The String Thing
String,
String a Song.
Make it Simple,
To Last Your Whole Life Long.
It seems simple enough to use strings of proportional lengths to create pleasant combinations of tones. And I guess it is.
In order to accomplish this and define these proportions, Pythagoras of Samos (his full proper name) and his followers/students used a device called a monochord – a resonating box with one musical string whose length can be divided by a movable bridge.. similar to the fixed bridge on a violin. By utilizing two or more of these boxes, they were able to calculate the proportions of each string, that, when sounded together, produced a pleasing sound, or consonant chord.
But Pythagoras, being of sound mind, realized that in order for this phenomenon to be of use in making music, a much more detailed, expanded and refined scale (compared to his initial calculations as described in my previous post) was needed to define the frequencies of additional tones – both within the octave as well as those in adjacent octaves.
He began to extrapolate.
The task at hand was to divide the scale (between the octave frequencies) so as to produce a series of tones which, when played together, would produce the greatest number of consonant chords…. creating a subset of frequencies of optimum musical value.
I cannot stress this enough.. because THIS is at the heart of harnessing science and mathematics in order to produce an art form.. the human factor.. the scale is the basic tool in creating music.
It’s really what these discussions are all about.
It was commonly agreed by P & Co., that the note which would come to be known as a “Fifth”, when sounded with the original tone, seemed to be the most pleasing chord to the ear. What they found was when the length of the original string was decreased by 50%, a frequency equal to 1.5 times of the original tone was produced. Again, seems pretty straight forward: the frequency of the first tone, when multiplied by 1.5, produces a frequency that is the exact average of the first and the octave notes, right down the middle…AND…sounds awesome.
Sweet.
This frequency would become the Fifth Note in the Pythagorean Scale.. all important. So important in fact, that this Interval, a representation of the musical proportion between two notes, (in this case between the original tone/first note and this middle tone/fifth note of the scale) would become the foundation for every other note in all octaves in the Pythagorean Scale.
Play it Again, Samos!
Death of a Scalesman
Great Thinkers Die Hard.
Socrates was forced to drink hemlock.
Galileo was found guilty of heresy and died under house arrest.
Dana Plato died of an overdose in her car.
And Pythagoras, often referred to as the world’s first pure mathematician, was no exception.
The reports of his death are varied. He is said to have been killed by an angry mob, to have been caught up in a war between the Agrigentum and the Syracusans and killed by the Syracusans, or to have been burned out of his school in Crotona and exiled to Metapontum where he starved himself to death. At least two of the stories include a scene where Pythagoras refused to trample a crop of bean plants in order to escape, and because of this, he was caught.
Oh Boy.
Pythagoras studied odd and even numbers, prime and perfect numbers. He contributed to our understanding of angles, triangles, areas, proportion, polygons, and polyhedra. My hero.
Pythagoras also related music to mathematics. He had long played the seven string lyre, and learned how harmonious the vibrating strings sounded when the lengths of the strings were proportional to whole numbers, such as 2:1, 3:2, 4:3.
He was the first to create a musical scale: that subset of musically relevant frequencies, where the last frequency is exactly double that of the first. Every scale from every culture on earth uses this rule of a doubled frequency that we call an octave. It is based on the human perception that doubling or halving a frequency results in two tones that sound remarkably similar to each other – and that our modern western scale contains eight notes.
Everything in between however is where the fun begins.
So Pythagoras began to play.. and not just on his seven string lyre. He used strings of different lengths, in specific proportions, to create a six note scale.
I don’t believe that Pythagoras had any method of calculating the actual number of vibrations per second.. but I could be wrong. In any case, it really didn’t matter.. because as we will see, it is the proportions of the rate of frequencies to each other that accounts for the basis of a scale. That is to say, if he started with a string vibrating at 440 Hz, he could have doubled the frequency, by halving the length of the string, and wind up with the octave of 880 Hz. Had he started with a string vibrating at 500 Hz, he could have achieved an octave by the same method..producing a tone whose frequency was 1000 Hz. This proportion is 2:1 or conversely, 1:2.
For our example, let’s assume the first string had a frequency of 440 Hz, which corresponds to our present day label of the note known as Concert A.
By cutting another string 4/5 the length of the A, he came up with C, the next note of his Greek scale. To find the frequency of this new C Note, (I could use a few of those) you simply multiply the reciprocal of the string length by the frequency of the first string (5/4*440) to get 550 Hz as the frequency.
Applying the same formula, you can find frequencies for D with a measure of 3/4 of the A string, E – 2/3 of the A string, F – 3/5 of the A string and finally another A measuring 1/2 of the original.
A = 440 Hz (440 Hz*1/1)
C = 550 Hz (440 Hz *5/4)
D = 586.66 Hz (440 Hz*4/3)
E = 660 Hz (440 Hz*3/2)
F = 733.33 Hz (440 Hz*5/3)
A Octave = 800 Hz (440 Hz*2/1)
I guess with a name like Pythagoras, he really had to “Prove” himself.
Life’s a Pitch
Q: What’s the difference between an “A” and an “A#”?
A: Except for the A#, they are both derived from an object vibrating at a frequency of 440 Cycles per Second.
Q: What’s the difference between a frequency and a pitch?
A: Frequency refers to the rate of vibrations per second of an object…expressed as Hertz (Hz.). Pitch refers to the mental representation of the sound in relation to a standard of musically named frequencies.
Let me put it another way..
Q: What’s the difference between a tone and a note (aside from the obvious palindrome)?
A: Tone refers to the sound an object emits at a given frequency. A Note is the musical representation of a tone.. again in relation to a cultural, mathematical and arbitrary standard subset of frequencies, called a Scale.
So frequency and pitch, tones and notes are all very closely related.. and at times used interchangeably.. but in fact they are not the same. What we have here is a human effort to convert the physical world into the psychological… to utilize physics to create structured sound, so that we may define and manipulate science in the furtherance of art.
Q: How may pitches are there?
A: There’s the Screwball..oh, wait.. no comment on that one. Theoretically, there are an infinite number of pitches - because they are derived from the continuum of vibrating objects. For every two frequencies I can pair up, there is always a mathematical value between them. But this is hardly a practical model with which to work.
Indeed, humans are limited in their ability to perceive and distinguish such minute changes in pitch… to say nothing of the fact that such degrees of change are not, by any stretch of the imagination, musically useful.
Well, time’s up for today.. I’m afraid we have to stop for now. Pencils down. No talking. We will pick-up with Scales when we reconvene.
A Small Note: There is so much to write about on this subject – but I want to take it a Half-Step at a time.. so please be patient.
And above all, “Stay Tuned”.
A Pretty Girl Is Like A Melody….
That Haunts You Night and Day.
Well, if you really can’t get the tune out of your head, and it becomes an earworm, it’s really more of a “Malady!”.
So what makes a girl “Pretty”? – Is it just facial features and curves? Or, is it personality, intelligence, warmth or sense of humor? Or all of the above and more?
I never saw the fascination of “Beauties” like Marilyn Monroe, Elizabeth Taylor or Joan Rivers. I was never able to hold a decent conversation with any of them…But give me Susan Boyle any day.
I think most would agree, you’ve got to consider the whole package in a friend or a mate.
So, what makes a melody “Pretty”? – Is it just an appealing succession of musical pitches over time? Or is it a feeling of personality, warmth or humor evoked by the performance of a piece of music by a person that you are already attracted to by virtue of their human traits? Or all of the above and more?
I would no sooner want to hear Frank Sinatra singing The Lemon Song than I would want to hear Led Zeppelin playing Luck Be a Lady. I have absolutely no desire to hear The Mills Brothers singing Sunshine of Your Love or Cream playing Cab Driver. It just doesn’t make sense. I like all those melodies.. but those packages suck.
Maybe that’s why popular songs covered by other artists often don’t have the same appeal as the original performance. When I was young, my sister gave me a record of a compilation of Beatles songs. I was very excited.. it was one of the first records I ever owned. When I took it home and played it, I almost cried. When I looked closer at the album “cover”, it was indeed a “cover”… some orchestral group called Silky Strings..playing, among other tunes, You’ve Got to Hide Your Love Away. Well, I hid that album away and never laid eyes on it again. Feh. Beautiful melodies, but the package was incomprehensible to a sensitive young boy. I started therapy the very next day.
So next time you listen to Pinball Wizard, think of what it would sound like performed by Barry White.
In the word of John Lennon: Help!
I’d Like To Have An Argument Please
Man #!: An argument isn’t the automatic negating of a statement by saying “No it isn’t!”
Man#2: Yes it is!
Man#1: No it isn’t!
This wonderful piece of logic comes to us courtesy of Monty Python. To my mind, it’s one of the wittiest perceptions of human behavior ever written. A few more of my Favorites:
Sign on a Door: Today’s Meeting of the Procrastinators Club Has Been Postponed!
Message on Answering Machine: “You’ve Reached The Department of Redundancy Department. Please Leave a Message. Please Leave a Message.”
My Motto: Just say “NO” to negativity.
Parent To Child: “I’ve told you a MILLION TIMES, DON’T EXAGGERATE!
So what does this have to do with music? Hmmm.. I’ll get back to you on that.
OK, seriously (or not), it’s the humor of circular logic and contradiction expressed through language… words. I love puns.. to the distress of many of my (former) friends. Puns express at least two thoughts in a phrase.. you can read my last posting for a few really silly ones.
Music certainly does communicate ideas and emotions to us ..though so often without words.. Reference the often quoted idea that music is “The Universal Language”. The previous post exemplified this through the imitation of natural sounds to convey the feeling of riding a horse and experiencing the flight of bees. Forgive me for stating the very obvious, but music can and IS interpreted in different ways by different people.
One of my favorite pieces of music is Beethoven’s Fifth Symphony.. arguably one of the most famous pieces of classical music ever penned. It’s played so much, because, in my humble opinion, it’s a masterpiece.. of musical style, construction, melody, instrumentation.. but most of all, emotion. There are many myths surrounding the origins of its opening theme:
Dah Dah Dah Dahhhhhhhhhhhh! Dah Dah Dah Dahhhhhhhhhhhhhhhhhh!
Four notes. Simply amazing. One of my clearest memories of the speculation of how this phrase came into being, was as a young child, watching a Disney (I believe) TV special on the Life and Good Times of Ludwig (his name as a Rapper.. here comes the pun.. stay with me now). Ludwig was behind in his rent, and the landlord, pretty damn angry, came up to his room, and “Rapped”, rather loudly on he door: Dah Dah Dah Dah..pause..Dah Dah Dah Dah. Ludwig, although on the verge of going deaf, as all rock musicians eventually do, went right to the piano and banged out this melody.. repeating the landlord’s anger and expressing his own pain. There’s that magic couplet again…The Redundancy Department in action!
True or not, (and I think not) I get chills when I hear a fine performance. I particularly love the recording by Leonard Bernstein and The Vienna Philharmonic… in part because the tempo is somewhat slower than most other renditions. For me, it leaves room for a tremendous expression of power. (Thanks for the CDs Sarah!)
So this piece of music is filled with anger and pain.. and YET, there is a most wonderful interpretation of the first movement which is a streak of genius humor. Thanks Jess for bringing it to my attention. I am not sure if the wordless comedy skit was choreographed to the music, or some young film editor saw the possibilities (I like to think the latter) but either way, it’s “Laughter in The Pain” (a rather obscure reference to a 1970′s Neil Sedaka Hit.. look it up!)
Thanks to Otto for bringing forth the following quote:
Woody Allen: “I can’t listen to Wagner. I get the urge to invade Poland.”
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Why The Long Face?
I know, I know.. It’s been a week and I promised a new post by “Tomorrow”.. But alas, tomorrow never comes. Just needed some August Summer Vacation.
Anyway, I know you’ve all been chomping at the bit to hear William Tell us about his horse. He doesn’t talk much… but, for all you “Neigh-Sayers”, I think this musical Tail will illustrate just how much we like to imitate natural sound in our music. I won’t Saddle you with a long Winded philosophical theory, so just feel free to Reign me in. OK.. here’s the Mane event:
Let’s Get Ready to Bumble!
The latest buzz on Rimsky Korsikov: He missed his connecting Flight and had to spend the night in Bee-oria. That’s OK, though, he had Roxanne on his iPod.. It was in his playlist for… (do I really have to say it????) …. Sting …. in Concert …. Hive at The Garden.
That’s all for now My Little Honeys.
Aren’t you glad?
Next Post
Tomorrow:
“Wiliam, Tell Me About Your Horse”
Today I’m hoofing it out to Jersey – “Happy Birthday Dave!”
Hey, I just realized, Happy Birthday to You is in 3/4 Timing.. Try counting threes and fours!
Audio Daily Double
OK. So yesterday I spoke of music based on a three beat cadence is most often actually constructed on couplet phrases, thereby giving it a basis in our theory of “Twos”. I made a chart to illustrate this idea – but today I will demonstrate the concept with my new found Audio Blog Powers. Below is a link to “The Man On The Flying Trapeze”. I have recorded myself counting the beats in several ways:
1. Three Beats per “Measure” (ie. the sub-division of notes defining the mathematical basis of the rhythm).
2. Six Beats, containing one couplet of three beat measures;
3. Fours Beats, containing two couplets, or, four measures of three.
The math is easy.
And, by the way, I am not writing to teach the rudiments of music to anyone, or to insult anyone’s intelligence by stating the obvious arithmetic – I simply aim to point out that though we may hear music based on a fundamental of odd beats per measure, the larger view appears to be founded upon our sense of even balance.
http://www.mach3audio.com/Trapeze
The next link is to Herman’s Hermits singing “Silhouettes”. This piece shows an even closer relationship between three, six and four beat cadences. The “Time Signature” calls for six beats per measure – twice as many as Trapeze. So right away, each measure contains an even number of beats, but seems to have two phrases of three beats contained within. Again, I have counted time in a few different ways. The guitars are playing six notes in the opening measures, the lyrics seem to be phrased more in groups of three and the piano (heard best at the end of the instrumental in the middle section) is playing in fours.
http://www.mach3audio.com/Silhouettes
The final link points to the theme to “Mission: Impossible”. The rhythm is based on FIVE notes per measure. Not very many pieces of music are written in this rhythm, because, at first, it doesn’t seem to fit our instinctual need for balance…and it does take some getting used to. BUT, even this rhythm, in the end, comes out evenly. Here, the melody is based on two measures of five – or a ten beat phrase. Not quite as neat as a four beat rhythm..there are indeed 2 1/2 four beats in a ten beat phrase. This is strictly based on two. As I said , it takes some getting used to, but as you can hear by my counting, each musical phrase is absolutely two measures of five beats.
One other interesting mathematical fact is also seen in 5/4 time: Each measure can be divided as as two beats plus three beats per measure.. five in all. So, for two measures, that would be two times two beats (4 – Even) plus two time three beats (6 – Even).. of course adding up to ten. So, no matter how you slice it.. it is still EVEN.
http://www.mach3audio.com/Mission_Impossible
There are other rhythms, especially in other cultures, that employ rhythms of varying numbers..perhaps most notably the 7/8 rhythm of Greek music. In general, music that does not fit comfortably into a two and four beat cycle is generally much more difficult to comprehend.. at least in Western and European cultures. Such rhythms can feel “syncopated” and hard to follow. The Fox Trot, The Waltz, The Watusi, The Twist, The Jitterbug, The Cha-Cha, The Merengue…. on and on.. are all based on our need to feel at ease with our two feet.
May I have this dance?
The Odd Couplet – When Three Is An Even Number
Because I have employed a visual aid, please click the link to view today’s post. It’s worth the effort. I promise.
You Say You Want A Resolution?
Stop me if you’ve heard this one: Mr. Jones lives in apartment 9A. Every night at 11 PM, he gets ready for bed. He sits in his chair, takes off his left shoe and drops it on the floor. He repeats this for the right shoe. Mr. Smith lives in apartment 8A. Every night he hears Mr. Jones drop his shoes on the floor. One day, unbeknownst to Mr. Smith, Mr. Jones slips at work and breaks his ankle. He is taken to the doctor who fits him with a cast. That night, Mr. Jones takes off his left shoe and drops it on the floor. Mr. Smith did not sleep a wink that night.
Humans want closure. We are thrown off-kilter when cycles, patterns and expected actions are not completed.
Try this:
1. “Four score and seven years…”
2. “There is nothing to fear, but fear….”
Ouch. The thought is incomplete.. it makes no sense.
Say these two sentences.. Don’t worry, no one is listening..
- “It’s time for dinner?”
- “It’s time for dinner!”
These two cadences have different melodies. Melodies? Melodies!
Let’s “define” melodies – Very simply: A series of tones heard over time created to establish a meaningful phrase. We have established that tones are structured, directed and easily remembered. A spoken word is a tone, as is a specific set of sound waves produced by an instrument.. the human voice being a beautiful one. A single tone has a “pitch”, defined in physics as the number of vibrations per second (emanating from the source, through the air and transferred to our inner ear). The more vibrations per second, the “higher” the tone. Enough of that.
Say the two phrases once again. I am sure you notice that in the question, the word “dinner” ends up in a higher tone than in the statement. Not only is it a different pitch, but that word just doesn’t seem to resolve for us! Why? Why Indeed! BECAUSE THE RESOLUTION IS IN THE ANSWER! We are waiting for the other shoe to drop.
The same with music. Sing these songs:
- “Twinkle, Twinkle little star, How I wonder what you.”
- “O’er The Land of the Free, And the Home of the.”
Painfully unsettling. In music, the “home” note, or pitch, is called the tonic. Once you are home, THEN you can take off your shoes.
In the words of John Lennon: “You know it’s gonna be… all.”
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Tomrrow:
The Odd Couplet – When Three is an Even Number Too (Two).
