Simply stated, the fundamental is the lowest and loudest frequency of a single string vibrating as a whole. It is the pitch by which we identify the letter name of the root, which is also known as the tone 1 scale degree. In Physics, a node is the exact point on a vibrating string where there is no vibration and therefore no sound! On the bass, the nut and the bridge are nodes.
Ratio is the relationship between the length of one whole string to the number of equal parts that it can be divided into. For example, a one-to-two ratio (1:2 ratio) means that one whole string has been divided into two equal parts. Now, if we consider the fundamental to be one string vibrating as one part, that would be a one-to-one relationship or a 1:1 ratio. This 1:1 ratio is also known as a unison interval. For this lesson, our fundamental reference frequency will be E, tone 1 at the nut and bridge of string 4.
Harmonics, also called overtones or partials, are higher frequencies produced by the vibrations of a string divided into any number of equal parts. Harmonics occur because strings not only vibrate as a whole, but they also vibrate in parts or fractions such as halves, thirds, fourths, fifths and so on. Now, whenever a fundamental is sounded, on any instrument whether it is sung, plucked, struck, blown or bowed, there is an entire harmonic series of frequencies that naturally vibrate with it at the same time. This harmonic series is a specific order of frequencies that climb like a ladder through a predictable series of intervals. And each harmonic step of the ladder is a precise multiple of the fundamental frequency.
Notice that tone 8, the 1st harmonic, is a 1:2 ratio that is produced by dividing the string into two equal parts directly over the 12th fret node. The node is the exact point that divides the string into equal parts. This forces the string to vibrate twice as fast as the tone 1 fundamental and to sound one octave higher in pitch, tone 8. Also notice that the same harmonic may be sounded on each side of the 12th fret: the nut side or the bridge side. For example: tone 15, the 3rd harmonic, may be sounded directly over both fret 5 and fret 24 because both frets are the same distance from the 12th fret! Also, notice that some of the harmonics occur directly above frets while other harmonics are in-between the frets or fractions such as 1/4. The reason for this has to do with just temperament, mean temperament and equal temperament, but that’s another lesson.
To produce a loud and clear harmonic on your bass, touch the string very lightly at the node. Do not push the string down toward the fret. Strike the string near the bridge with force. The string will then vibrate in smaller equal parts and this will produce the sound of the harmonic. New strings and perfect intonation, an adjustment at the bridge which assures that fret 12 is at the exact middle of the string, will also help. Be sure to quickly lift your finger off the string after sounding the harmonic, so that you don’t dampen the vibrating string.
Tone 12, which is tone 5 in the second octave, is the 2nd harmonic and has a 2:3 ratio that is produced by dividing the string into three equal parts directly above the fret 7 node. Fret 7 is 1/3 of the strings length. Since it is impossible for a string to vibrate in two unequal parts, the remaining 2/3’s of the string divides itself in half which produces three equal string lengths or parts!
As long as you continue to divide the string into smaller and smaller equal parts, an ever higher and higher series of harmonics will be produced. The sounds of which will be limited only by your strings, hearing and technique. Some harmonics are easy to play, others take more practice.
The important idea to take away from all this, is that the individual timbre (tam-burr) or tone quality of your bass results from the presents or absence of particular harmonics and their relative volumes. It is this balance between the fundamental and its harmonics that makes one bass sound different from another, even when the fundamental pitch they play is the same. In other words, no two basses, no matter how similar, have the same blend of harmonics or the same tone. Each bass is unique!
To learn more about the Bass, please visit: http://www.12tonemusic.com/bass/facts/
Image © C.Chris Peters 2010
Wouldn’t you like to be let in on something that other guitar players don’t know? Imagine what you would be able to do with this hidden information. Well, here are three little known, and rarely understood, secret fretboard angles that will change the way you view your guitar fretboard. Does the following look familiar to you? It should, it’s your fretboard!
Simply stated, X is the horizontal strings, Y is the vertical frets, including the nut and bridge, and Z is the diagonal octaves. This simple fretboard geometry is the key which unlocks your ability to instantly locate any letter on your fretboard. Let’s reveal the first secret angle, the horizontal strings with their letter names:
Next, let’s define the string’s length by revealing the second secret angle, the vertical nut and bridge:
The distance between two pitches, whether letters or tone numbers, is called an interval. The smallest interval is the half step, which on the guitar fretboard is one fret. Here’s a very old discovery made by Pythagoras, 6th Century BCE: when a string is divided in half, the frequency is doubled, and the octave is created. The interval of an octave is 12 half steps or 12 frets. Therefore, fret 12 divides the string in half and creates the 1st octave. Then, 12 frets higher, fret 24, the string is divided again and the 2nd octave is the result as follows:
We can now reveal the third secret angle, the diagonal octave. This simple angle enables us to locate any letter, on any string, faster than we every thought possible. Here are all the E’s within 12 frets connected by the diagonal octave angle:
Remember, although all the letters are the same, they are not all in the same octave, but that’s another lesson. So, let’s end this lesson by showing the 7 letters of music, A B C D E F G on 12 frets:
’til next time, have some fun connecting the diagonal octaves of all 7 letters because it’s a secret no more . . . I’ll be listening!
Music, whether it is seen or heard, has three fundamental parts: rhythm, pitch and dynamics. Each of these parts may be studied in great detail, but for this lesson, let’s simply define them as follows: rhythm is time, pitch is a sound that has a letter name, and dynamics is the degree of sound volume from quiet to loud. That all sounds simple enough, but what is sound?
Non-musical sound has but one name, noise. However, musical sound has many names. Here are a few: letter, tone number, harmony numeral, scale degree, timbre, pitch, octave, harmonics, vibration and frequency.
Let’s look at frequency more closely. The music we hear, in contrast to the silent music symbols we read, is the sound of air moving. This then begs the question, what is moving the air? On the guitar, it’s the strings that move the air. When you pick or pluck a guitar string it oscillates back and forth. It is these back and forth string motions, or vibrations, that are moving the air. These vibrations may be counted and then assigned a frequency number. We’ll skip over all the deep physics and acoustics data, like resonating bodies, air-pressure and amplitude, and instead, we’ll focus on frequency so that you may tune your guitar.
The speed of a string vibrating back and forth is steady, regular and predictable in time. The number of times a string oscillates in one second is called frequency. Frequency is measured as cycles per second (cps) which is also known as Hertz (Hz), named after the 19th century German physicist Heinrich Hertz. The frequency range of the human ear is approximately 20 Hz to 20,000 Hz. This may be written as 20 Hz to 20 kHz. Kilo (k) is the Greek prefix for thousand, 1 kHz = 1000 Hz. The question then becomes, how many times does a string move back and forth in one second? The answer to that question depends upon which string and fret is being played.
Any string played at any fret vibrates faster than the human eye can see and count. So, science has to count for us ~ think electronic tuner. Not only are the vibrations counted and given a frequency number, but they may also be assigned a letter, scale degree tone number, and a note location on the staff. For example, string 5 at fret zero vibrates 110 times a second (110 Hz) and is called A (A). On the guitar, Fret Zero is also known as the nut or open.
An octave is created by doubling the frequency. Therefore, string 3 fret two, which vibrates at 220 times a second (220 Hz), is one octave higher than string 5 fret zero, and is called prime A (1A). String 1 fret five vibrates 440 times a second (440 Hz), is two octaves higher than string 5 fret zero, and is called squared A (2A). Remember, faster frequencies sound higher in pitch, while slower frequencies sound lower in pitch ~ think treble and bass.
It helps if we consider pitch as being Absolute, Perfect or Relative.
Absolute Pitch is an external reference to a definite pitch of a specific frequency upon which everyone agrees. Absolute Pitch may be assigned a letter, scale degree tone number, harmony numeral, or staff note. Here are a few good sources of Absolute Pitch: a tuning fork, an electronic tuner, and a “tune-up pitch” from a play-along CD. Remember, audio tapes are not a good source of Absolute Pitch due to the varying playback speeds of different tape players. Here’s something interesting. The International Agreement, which made A = 440 Hz the Absolute Reference Pitch for the entire world, wasn’t agreed upon until 1939!
Perfect Pitch is the ability of a person to identify a given musical pitch without the benefit of an external Absolute Pitch reference. Those who have Perfect Pitch exhibit some or all of the following capabilities: identify individual pitches by name when played on various instruments; name the key letter of a given piece of tonal music just by listening without reference to an external Absolute Pitch; identify and name all the tones of a given chord or tone cluster; accurately sing any given pitch without an external Absolute Pitch reference; and name the pitches of common everyday sounds such as the honk of a car horn, the ring of a telephone, the chime of a doorbell, or the hum of a refrigerator. Many believe that you must be born with Perfect Pitch because it seems as though it can’t be learned. In contrast to Perfect Pitch, Relative Pitch can be learned.
Relative Pitch, as the name implies, relates to Absolute Pitch and is what is meant when someone says they “play by ear.” In other words, Relative Pitch is the ability to hear an Absolute Pitch, store it in auditory memory, and then match that pitch or relate it to another pitch. In music school they call this ear training. And while it’s true that Relative Pitch can be learned, it sure takes a lot of practice! However, the benefit of Relative Pitch is that it makes tuning your guitar so much easier and faster. The Relative Pitch method of tuning your guitar by matching or duplicating pitches is known as Unison Tuning.
Uni means one and sonus means sound, so, unison means one sound. Said a different way, in Latin uni means one and in Greek iso also means one, therefore, uni + iso(n) = unison. In other words, unison means more than one as one. No matter how we define it, unison is really a coincidence, that is, multiple events occurring at the same time. The prefix “co” means together (two or more as one), and an “incident” is an event. With this said, unison may be further defined to mean: two tones of the same pitch sounding at the same time. One of the greatest things about the guitar is that by playing the same pitch, on two or more different strings of the same guitar at the same time, unison is possible. On the piano, woodwinds, brass and voice, unison is impossible!
Okay, now that we have an elementary understanding of the science of sound, let’s apply it to the tuning of the guitar. We’ll begin by giving each of the six strings of the guitar a letter name and frequency number: string 6 E = 82.41 Hz, string 5 A = 110 Hz, string 4 D = 146.83 Hz, string 3 G = 196 Hz, string 2 B = 246.94 Hz and string 1 E = 329.62 Hz. What follows is the method of Unison Tuning that will enable you to tune your guitar by ear!
From an Absolute Pitch source, determine the reference pitch. For example, if you want to tune string 6, you’ll need to hear the Absolute Pitch of E = 82.41 Hz. Here’s something important. Be sure to use a tuning fork, or some other source of Absolute Pitch and not a piano, as the piano may not be in tune! Think of it this way, why would you tune a tunable instrument to another instrument that needs to be tuned?
Next, assuming that string 6 is in tune, play A = 110 Hz on string 6 fret five. This reference pitch should match string 5 fret zero. If string 5 fret zero, sounds the same as string 6 fret five, they’re in unison and your guitar is in tune. If they don’t match and sound the same, you’ll need to adjust string 5 higher or lower in pitch until they do. Be patient, this is Unison Tuning ~ and the more you practice it, the better you’ll get!
After string 5 is in tune, play D = 146.83 Hz on string 5 fret five. This should sound the same as string 4 fret zero. Again, if it doesn’t, you’ll need to adjust string 4 higher or lower to match the string 5 reference pitch.
After string 4 is in tune, play G = 196 Hz on string 4 fret five. This should sound the same as string 3 fret zero.
Now, once string 3 is in tune, in order to play B = 246.94 Hz on string 3 you must play fret four. This should sound the same as string 2 fret zero. Remember, the B = 246.94 Hz reference pitch on string 3 is located on fret four. This is a different fret than any of the other strings.
Finally, after string 2 is in tune, play E = 329.62 Hz on string 2 fret five. This should match and be in unison with string 1 fret zero.
Congratulations! Your guitar sounds so much better now that it’s in tune.
’til next time, keep playing and have fun… I’ll be listening!