Basic theory stuff...a question or three

[NewGuyonGuitar]

Trying to understand some basic musical concepts and theory. The source of my understanding is "This is your brain on music" by Daniel Levitin

 

He says "Here is a fundamental quality of music. Note names repeat because of a perceptual phenomenon that corresponds to the doubling and halving of frequencies. When we double or halve a frequency, we end up with a note that sounds remarkably similar to the one we started out with. This relationship, a frequency ratio of 2:1 or 1:2 is called an octave."

 

Cool, I wasn't aware of that fact.

 

He then talks about intervals. "The octave in Western music is subdivided into twelve (logarithmically) equally spaced tones. The intervallic distance between A and B is called a whole step." and "The smallest division in our Western scale system cuts a whole step perceptually in half: the half step or semitone, which is one twelfth of an octave."

 

He then lists a table that has intervals. Distance in semitones of 0 has the interval name of unison, distance in semitones of 1 has an interval name of minor second, etc.

 

Question: Is it useful to memorize this table?

 

He then talks about why there are only 7 letters used to describe the 12 named notes of an octave and said it adds complication to the system, but doesn't have a reason for why we came up with using sharps and flats.

 

Question: Anyone know the history of using the 7 letters and using the sharps and flats?

 

He then points out that "The frequency of each note in our system is approximately 6 percent more than the one before it. and "...when we increase each step by 6 percent twelve times, we end up having doubled our original frequency" Thereby moving up an octave.

 

He also says "In Western music we rarely use all the notes of the chromatic scale in composition; instead, we use a subset of seven (or less often, five) of those twelve tones.

 

Question: I don't understand this. Why don't we compose using the whole range of the chromatic scale? Doesn't it give us more options? I know this is an ignorant question....but I just don't get it.

 

Finally, he talks about different patterns of whole steps and half steps. He then talks about the C major scale and the A minor scale which both use only the white notes on a piano....and that "If a musician is playing white keys, how do I know if he is playing the A minor scale or the C major scale? The answer is that --entirely without our conscious awareness--our brains are keeping track of how many times particular notes are sounded, where they appear in terms of strong versus weak beats, and how long they last"

 

Wow. I can't say I was aware of this at all. I currently have no concept of this. I can't imagine myself knowing the difference between two scales using the same notes.

 

Question: For those musicians out there with lots of experience, you can tell what scale a peace is being played in by just hearing it?

[jeremy_green]

 

Finally, he talks about different patterns of whole steps and half steps. He then talks about the C major scale and the A minor scale which both use only the white notes on a piano....and that "If a musician is playing white keys, how do I know if he is playing the A minor scale or the C major scale? The answer is that --entirely without our conscious awareness--our brains are keeping track of how many times particular notes are sounded, where they appear in terms of strong versus weak beats, and how long they last" Wow. I can't say I was aware of this at all. I currently have no concept of this. I can't imagine myself knowing the difference between two scales using the same notes. Question: For those musicians out there with lots of experience, you can tell what scale a peace is being played in by just hearing it?

 

 

Yes I can usually tell... although I often need a guitar to check my work. I never used to understand the difference between C Major and A minor (like you). If it's the same notes, hell, what's the difference right? Well there is a very big difference. One Centers around the note C as it's central chord. The other centers around A - BIG Difference. If you know the main chord LIKELY the key is that. Genres also help, if it is Blues based rock it is almost always minor. Country is often Major etc. If it sounds kind of happy it is likely major, sadish or somberish likely minor. So certain assumptions are made to assist your guess.

[JonR]

He then talks about why there are only 7 letters used to describe the 12 named notes of an octave and said it adds complication to the system, but doesn't have a reason for why we came up with using sharps and flats. Question: Anyone know the history of using the 7 letters and using the sharps and flats?

Good question! Which translates as "I don't know the answer"
But you can get some way towards it by looking at the fretboard again.
12th is half-way. That's the 2:1 octave.
Now measure 1/3 of the distance from nut to bridge. You'll find yourself at 7th fret. That's the next most important division after the octave, and a perfect 5th is the next most consonant interval after the octave (and unison). It has a 3:2 ratio with the open string.
The next important ratio (the next simplest) is 4:3. This means 1/4 of the distance from nut to bridge. That's 5th fret, giving a perfect 4th above the open string.
Those marker points - 0, 5, 7, 12 - are the critical string divisions. Ie, it's not so much about dividing the octave equally, as dividing a string. The octave itself is actually the first division.
In Ancient Greek theory (based supposedly on Pythagoras's experiments with strings and weights), the wide intervals between 0-5 and 7-12 were the basis of their modal system. IOW, they pinpointed the positions of what we would call 5th and 7th and 12th fret, giving them 4 notes. But they considered the two halves of the scale separately. Either of those spaces (0-5 or 7-12) were taken as a unit, and they placed 2 other notes in the gap - but judging by ear, not by math any more. (They had no frets, of course, and didn't consider 12 equal octave divisions.) These small 4-note groupings were called "tetrachord" - literally "4-string", after the small lyres they used tuned to those notes.
The placement of those 2 internal notes in each tetrachord was not fixed exactly (according to our fret positions). Most of the choices were combinations of (approximate) whole and half-steps, but they also used quarter-tones.
Their modal system was derived by combining two tetrachords together to make a full octave. The two tetrachords could have different internal note spacings, and could be "conjunct" (joined together) or "disjunct" (separated by one note). Eg, a conjunct scale might be A-D plus D-G. A disjunct one would be A-D plus E-A. Either one ends up as an octave, of course, with 7 different notes.

Why not more notes? VERY good question... (yep same translation as above;)). Guesswork and common sense suggests that the space between what we call (on the A string) D and E - which is a 9:8 ratio (between 4:3 and 3:2) represented a useful unit of division between neighbouring notes. As your fretboard will confirm, we can insert 2 of those into each 5-fret tetrachord, with a half-unit left over in each. So it's kind of neat.

In addition - and perhaps more pertinently - if we continue our mathematical calculations, dividing up the string in further ratios, we progress as follows (using the A string as an example):
A x 2:1 = A (octave)

A x 3:2 = E (5th)
A x 4:3 = D (4th)

E x 3:2 (see what we're doing here?) = B (IOW, fret 14 is 2/3 of the distance between 7th fret and bridge - of course we double the string length to bring it back down into our 0-12 octave, giving us fret 2)
D x 4:3 = G (5 frets up from D)

B x 3:2 = F# (7 frets up from B)
G x 3:2 = C (5 frets up from G)

So after a few simple calculations - not moving too far from our starting place and keeping the numbers as simple as possible (factors of 2 and 3 only, as Pythagoras said) - we get a 7-note scale: what we call "A dorian mode". It sounds quite nice, with an appealing mix of whole and half-steps. Why would you go further? (It's significant that Indian raga - an ancient tradition not connected with Western music - also uses 7-note scales, many of them aligning with our modes. Of course, raga may also have derived, via a different route, from ancient Greece...)

Many centuries later, when the Catholic church was standardising its musical system (around 600 AD), they took a version of the Greek modal system. For various reasons, it differed from the original Greek system. (Eg, the Greeks thought of their scales as descending, not ascending, which led to confusion over which mode was which. The mode names we use now - since 600 AD - are different from the Greek modes of the same name.)

The medieval modal system - sometimes known as plainchant, or Gregorian chant, after Pope Gregory who started it - only used 4 modes: Dorian, Phrygian, Lydian, Mixolydian (which align exactly with the C major scale modes of the same names that we know now). But they allowed versions of each mode which ranged down 4 notes below their keynote; these they called the "plagal" modes (as opposed to the main four "authentic" ones), and gave each a "Hypo" prefix (which only means "low", or "below").
This meant that the lowest note in the modal system was 4 notes below the root of Dorian mode - IOW, the note we call "A". That is, "Hypodorian mode" ran A B C D E F G A - like our natural minor scale, but D was the keynote, or "final".

This 8-mode system lasted - astonishingly - for nearly 1000 years, although it did (naturally) undergo steady evolution over that time. The evolution was slow, because - of course - it was church music, in which innovation would typically have been regarded as outrageous or even evil. God's music ought to be perfect - so why would anyone want to change anything?

An important consideration to remember is SOUND. We often seem to forget that music is sound, and is not wholly explainable in words or numbers. (They didn't even have any exact notation before 1000 AD - everything was passed on aurally.) Eg, one of the first innovations that the modal system permitted (even demanded) was that, now and then, one or two scale notes had to be ALTERED, in order for the crude harmonies of the time to work properly. The earliest plainchant had no harmony - it was all unison or octaves. Then they allowed 2-part harmony, but only in 4ths and 5ths - the next most pure-sounding intervals after the octave. Remember this music was sung by choirs in enormous church naves - think of the long reverberation times, and you can see why very simple harmonies were preferred: the voices had to blend as much as possible, to sound "pure".
However, the 7-note modes contain one nasty interval: the tritone between F and B (or B and F). This was known as "diabolus in musica" - not literally satanic, but certainly to be avoided at all costs. So if the separate melodic lines threatened to land on F and B simultaneously, what would happen is that either they'd raise the F to F#, or lower the B to Bb (depending on where the melody was going next). That way, the tritone was softened into a perfect 4th or 5th. And civilisation was safe to breathe again.
And that was how (and why) accidentals started to appear. For some time, they only needed Bb and F#, but gradually - as harmony inevitably got more intricate, with things like major 3rds and 6ths being permitted, and the idea of transposable scales starting to gain ground - more accidentals started to creep in. But it was still many centuries before we ended up with a full set.

(Also remember that before equal temperament - our modern system of exactly equal half-steps - F# was a different note from Gb; Bb was different from A#; etc.... This was because tuning was by ear (or by simple ratio), to pure intervals. Equal temperament is an artificial mathematical compromise, in which every note is a little out of tune.)

[JonR]

[CONT]
And it wasn't until 1549 that a music theory treatise finally admitted the existence of Ionian and Aeolian modes (and their "hypo" variants) as suitable for serious music. (Locrian was also included then, although it was recognised as theoretical only; you couldn't really make music with Locrian.) Ionian had - apparently - been in use for some time in folk music (troubadours etc), but of course that association meant that the church regarded it as "licentious" and "vulgar". Still, fashion moves on, even in the Catholic church... And of course, Ionian ended up taking over the whole show, as it proved ideal for harmonising in the new tertian style (stacked 3rds), and the old modal system fell into disuse, to be replaced by KEYS.
The key system - aka the major-minor system, or functional harmony, or CPP ("common practice period") lasted around 300 years before it basically wore itself out, in the music of Wagner. Popular music has kept it alive (we love the singalong choruses that major and minor keys give us), and it has made a kind of grudging return in "serious" music. Composers such as Schoenberg attempted various systems of using all 12 notes, avoiding the idea of key, but - while that clearing of the decks around 1900 and after had to happen - it never really caught on with the wider public, or even much with serious music fans.

Question: For those musicians out there with lots of experience, you can tell what scale a peace is being played in by just hearing it?

Well, I can tell (and so can you, probably) whether a piece is in a major or minor key.
This comes down to recognition of one single interval: the one between the keynote and the 3rd of the scale. Major 3rd = major key, minor 3rd = minor key.
Other intervals tend to support that; eg, a major key is likely to have a major 6th and 7th too, and a minor key will have a minor 6th (tho not always a minor 7th). But the 3rd rules.
Eg, the melodic minor scale has all the same intervals as the major scale - except for the (minor) 3rd. And indeed, we can have problems recognising it as minor, which is why classical music tended to use melodic minor ascending only (to make a strong resolution to the upper tonic), reverting to natural minor (with its b6 and b7) on the way down.

Of course, you have to be able to recognise which note is keynote! This is done partly by how Levitin describes (frequency of use of keynote and similar factors), but also - at least with the major key - by the natural frequency relationships between the notes, and lastly by cultural familiarity with certain conventions.

Eg, if one were to play totally randomly on the white notes of the piano, it's likely that C would be the one most listeners would favour as the strongest final note - the one that seems to rule the others. We can (using the strategies Levitin describes) make almost any note (of those 7) sound like the keynote, but it's much easier to do it with C than the others. That's because the major key has been our standard default tonality for a few hundred years. (The minor key is a secondary sidekick, the moon to the major key's sun, if you like; we even alter the minor key to behave more like major, by raising its 7th and/or 6th to make harmonic and melodic minor.)
The next easiest notes - out of the natural 7 (ABCDEFG) to make sound like a keynote are the common modes (common in folk and rock):
A (aeolian)
D (dorian)
G (mixolydian)
Next comes phrygian (keynote E), perhaps equal with Lydian (keynote F).
Lastly Locrian (on B), which is pretty impossible to make sound like a key.

[NewGuyonGuitar]

Holy moly. Brain.....can't....assimilate....all.....that........information..... lol

[NewGuyonGuitar]

...and a perfect 5th is the next most consonant interval after the octave

 

 

Consonant interval? That means a sound that's pleasing to us?

[Jed]

 

Consonant interval? That means a sound that's pleasing to us?

 

 

Yes.

Consonant = pleasing

Dissonant = not pleasing

 

Note that everything is relative so what may be described as consonant to one person may be dissonant to another. At the extreme's there is little confusion.

 

cheers and welcome . . . don't worry it gets easier as you get used to the language.

 

Jed

[gennation]

Also, dissonance in the correct context can be pleasing

[JonR]

Consonant interval? That means a sound that's pleasing to us?

More or less, yes.
Dissonant is not so much "unpleasant" as "tense", or "clashing".
As Jed says, there are degrees (a kind of spectrum between extremes), and differing tastes.
So we can point to simple physical factors at the extremes - "consonant" sounds come from notes which share overtones - but in between perceptions vary. We can feel "tension" as either uncomfortable or exciting, depending on context.

Also, what seems to happen is that as we get used to dissonance, its effects diminish. In medieval times, a tritone sounded simply terrible - they couldn't bear it, weren't prepared to countenance its use in any form. (I doubt anybody really thought it would summon the devil, as some metal fans would lke to believe.) But in modern harmony- and even as far back as the early classical period - the tritone is an essential part of the key system. It's still "dissonant", but we hear it as a pleasant (mild) tension, creating an expectation. In conventional key harmony, dissonance is used to create anticipation - to signal the way a chord sequence is going; and then sometimes to tease us by delaying the expected resolution.
However, it becomes less pleasant if those expectations are not fulfilled - if the resolution it promises doesn't appear, or if what follows seems unrelated. On its own, the tritone can sound quite stark and confrontational (think of the intro to Jimi Hendrix's "Purple Haze"). But in the middle of a dominant 7th chord, going to a tonic chord - no problem! It's as bland and safe as can be:

-2-----3---
-1-----0----
------------
-0-----------
-----------
-------3-----

The tritone is on the top 2 strings. The D bass makes it part of a D7 chord. The tritone resolves by each note moving a half-step: the F# up to G and the C down to B - making part of a G chord. The G is like the "answer" to the "question" posed by the D7.
If the question isn't answered, we'd experience the D7 (on its own) as less satisfying, perhaps mysterious - "hanging" or "waiting". But we'd hardly be horrified, or look nervously around for the Devil to appear!
In addition, unresolved tritones (as part of dom7 chords) are a normal part of blues, where we experience them as "colour", as part of the normal funkiness of the music. No resolution required.

That's what I mean by context - and also personal taste, in that maybe some people can't stand the blues, because of all that unresolved tension!

[jonfinn]

I tend to think of consonance/dissonance in terms of "stable" (wants to rest in it's current state) and "unstable" (wants to move), rather than pleasing/displeasing. Jed and Mike are both right. There are are different opinions about what constitutes dissonance. Also, some dissonances can be very pleasing in their musical effect (i.e. tension/release).

 

I didn't answer the other questions because I'd just be repeating what others have already said.

[Jed]

I tend to think of consonance/dissonance in terms of "stable" (wants to rest in it's current state) and "unstable" (wants to move), rather than pleasing/displeasing. Jed and Mike are both right. There are are different opinions about what constitutes dissonance. Also, some dissonances can be very pleasing in their musical effect (i.e. tension/release).

 

 

Agreed - "stable" and less stable leading to "unstable" is better terminology.

[JonR]

Holy moly. Brain.....can't....assimilate....all.....that........information..... lol

LOL. Take your time!

If your time (and curiosity) stretches even further, this is a good site, on both the physics and history of scales:

http://www.midicode.com/tunings/index.shtml

[Jed]

 

Finally, he talks about different patterns of whole steps and half steps. He then talks about the C major scale and the A minor scale which both use only the white notes on a piano....and that "If a musician is playing white keys, how do I know if he is playing the A minor scale or the C major scale? The answer is that --entirely without our conscious awareness--our brains are keeping track of how many times particular notes are sounded, where they appear in terms of strong versus weak beats, and how long they last" Wow. I can't say I was aware of this at all. I currently have no concept of this. I can't imagine myself knowing the difference between two scales using the same notes. Question: For those musicians out there with lots of experience, you can tell what scale a peace is being played in by just hearing it?

 

 

You can already hear the difference (or you would not have any interest in music), you just don't know how to identify the differences. Any musician can hear these differences - but it takes time and effort to equate the different sounds to the terminology.

[Jed]

Holy moly. Brain.....can't....assimilate....all.....that........information..... lol

 

 

So apparently, you've just met our resident encyclopedia . . . thank your deity of choice that you didn't ask more complicated questions.

 

In general, music theory has a unique language because it describes unique systems that don't really mate well with westerners' concepts of symmetry. So while the terms may sound familiar, the meanings are unique and the ramifications profound - and very little of this makes sense from a physics perspective (although that won't stop people from trying to use physics to explain it).

 

Cheers and welcome to the Loft.

[JonnyPac]

He then talks about intervals. "The octave in Western music is subdivided into twelve (logarithmically) equally spaced tones. The intervallic distance between A and B is called a whole step." and "The smallest division in our Western scale system cuts a whole step perceptually in half: the half step or semitone, which is one twelfth of an octave." He then lists a table that has intervals. Distance in semitones of 0 has the interval name of unison, distance in semitones of 1 has an interval name of minor second, etc. Question: Is it useful to memorize this table? He also says "In Western music we rarely use all the notes of the chromatic scale in composition; instead, we use a subset of seven (or less often, five) of those twelve tones. Question: I don't understand this. Why don't we compose using the whole range of the chromatic scale? Doesn't it give us more options? I know this is an ignorant question....but I just don't get it. Finally, he talks about different patterns of whole steps and half steps. He then talks about the C major scale and the A minor scale which both use only the white notes on a piano....and that "If a musician is playing white keys, how do I know if he is playing the A minor scale or the C major scale? The answer is that --entirely without our conscious awareness--our brains are keeping track of how many times particular notes are sounded, where they appear in terms of strong versus weak beats, and how long they last" Wow. I can't say I was aware of this at all. I currently have no concept of this. I can't imagine myself knowing the difference between two scales using the same notes. Question: For those musicians out there with lots of experience, you can tell what scale a peace is being played in by just hearing it?

I can't answer some of the more science based questions, but it looks like JonR and the crew hit it pretty well... On that note, I really dig the brain music stuff that is hitting the scene lately.

THIS thread has some great links:
http://forums.allaboutjazz.com/showthread.php?t=48650 ...really interesting!!

On the intervals: I think they are the most important thing you can understand as a musician (or educated listener). I have every basic interval, chord quality, scale/mode memorized, and, yes, I can tell what is being used in a given piece (with some exceptions). Learning intervals by ear (relative pitch) and on the fretboard will open more creative door than anything else, IMHO. As far as C vs Am or any other mode within the diatonic scale, it is not re-occurrence of certain pitches, but the placement and emphasis. Even melodies that use call 12 notes can de-emphasize them in such a way that the selected "inside" notes are what we focus on; like magic tricks, chromaticism can be akin to slight of hand! True 12 tone music with equal weight given to each tone is pretty hard to handle, and seems to only be popular in certain academic circles, not the general public or average musician.

Anyhoo, check out the link I posted. Listen to the Radiolab episode, and watch all of the Hal Galper classes. The book Music the Brain and Ecstasy is also a fascinating read.

[JonR]

You can already hear the difference (or you would not have any interest in music), you just don't know how to identify the differences.

Precisely.

Everyone can hear music the same way. Everyone (musicians and non-musicians alike) - unless there is something physically wrong with their ears or auditory system; - hears the same things; it's just that non-musicians don't know what they're called (and don't care).

Becoming a musician is a matter of refining one's focus and attention (ear training), learning terminology (ie theory), and learning technique (how to play an instrument, or to sing).

[JonR]

In general, music theory has a unique language because it describes unique systems that don't really mate well with westerners' concepts of symmetry.

I thought I'd come back on that idea of symmetry. (The following is not directed at you personally - just some general observations.)
I think a lot of the time, people look (in music) for mathematical or physical simplicity in the wrong places. There is simplicity and symmetry there, but not where one often expects it.
Eg, it's a common mistake to look at the 12 divisions of the octave (they're equal after all!) and wonder why we don't make scales by dividing it neatly, eg, into 6 whole tones, or whatever. But that's the wrong question; it's looking at it from the wrong place.
That the octave seems to divide into 12 is coincidental - the result of how the scales we use actually came into being.
When you calculate dorian mode by working outward from the keynote in 5ths, you do see a symmetry, and (IMO) a musically relevant one.

, forget it.)
And of course, guitar is something else again. Every time we fret a note, we risk putting it out of tune by pressure.
But the EAR is our saviour. Yes, it can be fooled, but "if it sounds right, it IS right" - that's the fundamental musical rule that over-rides all others. We don't have to care WHY something sounds right (it's an unanswerable question anyway). All we need are the music theory formulas to get those sounds - which are totally non-scientific, but (in comparison) refreshingly simple.

[NewGuyonGuitar]

On the intervals: I think they are the most important thing you can understand as a musician (or educated listener). I have every basic interval, chord quality, scale/mode memorized, and, yes, I can tell what is being used in a given piece (with some exceptions). Learning intervals by ear (relative pitch) and on the fretboard will open more creative door than anything else, IMHO. As far as C vs Am or any other mode within the diatonic scale, it is not re-occurrence of certain pitches, but the placement and emphasis. Even melodies that use call 12 notes can de-emphasize them in such a way that the selected "inside" notes are what we focus on; like magic tricks, chromaticism can be akin to slight of hand! True 12 tone music with equal weight given to each tone is pretty hard to handle, and seems to only be popular in certain academic circles, not the general public or average musician.

 

 

I understand what you're saying about placement and emphasis, but can't recall experiencing it consciously and I'm looking forward to the time when I do!

[NewGuyonGuitar]

and a perfect 5th is the next most consonant interval after the octave (and unison). It has a 3:2 ratio with the open string. The next important ratio (the next simplest) is 4:3. This means 1/4 of the distance from nut to bridge.

 

Sorry, but the ratio thing is escaping my feeble brain capacity. I don't see how 4:3 correlates to 1/4 of the distance.....

[Jed]

 

OK, is the measuring the physical distance of the string important? If so, what makes 1/3rd of the way along important? Why not 2/3rds or 7/8ths?

 

 

In to the deep end . . . . Harmonic Oscillation. The science of vibrating strings (and things). These are the mechanism of how sound is produced in the family of instruments known as "Stringed Instruments" of which the guitar is one example. Have a look at this Wiki article and you can imagine how string length affects pitch / why fractions are important and what "nodes" are relative to vibrating strings. Possibly more than you want to know, but you asked. FWIW the physics of music has very little to do with learning how to make music.

 

http://en.wikipedia.org/wiki/Overtone

 

Which leads of course to the Harmonic Series (which does have musical importance)

 

http://en.wikipedia.org/wiki/Harmonic_series_(music)

 

There are four main families of instruments: Strings, Woodwinds, Brass & Precussion

Each family uses slightly different means to produce specific sounds (pitches) of varying character (attack / decay / sustain / release rate) and overtone content (timbre). If you find this stuff interesting you should examine some books on music synthesis. The Synth players have been tearing the concepts of musical tones apart for decades in an effort to reproduce the sounds of regular musical instruments.

 

http://en.wikipedia.org/wiki/Timbre

[Jed]

Sorry, but the ratio thing is escaping my feeble brain capacity. I don't see how 4:3 correlates to 1/4 of the distance.....

 

 

After your read about the Harmonic Series, note that:

 

The fundamental has some frequency (x)

The ratio of frequencies from fundamental to fundamental = x:x = 1:1 (in music this is called a unison)

 

 

The 1st overtone / 2nd harmonic has a frequency of (2x)

The ratio of frequencies of the 2nd harmonic to the fundamental = 2x:x = 2:1 (an octave)

 

The 2nd overtone / 3rd harmonic has a frequency of (3x)

The ratio of frequencies of the 3rd harmonic to the fundamental = 3x:x = 3:1 (an octave plus a perfect 5th)

The ratio of frequencies of the 3rd harmonic to the 2nd harmonic = 3x:2x = 3:2 (a perfect 5th)

 

The 3rd overtone / 4th harmonic has a frequency of (4x)

The ratio of frequencies of the 4th harmonic to the fundamental = 4x:x = 4:1 (two octaves)

The ratio of frequencies of the 4th harmonic to the 2nd harmonic = 4x:2x = 2:1 (an octave)

The ratio of frequencies of the 4th harmonic to the 3rd harmonic = 4x:3x = 4:3 (a perfect 4th)

 

The 4th overtone / 5th harmonic has a frequency of (5x)

The ratio of frequencies of the 5th harmonic to the fundamental = 5x:x = 5:1 (2 octaves plus a major 3rd)

The ratio of frequencies of the 5th harmonic to the 2nd harmonic = 5x:2x = 5:2 (an octave plus a major 3rd or a major 10th)

The ratio of frequencies of the 5th harmonic to the 3rd harmonic = 5x:3x = 5:3 (a major 6th)

The ratio of frequencies of the 5th harmonic to the 4th harmonic = 5x:4x = 5:4 (a major 3rd)

 

The 5th overtone / 6th harmonic has a frequency of (6x)

The ratio of frequencies of the 6th harmonic to the fundamental = 6x:x = 6:1 (two octaves plus a perfect 5th)

The ratio of frequencies of the 6th harmonic to the 2nd harmonic = 6x:2x = 3:1 (an octave plus a perfect 5th)

The ratio of frequencies of the 6th harmonic to the 3rd harmonic = 6x:3x = 2:1 (an octave)

The ratio of frequencies of the 6th harmonic to the 4th harmonic = 6x:4x = 3:2 (a perfect 5th)

The ratio of frequencies of the 6th harmonic to the 5th harmonic = 6x:5x = 6:5 (a minor 3rd)

 

. . . and on and on it goes

 

In summary:

1:1 is called a unison

2:1 is called an octave

3:2 is called a perfect 5th

4:3 is called a perfect 4th

5:4 is called a major 3rd

6:5 is called a minor 3rd

 

Ultimately consonance is strongest at unison (1:1) and octaves (2:1)

Slightly less consonant is the perfect 5th (3:2)

Slightly less consonant than that is the perfect 4th (4:3)

Less consonant than that is the major 3rd (5:4)

Slightly less consonant than that is the minor 3rd (6:5)

Less consonant than that is the major 2nd (7:6)

Less consonant than that is the minor 2nd (8:7)

 

As you move further up into smaller and smaller intervals, the two notes played together become less and less consonant until at some point they are heard as dissonant. Also since every string vibrates at multiple frequencies at the same time we could look at adding up all these frequencies of the individual strings and see how the various harmonics interact with each other. This kind of things gets very complicated very quickly.

[jeremy_green]

This stuff makes my head want to explode ... when science meets art I have a hard time gripping the point.

I respect it... and appreciate that people do it... but want little part in it : )

[Jed]

 

. . the safest plan for any musician is simply not to think about the science at all. That way lies madness.

 

 

Just because someone can try to explain music with science, doesn't mean they should. Better to leave music as art and deal with it on it's artistic terms.

[JonR]

This stuff makes my head want to explode ... when science meets art I have a hard time gripping the point. I respect it... and appreciate that people do it... but want little part in it : )

There was once a centipede that was told that it had 100 legs. It found it could no longer walk.

For a musician, the science is, at best, beside the point. If you happen to be curious about it, that's fine. But it's only like being curious about nature in general; it's just brain exercise.
It's certainly a mistake to attempt to attach science to music theory, to use it to "explain" music theory.
If you want to learn a foreign language, it's not a lot of use to study the physiology of your vocal chords or your breathing. You know how to use your mouth. What you need is the vocabulary and the grammar.
Music is similar: you know what it sounds like. You just need to know how to make those sounds, not where they come from, or why we hear them the way we do.

[JonR]

Sorry, but the ratio thing is escaping my feeble brain capacity. I don't see how 4:3 correlates to 1/4 of the distance.....

It's the ratio of the full length (4/4) to 3/4 of the length. IOW, the ratio of the distance from nut to bridge, with the distance from 5th fret to bridge.

 

One possibly confusing thing is that when we fret the A string at 5th fret, we get D, not A. (The harmonic there gives us A). This is because the string length from 5th fret to bridge (3/4 full length) is vibrating at 4/3 x the open frequency.

4/3 x 110 = 146.67 = D.