| Just Intonation Tutorial |
The fundamental basis of harmony.
I created a thread asking a question about Just Intonation vs Equal Temperment and I think it might have been above the heads of most people on this forum. So, here is a light introductory tutorial to what all this is. Some of this is grossly oversimplified for the sake of the venue in which it is presented, none-the less I think all the important concepts are here.
Sound Physics Basics:
All sounds are comprised of waves.
There are two types of sounds: Harmonic and Chaotic. Examples of Harmonic sounds include the human voice, vibrating strings, lawn mowers, jet engines, jack hammers, cars and fans. Examples of Chaotic sounds include crumpling paper, blowing leaves, sneezes, the background conversational din in a restaraunt and firecrackers.
Waves have these characteristics: Wavelength. Wavelength is the size of one full cycle of the sound wave lengthwise through time. Volume is the amount of air pressure applied by the sound, measured in diameter of the wave. Frequency is how many times the wavelength occurs in one second. Frequency is measured in Hertz, abbreviated Hz, the namesake of Heinrich Rudolf Hertz. Hertz means "times per second" and many things are measureable in Hertz. The second hand on a clock moves at 1Hz, or one time per second. Electrical lights usually vibrate at 60Hz, or 60 times per second.
Sound combinatorics: Two waves which are exactly opposite, when combined, form silence. This is theoretically speaking, in reality this rarely happens. It is known as being out of phase. This is however the basis of some new breeds of "noise cancelling" headphones manufactured by people such as Sony and BOSE. These headphones only work with Harmonic noise. Two waves which are exactly identical, when combined, form a much louder but rather identical wave. Any time any two other types of sound waves combine, a "beat" is created. This beat, which is audible, is the increase and decrease of volume created by the addition and subtraction of the two waves from each other.
Harmonic sounds have overtones: every harmonic tone, called a fundamental, generates other harmonic tones which are higher in frequencey than the fundamental. These tones being higher or "above" the fundamenal are called "over" tones and are also called simply "Harmonics". You guitarists out there are probably familiar with harmonics where you place your finger very lightly on a string over the 5th 7th or 12th frets and then play it and what results is a totally different higher pitched note.
Just Intonation:
Fibonacci sequence: The Fibonacci Sequence is a mathematical ratio which generates the classic spiral. It is evident in many many things such as, nautilus shells, the growth pattern of thorns on a rose, davinci's paintings and drawings, the pyriamids and even sound. The distances between the overtones of a note progress and change in the ratio of the Fibonacci Sequence.
Resonance. Any two sound waves combined, if properly aligned frequency wise, will create resonance. Essentially, two tones which are "in tune" resonate, while two tones which are not in tune do not resonate. Resonance also gives an increase in volume and pleasant perceptics. Resonance is generally aesthetic.
Overtone ratio sequence: Now this section will require a small amount of knowlege of music theory concerning intervals and the major scale. The overtones of any note occur in exact whole number multiples, 1 2 3 4 5 6 and so on. All the intervals of the modern musical scales come from this overtone series.
The first over tone lives at multiple 2. This overtone happens to be the octave of a major scale, or any scale for that matter. Why is it the octave? I don't know, it just is. The next over tone happens to be at multiple 3. This overtone happens to be the Perfect Fifth of a major scale. The next overtone is multiple 4, which is the octave again. Anyway, the ratios of these latest two overtones, 4:3 gives us a Perfect Fourth. The next overtone is at the multiple of 5. This is the Major Third of the major scale. The next overtone is at 6, this again, like 3, is a Perfect Fifth.
And so on down the line. Interestingly, all 12 intervals on the piano can be derived from jumps of only Perfect Fifths and Major Thirds. Notice that the names of the intervals and the corresponding numbers in their ratios are absolute coincidence. The names have to do with the major scale, the ratios have to do with the overtones. From the study of these intervals we can define a very nice orderly progressional sequence of ratios which provides us with all the primary intervals in existence.
2:1
3:2
4:3
5:4
6:5
7:6
8:7
9:8
and so on to infinity.
The Justly Intoned major scale intervals are generally considered to be as follows.
C:C = 1:1 Unison
D:C = 9:8 Whole step
E:C = 5:4 Major Third
F:C = 4:3 Perfect Fourth
G:C = 3:2 Perfect Fifth
A:C = 27:16 Major Sixth
B:C = 15:8 Major Seventh
C:C = 2:1 Octave
These ratios are what form the most basic of basic intervals in our musical scale, and it is these intervals which are termed "Just" in fact, any interval which exists in a perfect mathematical ratio can be termed Just, for example 1395:4287 is a just interval, however, in practice only such simple ratios as those above are termed "just".
Overtone combinatorics: When two waves combine which share overtones or whose fundamental notes and or overtones combine to form perfect ratios as above. These combined notes are said to be Resonant, They are also said to be "Justly Intoned"
Just Intonation is not without it's problems: For example the basic unaltered diatonic scale is considered to be 1/1, 9/8, 5/4, 4/3, 3/2, 27/16, 15/8, 2/1. Theoretically the major sixth plus the major third should give a perfect forth, count it, that's four notes, 6, 5, 4, 3... that's four notes, a pefect fourth. A perfect fourth (check the chart) has a ratio of 4/3 however since (27/16):(5/4) = 27:20 and (27/16):(5/4) does not equal 4/3 we've got problems. 27:20 is called a "wolf interval" and don't worry if you didn't follow that at all, just try looking at the chart of the major scale interavals again above, otherwise just go on. The point is that - within Just Intonation - ratio's get screwy when you try to combine more than the fundamental and one other note.
Equal Temperment:
Accomplishments and why: Equal temperment was invented to solve some of the problems of Just Intonation. It allows you to play music in all major scales and with multiple notes but with an equal amount of detuning on each note in each scale. This is good. In Just Intonation the balance of unequal tuning can be so lopsided that things sound bad. In Equal Temperment, everything is fully and proportionately out of tune compared to everything else.
Mathematical basis: The octave is a multiple of two. There are twelve notes in modern harmony these days, thus two divided by twelve is: The Twelfth Root of Two. Every half step is exactly The Twelfth Root of Two apart from the next half step.
Where do you find equal temperment? Well basically every modern instrument in the world is tuned with equal temperment. Particularly pianos are tuned with equal temperment. Guitars are fretted with equal temperment, saxophones and flutes and clarinets and such are manufactured to equal temperment. Violins are general meant to be played inside of equal temperment. The human voice is really the one main instrument that has the option of playing in either euqal temperment or just intonation. It is certainly possible to tune other instruments such as guitars and violins into just intonation as well, but is more difficult.
End of Story. Hope that was informative and also readable.
Any smarty pantsers out there? Did I miss anything?
Questions, comments, confused?
Did anyone actually get this far? 
I created a thread asking a question about Just Intonation vs Equal Temperment and I think it might have been above the heads of most people on this forum. So, here is a light introductory tutorial to what all this is. Some of this is grossly oversimplified for the sake of the venue in which it is presented, none-the less I think all the important concepts are here.
Sound Physics Basics:
All sounds are comprised of waves.
There are two types of sounds: Harmonic and Chaotic. Examples of Harmonic sounds include the human voice, vibrating strings, lawn mowers, jet engines, jack hammers, cars and fans. Examples of Chaotic sounds include crumpling paper, blowing leaves, sneezes, the background conversational din in a restaraunt and firecrackers.
Waves have these characteristics: Wavelength. Wavelength is the size of one full cycle of the sound wave lengthwise through time. Volume is the amount of air pressure applied by the sound, measured in diameter of the wave. Frequency is how many times the wavelength occurs in one second. Frequency is measured in Hertz, abbreviated Hz, the namesake of Heinrich Rudolf Hertz. Hertz means "times per second" and many things are measureable in Hertz. The second hand on a clock moves at 1Hz, or one time per second. Electrical lights usually vibrate at 60Hz, or 60 times per second.
Sound combinatorics: Two waves which are exactly opposite, when combined, form silence. This is theoretically speaking, in reality this rarely happens. It is known as being out of phase. This is however the basis of some new breeds of "noise cancelling" headphones manufactured by people such as Sony and BOSE. These headphones only work with Harmonic noise. Two waves which are exactly identical, when combined, form a much louder but rather identical wave. Any time any two other types of sound waves combine, a "beat" is created. This beat, which is audible, is the increase and decrease of volume created by the addition and subtraction of the two waves from each other.
Harmonic sounds have overtones: every harmonic tone, called a fundamental, generates other harmonic tones which are higher in frequencey than the fundamental. These tones being higher or "above" the fundamenal are called "over" tones and are also called simply "Harmonics". You guitarists out there are probably familiar with harmonics where you place your finger very lightly on a string over the 5th 7th or 12th frets and then play it and what results is a totally different higher pitched note.
Just Intonation:
Fibonacci sequence: The Fibonacci Sequence is a mathematical ratio which generates the classic spiral. It is evident in many many things such as, nautilus shells, the growth pattern of thorns on a rose, davinci's paintings and drawings, the pyriamids and even sound. The distances between the overtones of a note progress and change in the ratio of the Fibonacci Sequence.
Resonance. Any two sound waves combined, if properly aligned frequency wise, will create resonance. Essentially, two tones which are "in tune" resonate, while two tones which are not in tune do not resonate. Resonance also gives an increase in volume and pleasant perceptics. Resonance is generally aesthetic.
Overtone ratio sequence: Now this section will require a small amount of knowlege of music theory concerning intervals and the major scale. The overtones of any note occur in exact whole number multiples, 1 2 3 4 5 6 and so on. All the intervals of the modern musical scales come from this overtone series.
The first over tone lives at multiple 2. This overtone happens to be the octave of a major scale, or any scale for that matter. Why is it the octave? I don't know, it just is. The next over tone happens to be at multiple 3. This overtone happens to be the Perfect Fifth of a major scale. The next overtone is multiple 4, which is the octave again. Anyway, the ratios of these latest two overtones, 4:3 gives us a Perfect Fourth. The next overtone is at the multiple of 5. This is the Major Third of the major scale. The next overtone is at 6, this again, like 3, is a Perfect Fifth.
And so on down the line. Interestingly, all 12 intervals on the piano can be derived from jumps of only Perfect Fifths and Major Thirds. Notice that the names of the intervals and the corresponding numbers in their ratios are absolute coincidence. The names have to do with the major scale, the ratios have to do with the overtones. From the study of these intervals we can define a very nice orderly progressional sequence of ratios which provides us with all the primary intervals in existence.
2:1
3:2
4:3
5:4
6:5
7:6
8:7
9:8
and so on to infinity.
The Justly Intoned major scale intervals are generally considered to be as follows.
C:C = 1:1 Unison
D:C = 9:8 Whole step
E:C = 5:4 Major Third
F:C = 4:3 Perfect Fourth
G:C = 3:2 Perfect Fifth
A:C = 27:16 Major Sixth
B:C = 15:8 Major Seventh
C:C = 2:1 Octave
These ratios are what form the most basic of basic intervals in our musical scale, and it is these intervals which are termed "Just" in fact, any interval which exists in a perfect mathematical ratio can be termed Just, for example 1395:4287 is a just interval, however, in practice only such simple ratios as those above are termed "just".
Overtone combinatorics: When two waves combine which share overtones or whose fundamental notes and or overtones combine to form perfect ratios as above. These combined notes are said to be Resonant, They are also said to be "Justly Intoned"
Just Intonation is not without it's problems: For example the basic unaltered diatonic scale is considered to be 1/1, 9/8, 5/4, 4/3, 3/2, 27/16, 15/8, 2/1. Theoretically the major sixth plus the major third should give a perfect forth, count it, that's four notes, 6, 5, 4, 3... that's four notes, a pefect fourth. A perfect fourth (check the chart) has a ratio of 4/3 however since (27/16):(5/4) = 27:20 and (27/16):(5/4) does not equal 4/3 we've got problems. 27:20 is called a "wolf interval" and don't worry if you didn't follow that at all, just try looking at the chart of the major scale interavals again above, otherwise just go on. The point is that - within Just Intonation - ratio's get screwy when you try to combine more than the fundamental and one other note.
Equal Temperment:
Accomplishments and why: Equal temperment was invented to solve some of the problems of Just Intonation. It allows you to play music in all major scales and with multiple notes but with an equal amount of detuning on each note in each scale. This is good. In Just Intonation the balance of unequal tuning can be so lopsided that things sound bad. In Equal Temperment, everything is fully and proportionately out of tune compared to everything else.
Mathematical basis: The octave is a multiple of two. There are twelve notes in modern harmony these days, thus two divided by twelve is: The Twelfth Root of Two. Every half step is exactly The Twelfth Root of Two apart from the next half step.
Where do you find equal temperment? Well basically every modern instrument in the world is tuned with equal temperment. Particularly pianos are tuned with equal temperment. Guitars are fretted with equal temperment, saxophones and flutes and clarinets and such are manufactured to equal temperment. Violins are general meant to be played inside of equal temperment. The human voice is really the one main instrument that has the option of playing in either euqal temperment or just intonation. It is certainly possible to tune other instruments such as guitars and violins into just intonation as well, but is more difficult.
End of Story. Hope that was informative and also readable.
Any smarty pantsers out there? Did I miss anything?
Questions, comments, confused?
Did anyone actually get this far? 
Yeh, I got that far!
I just read and listened to substitutions too. This is a good tutorial / intro - more of this would be great. Anyhow, being a theory ignoramus I can relate to substitutions by ear and I can hear that it works - I think I can apply the ideas (I think I already do to some extent), but I don't know why it works or the reasoning behind how you derived those particular chords - I can see the recommended reading.
I play my acoustic a lot and I use all sorts of chords - don't know what half of them are called but they make sense to my ear. Sometimes I think that ignorance is bliss when I consider the weight of theory behind it all, the erminology etc...
However, I think that there are many people here that would really benefit from the odd entry level theory explanation here and there, so keep posting this kind of stuff!
I just read and listened to substitutions too. This is a good tutorial / intro - more of this would be great. Anyhow, being a theory ignoramus I can relate to substitutions by ear and I can hear that it works - I think I can apply the ideas (I think I already do to some extent), but I don't know why it works or the reasoning behind how you derived those particular chords - I can see the recommended reading.
I play my acoustic a lot and I use all sorts of chords - don't know what half of them are called but they make sense to my ear. Sometimes I think that ignorance is bliss when I consider the weight of theory behind it all, the erminology etc...
However, I think that there are many people here that would really benefit from the odd entry level theory explanation here and there, so keep posting this kind of stuff!
ooo, that's going to cost you, hehe
that was one of his shorter posts
it may not be flashy and "exciting", but if you can't sit and make honest efforts to learn it, you can't always get as far with this silly "music" thing. this is an underlying idea behind music that makes it what it is like history, it may be boring and it may not be completely manditory to know, you're better off for understanding it, hehe
anyway, just wanted to say something
that was one of his shorter posts
it may not be flashy and "exciting", but if you can't sit and make honest efforts to learn it, you can't always get as far with this silly "music" thing. this is an underlying idea behind music that makes it what it is
like history, it may be boring and it may not be completely manditory to know, you're better off for understanding it, heheanyway, just wanted to say something
that helped a little.... bvut how would you start a root note? Also Just intonation is just the way of writing???
I thought there were some differences in the actual # of cycles? sorry im a dumbass...
I thought there were some differences in the actual # of cycles? sorry im a dumbass...
JustlyHarmonized wrote…
that helped a little.... bvut how would you start a root note? Also Just intonation is just the way of writing???
I thought there were some differences in the actual # of cycles? sorry im a dumbass...
you start a root note like you start any other note. you play it, sing it, pluck it. and no just intonation is not simply a way of writing things, it refers to the actual physics of sound and how we perceive it and it's mathematical characteristics.mostly it refers to the mathematical ratio between two notes which is perceivable by the human ear.
| you hit the nail head with equal temperment |
Good post I'm lazy, I woulda just sent a link to a website that already had this.
It's been a few years since I had studied theory but with post like this it seem to be coming back to me.
At school it seemed there were three groups, performance majors, composition majors, and theory majors, sometimes the last 2 were combined. It's good to know this, but as far as song writing goes it's not a must. It does help if you know how to read and write music. Imho
It's been a few years since I had studied theory but with post like this it seem to be coming back to me.
At school it seemed there were three groups, performance majors, composition majors, and theory majors, sometimes the last 2 were combined. It's good to know this, but as far as song writing goes it's not a must. It does help if you know how to read and write music. Imho
From my last post, I have another question. If the ratio's like 3/2; 15/14; 5/4; 4/3.... are going to be the same fractions for whatever the open note is, then why would the frets be in different places for the other strings. Is it due to string diameter, or the specific pitch being played??
| good question |
i honestely don't know the answer. that's a very ambitious project you've got there. i would try to check amazon or something for books on the subject as i doubt the information is readily available on the internet.
i do know that the sitar, the funny sounding guitar like instrument played in india has movable frets, so you might want to just get yourself a sitar and play with that to start.
i do know that the sitar, the funny sounding guitar like instrument played in india has movable frets, so you might want to just get yourself a sitar and play with that to start.
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