of allegiance! *hand over heart*
So 12-tone Equal Temperament, it's the guy we in the Western World look to for our musical tuning system. It caught on like wildfire when the concept appeared toward the end of the Baroque era. So what's so fab about it? First off, it approximates Perfect 5ths almost perfectly. In this case, "almost" is a VERY good thing. You recall last time I said people were looking for a way to close the Spiral of 5ths? Yeah, 12-TET did that. Like a boss. To close the Spiral into a Circle of 5ths, each 5th used in the circle was made ever-so-slightly flat. That way, when you start on A and go up 12 Equal Tempered 5ths, you land on the same A as if you'd gone up 7 Octaves. This notion of a perfect acoustic circle was too good to be true, and it quickly began to dominate the world of instrument tuning /even though/ some people thought it sounded weird at the time.
Alright, we make the Perfect 5ths a little flat. But how much flatter than a real one? The answer: almost exactly 2 Cents flatter.
Wait WAT? What the devil are "Cents"? Well Cents are a notion that only came into being once 12-TET took over the world. There are 12 chromatic pitches in an octave, right? Well, each one of those pitches in 12-TET is equidistant from its neighbors. That is to say, the distance between C and C# or Gb and G is 100 cents. In this way, 12 notes having 100 cents distance between each other make an octave comprising 1200 cents, and 12 notes. Not difficult to process, is it? This also gave us a new way of quantifying intervals from a Root, other than talking about pure harmonic relationships.
Here are the different Chromatic intervals, showing their Cent values:
0 c. = Root
100 c. = half step / semitone
200 c. = whole step / whole tone
300 c. = minor third
400 c. = Major third
500 c. = Perfect Fourth
600 c. = Tritone (I'll talk about this guy)
700 c. = Perfect Fifth
800 c. = minor 6th
900 c. = Major 6th
1000 c. = minor 7th
1100 c. = Major 7th
1200 c. = Perfect Octave (fun fact: in some Meantone temperaments the octaves were NOT perfect!)
These cent values are convenient and easy to remember, perhaps easier to remember than the specific numbered harmonic that is ascribed to each note. Is it easier to remember a Major 6th as 900 cents or the 27th harmonic? Maybe that depends on the person; I know not. In any case, those intervals are the ones we (read: the entire Western world and all "modernized" countries) use on our instruments, our synthesizers, our pitch pipes and all the rest. 12-EQT is actually a bold compromise, as the only TRULY perfect interval is the octave while the 4th and 5th are almost exact but not quite. The Major 3rd we use is 14 cents sharp; our Major 7th is 12 cents sharp; our "harmonic seventh" is 31 cents sharp of the true one, and that's a lot.
Then we have the lovely little lady called the Tritone.
Throughout history the interval of the Tritone was called by many unpleasant names. The most famous of these is "diabolus in musica" -- which means "THE DEVIL IN MUSIC." Why did it get such a bad rap? Quite simply, it was almost impossible to sing in tune. This is partly because there are many different "tritones" in the Harmonic series, rather than a single example to emulate.
The Equal Tempered Tritone, whose ratio is the square root of 2 (1.414...), became the quintessential version of the interval. Because it's an equal division of the octave, it has a strange kind of "stability" while being nonetheless dissonant to our ears (dissonant =/= ugly!). Our tritones have become on a wide scale much easier to sing after years and years of acoustic conditioning. I am personally so comfortable with tritones now that they are no longer dissonant to my ears. Not everyone has this luxury; some still find them hideous or unpleasant.
I suppose now's as good a time as any to begin on Hertz. Hertz are especially easy to understand if we're talking about Just Intonation. The word Hertz refers to a unit of measurement, and what the unit measures is Cycles-Per-Second. This basically means, in acoustic terms, the frequency of a wave. A sound wave.
Every note has a value in Hz, and the best example to begin with is A. In our tuning, we tune the note A to 440 Hz. Thus, every other note A is an octave up or down from 440, which means dividing or multiplying by 2. There's A 440, A 880, A 1760...and on the other hand, A 220, A 110, A 55. This direct, perfect relationship of doubling or halving a frequency is what Octaves are all about.
In J.I. you can take the hertz values even further in their relationship. Let's consider A 110. Recall the harmonic series: 1, 2, 3, 4, 5 etc. If we treat A 110 as the Root, or first harmonic, this is what follows:
110, 220, 330, 440, 550
Which from the Root spells an Octave, Perfect 5th, Double Octave, and Major Third.
Isn't Math fun? :)
Hz and Cents can both be used to compare the similarities and differences of different types of tuning. They can show numerically how the 12-EQT Major Third is considerably sharper than the acoustic/numeric reality, and the slight, nigh-imperceptible difference between an acoustic P5 and a tempered one. I often check the cent values of my tempered notes against the true harmonic series' pitches when I am crafting new chords for my compositions. I also use a waveform analyzer to prove to me visually whether a chord is consonant or dissonant, and in what way.
I took a course on the Physics of Music, and I took from it a strong understanding about Cents and Hertz, on a level that I can't fully communicate here. If any of you want to ask me about it in person, I can show you materials I still have from that class which would back up everything I've said here and reinforce it in new ways. Otherwise, if you're curious enough you can take the mindspark I've shared here and ride it across the web searching for answers I haven't provided.
That wraps up my rant on Temperament, Cents, and Hertz. I hope you found it on some level informative--and once again, if I'm missing any crucial information or didn't cover something you've wondered about just let me know and I'd gladly make an addendum to this post.
Next time: ??? - I dunno, maybe Combination Tones and their relevance to harmony. We'll see. :)
Friday, August 3, 2012
The Circle of 5ths and introduction
Hello all. I'm Dave, and this will be my first time posting to a blog. For those who know me, HI! For those who don't know me, you're probably not reading this blog. But in any case, I will share a bit about myself.
I have been trying my hand as a serious composer since the age of ~14, when I began teaching myself piano and learned to play/sing/write music by ear. I have written in myriad styles through the years, from the diversity of a video game soundtrack to attempted stage musicals to Choral music performed live by my college choir. I feel I've come very far and learned a lot in my time pursuing music, and I've got a long way to go for sure.
But I've decided I'd like to share some of what I learned, and here is where I'll do it.
Today's topic is the Circle of 5ths. If you're a music student, you've heard of this Circle before, and you probably know what a Perfect 5th is in some sense. In case you're not a music student, I will start from the beginning.
When most music teachers start from "the beginning" they say:
"This here's the circle of 5ths. It's the foundation of our system of music and shows how all 12 of our chromatic pitches are related. You must memorize this or you'll do poorly in my class."
Which is a horrid approach that completely misses the point. If you want your students to really learn what it is, the REAL beginning is the best place to start. If you approach it from there, you will learn some things that make the Circle of 5ths much more interesting.
For starters, it isn't really a circle.
Some of you may remember a man called Pythagoras. He lived in ancient Greece, and he was basically a wizard at everything. He (or his followers, whatever; I'll just call it Pythagoras) invented the Pythagorean Theorem familiar to anyone who's taken Math in high school. It stated that the relationship between the lengths of the three sides of a Right Triangle was a^2+b^2=c^2. This Theorem is so pervasive that everyone who hears the name Pythagoras thinks of it before anything else.
To me this is kind of a crime. Certainly the Theorem is of huge importance in mathematics, and we owe him a debt of gratitude for that alone. But another monumentally important mathematical principle helped Pythagoras to do something else extraordinary.
What Pythagoras did became the foundation of the system of Western music we use today. It's seriously that enduring; after thousands of years we still utilize the founding principles he established about music. And what was it he established? It begins with something called a Monochord.
A Monochord is a simple device: one very long string (as on a guitar) stretched across a single structure of wood. Pythagoras conducted an amazing experiment with it: when he plucked the string on its own, he could hear not only the tone the string was tuned to make, but also some higher tones that related to the string harmoniously. After noticing this, he experimented to find what those tones were, and how they related to the root note to which the string was tuned. Forgive me for what seems like a diversion, but now I must discuss something vital to the understanding of the rest of Pythagoras' story: The Harmonic Series.
Can you count? 1, 2, 3, 4, 5, 6, 7, 8 -- let's start with that. The harmonic series is as easy as counting; it establishes that most tones in nature have an acoustic relationship with other tones called harmonics. Perhaps that isn't the best way to phrase it, but bear with me. Here is a table of what the harmonics mean, musically:
1 = Root
2 = Octave
3 = Perfect Fifth
4 = Double Octave
5 = Major Third
6 = Octave-Fifth
7 = Harmonic Seventh
8 = Triple Octave
The Root is also called the Fundamental; it is the basis of a Chord. Chords are built out of tones that relate harmonically. All even-numbered harmonics are forms of the Root: 1, 2, 4, 8, 16, 32, etc. ; tones with this relationship will always be in perfect harmony with one another, because they're all powers of two. To make it more simple, if the harmonic "1" or Root is tuned to the note C, then every even-numbered harmonic will also be the note C.
On the other hand, odd-numbered harmonics will always be tones distinct from the fundamental tone. They provide us musically with unique pitches that still relate mathematically to the Root. With a combination of harmonics, be they odd or even, you create acoustic sounds that are mathematically, naturally, harmonious and pleasant to the ear.
A simple example would be to take the first three odd harmonics: 1, 3, and 5. Anybody remember Solfege? That system of giving each note in a scale a short name to use when you sing the scale? It goes like (duh): Do-Re-Mi-Fa-Sol-La-Ti-Do. These same scale degrees also have letter names, like C. If we made Do into C, the scale would become C-D-E-F-G-A-B-C. I'll tell you right now that the harmonics 1, 3, and 5 are the basis of what is called a Major chord. Most of you are probably familiar with the difference in sound between Major and Minor chords. If you're not, that's fine, just keep reading.
Let's say we want to make a Major chord out of notes in the scale C-D-E-F-G-A-B-C. Let's also say we want to use "C" as the Root of the chord. We need three pitches, and "C" is the Root, 1. What do you say we take the third, and 5th notes of the scale and add them above C. Does that make a Major chord?
C - E - G = Major chord
What do you know! C - E - G is a Major chord! And,
1 - 3 - 5 = Major chord
Brilliant! So that must mean that C - E - G is the same as 1 - 3 - 5!
C - E - G =/= 1 - 3 - 5
Except no, it isn't. Not really. The truth is, both are certainly Major chords in every sense. BUT, the problem is that E is not the third harmonic and G is not the 5th harmonic. It's the other way around.
The REAL way to spell C - E - G is with harmonics 4 - 5 - 6. This is because 4th harmonic is an Octave of the Root, 5th harmonic is the Major Third, and 6th harmonic is an Octave of the Perfect Fifth. Confused yet? Let me clear up something else important.
When we say "Major third" it doesn't refer to the 3rd harmonic. It refers to the third note in OUR musical scale, not harmonic order. And the third note in the Major scale is Mi; or, if we're using note names, when C is the first note of the scale the third note (the Major Third) is E. And when C is the first note, the fifth note of the scale (the Perfect Fifth) is the note G. So C - E - G is a Major chord spelled in a Triad (Triad meaning that the pitches are distances to each other by some form of Third, Major or Minor). And, 4 - 5 - 6 is the harmonic spelling of the Major Triad, with C - E - G being a C Major Triad.
Still, the most accurate representation of the Major chord, acoustically, utilizes its harmonics in the order they occur in series. This means that the first appearance of the third harmonic (the Perfect Fifth) must occur before/beneath the first appearance of the 5th harmonic (the Major Third) in order for a chord to be in true harmonic alignment and have a pure sound. For this reason 1 - 3 - 5 seems a viable candidate for the Major chord, the only issue being that the tones are actually pretty far apart acoustically.
I personally think that 1 - 2 - 3 - 5 - 8 is one of the finest Major chords you can make, but I'll save the reason why for another lecture.
So we see how a combination of different harmonics, like 4 - 5 - 6, can make a harmonious chord like a Major chord. By the same token, the harmonics 1, 3, 5, 7, 9 create what's called a Dominant 9 chord. Sadly, because of the way our instruments are tuned it's impossible to play most odd harmonics, like 11 and 13; we actually have approximations in our scale that can be used for 15, 17, and 19, but I'm getting ahead of myself.
Back to Pythagoras. He understood the mathematics of the harmonic series and wanted to harness it to create musical instruments. He experimented with his Monochord to try and capture the higher tones he heard when he plucked the open string. So he would place his finger on part of the string, pressing it down, and pluck the string at the length that wasn't pressed down. This changed the pitch, because shortening the length of the string increased the frequency of its vibrations. Pythagoras found that when the string was held down at certain points - point correlating to harmonics - he would hear the higher tones the string had been making on its own.
Specifically, when he held the string down at points correlating to Small Integer Ratios. A ratio is something like 1/2; one over two is one half or 0.5. But some specific ratios Pythagoras found important were these:
3/2, 5/4, 7/4
Because 3/2 is a Perfect Fifth, 5/4 is a Major Third, and 7/4 is a Harmonic Seventh. These are ratios that take the odd-numbered harmonics and reduce their distance from the Root (by dividing them by powers of two, like /2 and /4). With harmonic intervals using the right ratios, it would be possible to compress harmonics into the width of a single octave, creating a musical scale.
If you took only the ratios 4/4, 5/4, 6/4, and 7/4, this is the scale of notes you would get:
C - E - G - ~Bb (why the ~ is for another day, when I talk about Temperament)
This is a fine scale, but with only four notes it's very limiting. You can't make good melodies using only a scale with each note so acoustically distant from the others. So Pythagoras thought, "How can I make a scale with numerous notes, all of which have some harmonic relationships, which fits in the span of a single octave and would be practical for instruments to play?"
Which is a fine question! Eventually Pythagoras realized something. Apart from the Octave, the Third Harmonic/Perfect Fifth was the smallest integer ratio and therefore the most stable interval aside from the octave. If Pythagoras could devise a scale based around the interval of the Perfect Fifth, the musical possibilities would be grandiose and far-reaching.
But how to make such a scale? Hmmm...Eureka came around eventually, and he said, "I will start on one note, my Root, and find its third harmonic; then I will find the third harmonic of that third harmonic, and so on, and so forth, yea and verily, til death do us part!"
And so he did. I've already mentioned that the third harmonic is the numerical value of 3 (duhhhh), and that the Octave is the numerical value of 2. To create a scale made of P5s that fits into one Octave, the only thing to do is multiply 1.5 (3/2) by itself and multiply 2 by itself until their values intersect. As it happens,
1.5^12 = 129.746
2^7 = 128
Well there you have it! Those numbers are extremely similar, so we can make a scale based on Perfect 5ths. It turns out that the acoustic distance of seven octaves is only slightly less than the distance of 12 P5s.
...only slightly less...that should be fine...won't cause any problems that endure for centuries to come...
So if there are 12 P5s in an octave, that means there are 12 distinct notes before the scale repeats. This is known today as the Chromatic Scale, which contains these 12 pitches:
C - C# - D - D# - E - F - F# - G - G# - A - A# - B ... and repeats at C, going further up or down as you like. Here (click) is the Wiki article on the Circle of 5ths, so you can see a picture (if you've never seen one before) of the notes arranged in an actual visible circle, rather than a metaphorical acoustic one.
"So in conclusion, Pythagoras invented the Circle of Fifths that we use today in modern music."
I would say, if that was accurate. But here's the thing.
That "slight" difference between 12 P5s and 7 Octaves is significant enough that it causes a huge, huuuuge problem. That problem is called the Pythagorean Comma, 23.46 cents (I have not yet explained cents, but I will in a later entry). Here's how I normally explain it:
If you begin on A, and go up seven octaves, you will arrive on another pure A.
If you begin on A, and go up the entire circle of fifths, you will arrive on an overly sharp "A".
If you have a note that IS A, played against a note that is ALMOST A,
you get something acoustically disgusting and discordant.
Octaves must be played in tune; either note of an "octave" being too sharp or too flat creates a frequency ratio that is overly complex and thereby dissonant. Dissonance is not bad in and of itself, but if what you're attempting to create is consonant harmony (which was Pythagoras' intention with the scale) then you must tune every interval accurately. The Pythagorean Comma meant an octave would be detuned, if the entire Circle of 5ths were used on an instrument.
In essence, this means that the "Circle" of 5ths is actually a Spiral of 5ths. If you use true acoustic Perfect 5ths and continue moving up, you will absolutely never arrive on a pitch that is a pure octave of the note you started on. The 5ths may have been perfect, but the scale they created had its issues.
So the first response was, "Hey, let's just not use the entire Spiral." So the Diatonic scale was born, a seven-note scale we know as Do-Re-Mi-Fa-Sol-La-Ti-Do. In this scale, instead of using the overly sharp Do from the end of the Spiral, the second higher Do was a pure octave of the first. So that solved that problem. There was another concern, though, namely the tuning of "Mi." The Major chord as I mentioned before is Do-Mi-Sol, but the Pythagorean Major Third is way, waaay too sharp. How sharp exactly?
The Harmonic Major Third is the ratio of 5/4.
The Pythagorean Major Third is the ratio of 81/64.
Its ratio's complexity makes it grating on the ear, and more simply it's just too sharp to create a truly pleasant, consonant sound. However, it was close enough that this Diatonic tuning endured for a long time. There was a Pythagorean Chromatic Scale as well, that used all 12 of the pitches of the Spiral of 5ths. But this one had even more harmonic trouble in use due to both its complexity and the frequent frequential disagreement of many of the scale's intervals.
Well, Pythagoras eventually passed away, but his followers carried on his work and promulgated his discoveries. The Spiral of 5ths as a musical system became widespread and popular, due to its reliance on clear mathematics and the acoustic functions of the natural world--and because it sounded pretty. The question of how to deal with the Pythagorean Comma and the other oddities of the Spiral of 5ths baffled people for many centuries, and they never stopped trying to find a way to solve that riddle. There had to be some way to fix the Spiral. To close it.
Ladies and gentlemen, I'll see you next time. For next time we delve. Next time we discover.
Next time, we close the Circle.
-Dave
Stay tuned for the upcoming entry on musical Temperament!
PS: I typed 99% of this from memory, so if you notice any errors in my information please let me know and I will add your corrections to the document. :)
First edit: So I browsed this after reading,
http://en.wikipedia.org/wiki/Pythagoras#Musical_theories_and_investigations
And apparently though Pythagoras did connect musical notes to the natural harmonic series, he was not responsible for inventing the Circle of 5ths. However, there ARE musical tunings called Pythagorean and Pythagorean Chromatic, probably devised using the small integer ratios of the harmonic series.
This is described in the article on Pyth tuning, for anyone interested.
http://en.wikipedia.org/wiki/Pythagorean_tuning
I have been trying my hand as a serious composer since the age of ~14, when I began teaching myself piano and learned to play/sing/write music by ear. I have written in myriad styles through the years, from the diversity of a video game soundtrack to attempted stage musicals to Choral music performed live by my college choir. I feel I've come very far and learned a lot in my time pursuing music, and I've got a long way to go for sure.
But I've decided I'd like to share some of what I learned, and here is where I'll do it.
Today's topic is the Circle of 5ths. If you're a music student, you've heard of this Circle before, and you probably know what a Perfect 5th is in some sense. In case you're not a music student, I will start from the beginning.
When most music teachers start from "the beginning" they say:
"This here's the circle of 5ths. It's the foundation of our system of music and shows how all 12 of our chromatic pitches are related. You must memorize this or you'll do poorly in my class."
Which is a horrid approach that completely misses the point. If you want your students to really learn what it is, the REAL beginning is the best place to start. If you approach it from there, you will learn some things that make the Circle of 5ths much more interesting.
For starters, it isn't really a circle.
Some of you may remember a man called Pythagoras. He lived in ancient Greece, and he was basically a wizard at everything. He (or his followers, whatever; I'll just call it Pythagoras) invented the Pythagorean Theorem familiar to anyone who's taken Math in high school. It stated that the relationship between the lengths of the three sides of a Right Triangle was a^2+b^2=c^2. This Theorem is so pervasive that everyone who hears the name Pythagoras thinks of it before anything else.
To me this is kind of a crime. Certainly the Theorem is of huge importance in mathematics, and we owe him a debt of gratitude for that alone. But another monumentally important mathematical principle helped Pythagoras to do something else extraordinary.
What Pythagoras did became the foundation of the system of Western music we use today. It's seriously that enduring; after thousands of years we still utilize the founding principles he established about music. And what was it he established? It begins with something called a Monochord.
A Monochord is a simple device: one very long string (as on a guitar) stretched across a single structure of wood. Pythagoras conducted an amazing experiment with it: when he plucked the string on its own, he could hear not only the tone the string was tuned to make, but also some higher tones that related to the string harmoniously. After noticing this, he experimented to find what those tones were, and how they related to the root note to which the string was tuned. Forgive me for what seems like a diversion, but now I must discuss something vital to the understanding of the rest of Pythagoras' story: The Harmonic Series.
Can you count? 1, 2, 3, 4, 5, 6, 7, 8 -- let's start with that. The harmonic series is as easy as counting; it establishes that most tones in nature have an acoustic relationship with other tones called harmonics. Perhaps that isn't the best way to phrase it, but bear with me. Here is a table of what the harmonics mean, musically:
1 = Root
2 = Octave
3 = Perfect Fifth
4 = Double Octave
5 = Major Third
6 = Octave-Fifth
7 = Harmonic Seventh
8 = Triple Octave
The Root is also called the Fundamental; it is the basis of a Chord. Chords are built out of tones that relate harmonically. All even-numbered harmonics are forms of the Root: 1, 2, 4, 8, 16, 32, etc. ; tones with this relationship will always be in perfect harmony with one another, because they're all powers of two. To make it more simple, if the harmonic "1" or Root is tuned to the note C, then every even-numbered harmonic will also be the note C.
On the other hand, odd-numbered harmonics will always be tones distinct from the fundamental tone. They provide us musically with unique pitches that still relate mathematically to the Root. With a combination of harmonics, be they odd or even, you create acoustic sounds that are mathematically, naturally, harmonious and pleasant to the ear.
A simple example would be to take the first three odd harmonics: 1, 3, and 5. Anybody remember Solfege? That system of giving each note in a scale a short name to use when you sing the scale? It goes like (duh): Do-Re-Mi-Fa-Sol-La-Ti-Do. These same scale degrees also have letter names, like C. If we made Do into C, the scale would become C-D-E-F-G-A-B-C. I'll tell you right now that the harmonics 1, 3, and 5 are the basis of what is called a Major chord. Most of you are probably familiar with the difference in sound between Major and Minor chords. If you're not, that's fine, just keep reading.
Let's say we want to make a Major chord out of notes in the scale C-D-E-F-G-A-B-C. Let's also say we want to use "C" as the Root of the chord. We need three pitches, and "C" is the Root, 1. What do you say we take the third, and 5th notes of the scale and add them above C. Does that make a Major chord?
C - E - G = Major chord
What do you know! C - E - G is a Major chord! And,
1 - 3 - 5 = Major chord
Brilliant! So that must mean that C - E - G is the same as 1 - 3 - 5!
C - E - G =/= 1 - 3 - 5
Except no, it isn't. Not really. The truth is, both are certainly Major chords in every sense. BUT, the problem is that E is not the third harmonic and G is not the 5th harmonic. It's the other way around.
The REAL way to spell C - E - G is with harmonics 4 - 5 - 6. This is because 4th harmonic is an Octave of the Root, 5th harmonic is the Major Third, and 6th harmonic is an Octave of the Perfect Fifth. Confused yet? Let me clear up something else important.
When we say "Major third" it doesn't refer to the 3rd harmonic. It refers to the third note in OUR musical scale, not harmonic order. And the third note in the Major scale is Mi; or, if we're using note names, when C is the first note of the scale the third note (the Major Third) is E. And when C is the first note, the fifth note of the scale (the Perfect Fifth) is the note G. So C - E - G is a Major chord spelled in a Triad (Triad meaning that the pitches are distances to each other by some form of Third, Major or Minor). And, 4 - 5 - 6 is the harmonic spelling of the Major Triad, with C - E - G being a C Major Triad.
Still, the most accurate representation of the Major chord, acoustically, utilizes its harmonics in the order they occur in series. This means that the first appearance of the third harmonic (the Perfect Fifth) must occur before/beneath the first appearance of the 5th harmonic (the Major Third) in order for a chord to be in true harmonic alignment and have a pure sound. For this reason 1 - 3 - 5 seems a viable candidate for the Major chord, the only issue being that the tones are actually pretty far apart acoustically.
I personally think that 1 - 2 - 3 - 5 - 8 is one of the finest Major chords you can make, but I'll save the reason why for another lecture.
So we see how a combination of different harmonics, like 4 - 5 - 6, can make a harmonious chord like a Major chord. By the same token, the harmonics 1, 3, 5, 7, 9 create what's called a Dominant 9 chord. Sadly, because of the way our instruments are tuned it's impossible to play most odd harmonics, like 11 and 13; we actually have approximations in our scale that can be used for 15, 17, and 19, but I'm getting ahead of myself.
Back to Pythagoras. He understood the mathematics of the harmonic series and wanted to harness it to create musical instruments. He experimented with his Monochord to try and capture the higher tones he heard when he plucked the open string. So he would place his finger on part of the string, pressing it down, and pluck the string at the length that wasn't pressed down. This changed the pitch, because shortening the length of the string increased the frequency of its vibrations. Pythagoras found that when the string was held down at certain points - point correlating to harmonics - he would hear the higher tones the string had been making on its own.
Specifically, when he held the string down at points correlating to Small Integer Ratios. A ratio is something like 1/2; one over two is one half or 0.5. But some specific ratios Pythagoras found important were these:
3/2, 5/4, 7/4
Because 3/2 is a Perfect Fifth, 5/4 is a Major Third, and 7/4 is a Harmonic Seventh. These are ratios that take the odd-numbered harmonics and reduce their distance from the Root (by dividing them by powers of two, like /2 and /4). With harmonic intervals using the right ratios, it would be possible to compress harmonics into the width of a single octave, creating a musical scale.
If you took only the ratios 4/4, 5/4, 6/4, and 7/4, this is the scale of notes you would get:
C - E - G - ~Bb (why the ~ is for another day, when I talk about Temperament)
This is a fine scale, but with only four notes it's very limiting. You can't make good melodies using only a scale with each note so acoustically distant from the others. So Pythagoras thought, "How can I make a scale with numerous notes, all of which have some harmonic relationships, which fits in the span of a single octave and would be practical for instruments to play?"
Which is a fine question! Eventually Pythagoras realized something. Apart from the Octave, the Third Harmonic/Perfect Fifth was the smallest integer ratio and therefore the most stable interval aside from the octave. If Pythagoras could devise a scale based around the interval of the Perfect Fifth, the musical possibilities would be grandiose and far-reaching.
But how to make such a scale? Hmmm...Eureka came around eventually, and he said, "I will start on one note, my Root, and find its third harmonic; then I will find the third harmonic of that third harmonic, and so on, and so forth, yea and verily, til death do us part!"
And so he did. I've already mentioned that the third harmonic is the numerical value of 3 (duhhhh), and that the Octave is the numerical value of 2. To create a scale made of P5s that fits into one Octave, the only thing to do is multiply 1.5 (3/2) by itself and multiply 2 by itself until their values intersect. As it happens,
1.5^12 = 129.746
2^7 = 128
Well there you have it! Those numbers are extremely similar, so we can make a scale based on Perfect 5ths. It turns out that the acoustic distance of seven octaves is only slightly less than the distance of 12 P5s.
...only slightly less...that should be fine...won't cause any problems that endure for centuries to come...
So if there are 12 P5s in an octave, that means there are 12 distinct notes before the scale repeats. This is known today as the Chromatic Scale, which contains these 12 pitches:
C - C# - D - D# - E - F - F# - G - G# - A - A# - B ... and repeats at C, going further up or down as you like. Here (click) is the Wiki article on the Circle of 5ths, so you can see a picture (if you've never seen one before) of the notes arranged in an actual visible circle, rather than a metaphorical acoustic one.
"So in conclusion, Pythagoras invented the Circle of Fifths that we use today in modern music."
I would say, if that was accurate. But here's the thing.
That "slight" difference between 12 P5s and 7 Octaves is significant enough that it causes a huge, huuuuge problem. That problem is called the Pythagorean Comma, 23.46 cents (I have not yet explained cents, but I will in a later entry). Here's how I normally explain it:
If you begin on A, and go up seven octaves, you will arrive on another pure A.
If you begin on A, and go up the entire circle of fifths, you will arrive on an overly sharp "A".
If you have a note that IS A, played against a note that is ALMOST A,
you get something acoustically disgusting and discordant.
Octaves must be played in tune; either note of an "octave" being too sharp or too flat creates a frequency ratio that is overly complex and thereby dissonant. Dissonance is not bad in and of itself, but if what you're attempting to create is consonant harmony (which was Pythagoras' intention with the scale) then you must tune every interval accurately. The Pythagorean Comma meant an octave would be detuned, if the entire Circle of 5ths were used on an instrument.
In essence, this means that the "Circle" of 5ths is actually a Spiral of 5ths. If you use true acoustic Perfect 5ths and continue moving up, you will absolutely never arrive on a pitch that is a pure octave of the note you started on. The 5ths may have been perfect, but the scale they created had its issues.
So the first response was, "Hey, let's just not use the entire Spiral." So the Diatonic scale was born, a seven-note scale we know as Do-Re-Mi-Fa-Sol-La-Ti-Do. In this scale, instead of using the overly sharp Do from the end of the Spiral, the second higher Do was a pure octave of the first. So that solved that problem. There was another concern, though, namely the tuning of "Mi." The Major chord as I mentioned before is Do-Mi-Sol, but the Pythagorean Major Third is way, waaay too sharp. How sharp exactly?
The Harmonic Major Third is the ratio of 5/4.
The Pythagorean Major Third is the ratio of 81/64.
Its ratio's complexity makes it grating on the ear, and more simply it's just too sharp to create a truly pleasant, consonant sound. However, it was close enough that this Diatonic tuning endured for a long time. There was a Pythagorean Chromatic Scale as well, that used all 12 of the pitches of the Spiral of 5ths. But this one had even more harmonic trouble in use due to both its complexity and the frequent frequential disagreement of many of the scale's intervals.
Well, Pythagoras eventually passed away, but his followers carried on his work and promulgated his discoveries. The Spiral of 5ths as a musical system became widespread and popular, due to its reliance on clear mathematics and the acoustic functions of the natural world--and because it sounded pretty. The question of how to deal with the Pythagorean Comma and the other oddities of the Spiral of 5ths baffled people for many centuries, and they never stopped trying to find a way to solve that riddle. There had to be some way to fix the Spiral. To close it.
Ladies and gentlemen, I'll see you next time. For next time we delve. Next time we discover.
Next time, we close the Circle.
-Dave
Stay tuned for the upcoming entry on musical Temperament!
PS: I typed 99% of this from memory, so if you notice any errors in my information please let me know and I will add your corrections to the document. :)
First edit: So I browsed this after reading,
http://en.wikipedia.org/wiki/Pythagoras#Musical_theories_and_investigations
And apparently though Pythagoras did connect musical notes to the natural harmonic series, he was not responsible for inventing the Circle of 5ths. However, there ARE musical tunings called Pythagorean and Pythagorean Chromatic, probably devised using the small integer ratios of the harmonic series.
This is described in the article on Pyth tuning, for anyone interested.
http://en.wikipedia.org/wiki/Pythagorean_tuning
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