# What determines the function of a chord?



## youngcapone

I’ve been trying to learn about functional harmony and I keep asking myself the above question. Each chord seems to have a general function (with exceptions) within scales, but why is that so? Were scales made with those functions in mind, or were those functions agreed upon or discovered after scales were already determined? Is it a result of having 7 notes in a scale? For example, if you were to choose 7 random notes within an octave...would the 5th chord always serve a dominant function?

DISCLAIMER: I know I’m asking a simple question that doesn’t have a simple answer and I don’t expect any response to be cut and dry. I also realize I’m making a lot of assumptions about the way music works that may or may not be true, however any insight at all would be helpful! Thanks!


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## Guest

I don't have time to write a detailed answer and I don't like writing reams of text. 
Try this: http://openmusictheory.com/harmonicFunctions.html


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## mikeh375

TalkingHead said:


> I don't have time to write a detailed answer and I don't like writing reams of text.
> Try this: http://openmusictheory.com/harmonicFunctions.html


What a good looking site TalkingH. An excellent resource for anyone interested.


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## millionrainbows

TalkingHead said:


> I don't have time to write a detailed answer and I don't like writing reams of text.
> Try this: http://openmusictheory.com/harmonicFunctions.html


I've added it to my theory bookmarks list. It is nice, though, to have a person explain it.


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## pianozach

I'll just point out that the "function of a chord" has changed over time.

If you go from, say, Bach to Beethoven to Mussorgsky, harmonic structure and progression is no longer the same. 
There's even further breakdown of that as the 20th century progressed.

No, I'd go with an educated explanation, like the one *TalkingHead* pointed out. It's short, simple, and a good place to *start*.


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## millionrainbows

pianozach said:


> No, I'd go with an educated explanation, like the one *TalkingHead* pointed out. It's short, simple, and a good place to *start*.


It is nice, though, to have a person explain it.


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## Guest

pianozach said:


> *I'll just point out that the "function of a chord" has changed over time.*
> If you go from, say, Bach to Beethoven to Mussorgsky, harmonic structure and progression is no longer the same.
> There's even further breakdown of that as the 20th century progressed.


Spot on, Pianozac.
From the website I posted above:
_Because tendency is style-specific, the same chord can have different functions *in different musical styles*. For instance, *the kinds of functions we find in classical music are different from those we find in pop/rock songs from the Billboard charts*. And though there are some general harmonic traits that are common to most eighteenth - and nineteenth-century Western composers (what we call the "common practice"), when we look in closer detail, *we find some significant differences in the way Bach, Mozart, Brahms, and others compose their harmonic progressions*_.



pianozach said:


> No, I'd go with an educated explanation, like the one *TalkingHead* pointed out. It's short, simple, and a good place to *start*.


 Thank you, Pianozach.


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## millionrainbows

TalkingHead said:


> Because tendency is style-specific, the same chord can have different functions in different musical styles. For instance, the kinds of functions we find in classical music are different from those we find in pop/rock songs from the Billboard charts. And though there are some general harmonic traits that are common to most eighteenth - and nineteenth-century Western composers (what we call the "common practice"), when we look in closer detail, we find some significant differences in the way Bach, Mozart, Brahms, and others compose their harmonic progressions.


I think that chord function is more consistently the same for any era, is based on the same phenomena, and that stylistic practices are much less relevant.

I think a chord's position in the hierarchy, i.e. what scale step it is built on, has everything to do with its perceived function or possible use.

You'd have to be a lot more specific than this to convince me otherwise; although it wouldn't surprise me to learn of some archaic practice which took place ages ago, which makes no sense and goes against what our ears tell us is right. It's happened before.

From the article: 
However, these rules are also related to laws, in as much as they represent one set of practices that mediate the various demands on music from basic principles of human auditory perception and cognition. For instance, the prohibition against parallel fifths is a specific way in which Western tonal composers have mediated the conflict between tonal fusion, goal-directed motion, and independence of line. There are many other similar cases.


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## millionrainbows

TalkingHead said:


> Spot on, Pianozac.
> From the website I posted above:
> _Because tendency is style-specific, the same chord can have different functions *in different musical styles*. For instance, *the kinds of functions we find in classical music are different from those we find in pop/rock songs from the Billboard charts*. And though there are some general harmonic traits that are common to most eighteenth - and nineteenth-century Western composers (what we call the "common practice"), when we look in closer detail, *we find some significant differences in the way Bach, Mozart, Brahms, and others compose their harmonic progressions*_.
> 
> Thank you, Pianozach.


But I'd be interested in knowing if there is an underlying principle which is common to all ideas of function. whether they are specific style-related ideas or not; in other words, a "unified field theory" of function.


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## Wes Lachot

millionrainbows said:


> But I'd be interested in knowing if there is an underlying principle which is common to all ideas of function. whether they are specific style-related ideas or not; in other words, a "unified field theory" of function.


Millionrainbows: I think that's a really interesting question, so interesting, in fact, that it caused me to join this forum. I'm wondering what others may have to say about this. My own experience is that it has been difficult to find a cohesive definition of the term "harmonic function" in the music theory literature, whether it's textbooks, scholarly books, white papers, or whatever. But I am all ears, so to speak, if anyone has come across such a definition that makes sense.


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## millionrainbows

Wes Lachot said:


> Millionrainbows: I think that's a really interesting question, so interesting, in fact, that it caused me to join this forum. I'm wondering what others may have to say about this. My own experience is that it has been difficult to find a cohesive definition of the term "harmonic function" in the music theory literature, whether it's textbooks, scholarly books, white papers, or whatever. But I am all ears, so to speak, if anyone has come across such a definition that makes sense.


Thanks, Wes. Here's my view, which is from a blog of mine:

In a scale, the pull towards a tonic is inherently determined by vertical harmonic factors, not horizontal "emphasis" by repetition or accent. That comes later.

1. minor seventh (C-Bb) 9:16
2. major seventh (C-B) 8:15
3. major second (C-D) 8:9
4. minor sixth (C-Ab) 5:8
5. minor third (C-Eb) 5:6
6. major third (C-E) 4:5
7. major sixth (C-A) 3:5
8. perfect fourth (C-F) 3:4
9. perfect fifth (C-G) 2:3
10. octave (C-C') 1:2
11. unison (C-C) 1:1

So a C major scale's horizontal functions correspond to these harmonic relations; and one can observe how these functions were derived:

I - 1:1
ii - 8:9
iii - 4:5
IV - 3:4
V - 2:3
vi - 3:5
vii - 8:15

Their importance in establishing the tonality can be ranked by the order of consonance to dissonance, with smaller-number ratios being more consonant.

I - 1:1
V - 2:3
IV - 3:4
vi - 3:5
iii - 4:5
ii - 8:9
vii - 8:15

Using this model, a "function" hierarchy can be applied to any scale, after the degrees of dissonance are ranked.

Whole Tone scale: C-D-D-F#-G#-A#

C - 1:1
D -8:9
E -4:5
F#- 45:32
G# - 8:5
A# - 16:9

Whether or not you attach Roman numerals to the above is optional; but by the numbers, one can see a ranking:

C - 1:1
E -4:5
G# - 8:5
D -8:9
A# - 16:9
F#- 45:32


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## BabyGiraffe

Wes Lachot said:


> Millionrainbows: I think that's a really interesting question, so interesting, in fact, that it caused me to join this forum. I'm wondering what others may have to say about this. My own experience is that it has been difficult to find a cohesive definition of the term "harmonic function" in the music theory literature, whether it's textbooks, scholarly books, white papers, or whatever. But I am all ears, so to speak, if anyone has come across such a definition that makes sense.


https://en.wikipedia.org/wiki/Function_(music)

Read the whole article. Btw, it assumes meantone temperament.
Pythagorean tuning is schismic (until 665 edo when schisma becomes a step), so the theory is different (diminished and augmented intervals are the 5-limit consonances = 317.595 cents, which is almost pure min3rd is Pythagorean augmented second and 384.360 cents, almost major third is Pythagorean diminished fourth).
Schismic, meantone and their difference (diashismic) and their sum (pythagorean comma) meet in 12 equal only (this can lead to potentially confusing or wrong notation systems)


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## Wes Lachot

million rainbows; Thanks, interesting and logical stuff. The hierarchy method for determining which scale degrees, are "functional" would tend to support only calling the I, IV, and V functional degrees, since the next one, 3:5, is way out of tune compared to ET, around 14 cents (I'm sure Babygiraffe could tell us precisely in less than a nanosecond). Then the next one, the III or 4:5 is off by a similar amount. Only after those two in the hierarchy do we get to the II, which is more in tune with ET and is the one some theorists let into the "functional" club right after the I, IV, and V.


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## Wes Lachot

BabyGiraffe said:


> https://en.wikipedia.org/wiki/Function_(music)
> 
> Read the whole article. Btw, it assumes meantone temperament.
> Pythagorean tuning is schismic (until 665 edo when schisma becomes a step), so the theory is different (diminished and augmented intervals are the 5-limit consonances = 317.595 cents, which is almost pure min3rd is Pythagorean augmented second and 384.360 cents, almost major third is Pythagorean diminished fourth).
> Schismic, meantone and their difference (diashismic) and their sum (pythagorean comma) meet in 12 equal only (this can lead to potentially confusing or wrong notation systems)


Thanks, interesting article. It does sum up pretty succinctly some of the differences in terminology, and one can infer the differences in the conception of what is meant by the term "function" within the various schools of thought. The term seems to mean different things to different people.

For instance, theorists will sometimes state that because a chord "stands in" for another chord in a chord progression, it must have the same "function" as the chord it replaced. An example would be the III occurring in a place where the I could be in the progression. Or lets say a N6 chord in place of a IV chord. The term "pre-dominant" seems shaky to me for this reason, as even the name itself seems to lock in the position in the chord progression--this in an art form that's all about defying expectations. Secondary (applied) dominants are also referred to as "functional" and "non-functional" depending on where they resolve to, and so on. And so began the search for clarity.


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## BabyGiraffe

Wes Lachot said:


> Thanks, interesting article. It does sum up pretty succinctly some of the differences in terminology, and one can infer the differences in the conception of what is meant by the term "function" within the various schools of thought. The term seems to mean different things to different people.
> 
> For instance, theorists will sometimes state that because a chord "stands in" for another chord in a chord progression, it must have the same "function" as the chord it replaced. An example would be the III occurring in a place where the I could be in the progression. Or lets say a N6 chord in place of a IV chord. The term "pre-dominant" seems shaky to me for this reason, as even the name itself seems to lock in the position in the chord progression--this in an art form that's all about defying expectations. Secondary (applied) dominants are also referred to as "functional" and "non-functional" depending on where they resolve to, and so on. And so began the search for clarity.


There will never be clarity, because there are no universal functions even in the same tuning or even in the same musical scale, if we change the modes... And meantone theory fails in other temperaments, supported by 12 equal like modes of limited transposition (which can be described as diminished, augmented, diaschismic) and pythagorean/schismic tuning - they have their own logic of chord changes.
Classical function theory assumes heptatonic diatonic in meantone temperament.


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## Wes Lachot

I hear you, BabyGirrafe. But couldn't we at least have more clarity with in the limited, admittedly out of tune but beautifully symmetrical 12 tone system? 

I think that the out-of-tuneness of the non-perfect intervals has had a really positive affect on music, if you look at the glass as half full. It caused jazz pianists to employ voicing made up mostly of 4ths, which sound expansively in tune compared to the same set of notes stacked in 3rds. There's none of the obnoxious roughness, just clean, modern sounding consonance. Sure, the 4ths are a little under 2 cents sharp, but who can hear that over the enharmonicity of the piano? It's more of a slow chorusing, which the piano is full of anyway if there's any real music being played. And the tuners know they have to stretch it back up somewhat, in fact it's often the entire 2 cents that they stretch it, in the next octave up (tuners realize that all of the harmonics can't be in tune at once, due to enharmonicity).

And ET has certainly caused heavy rock players to drop the 3rd from their voicings and stick to root, 5th, root chord structures that slide around in modal fashion. A Marshall stack just doesn't distort in a consonant way if you add 3rds to the bar chords, but if you stick to the perfect intervals the tube amps, with their bias toward even harmonics, tend to reinforce and enhance the open chords. Until it all sounds like a vast, open, modal wasteland. Not my particular cup of tea, but a lot of people like it, and ET is arguably to blame.


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## BabyGiraffe

You can't have clarity, because there is no clarity - chords don't have inherent function (meaning) in isolation, only in relation with each other in a real composition employing some kind of systematic language (like sticking to certain patterns in rhythm, melody or harmony, not playing random sequences as heard in pseudo-randomly generated computer music or serialism). 

(And this doesn't have anything to do with tuning. Equal tunings support different options like i said before. Choosing a specific tuning just means that you disqualify the other options. So, classical meantone means no enharmonic augmented, diminished and tritone substitutions and you need more than 12 keys to access some of these enharmonic modulations. If you tune to pythagorean, diatonic, pentatonic and many other scales have false relations etc.)


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## millionrainbows

From my studies of tonality and intervals, the 12 ET system is best for tonality because the fifths are almost perfect, only 2 cents off, and this encourages stable triads and root movement by fifths, not repeating the entire chromatic until (fifths=7 semitones) 7 X 12= 84, or the other direction (fourths=5 semitones) 5 X 12=60. These intervals are "built for travel" outside the octave into new areas.

The other intervals (M2, m3, M3, tritone) are smaller and recursive within the octave, not having to venture outside the octave to repeat. This also means they are more limited tonally, since none of them _(except the minor second)_ can generate the entire chromatic when multiplied; only fourths, fifths. and m2s can do that. The smaller intervals are "built for chromaticism" and staying-put within the octave, in localized areas.

So these smaller intervals are more suited for chromatic tonalities (as Bartok knew), not our tonality of fifths-relations. BabyGiraffe should figure this in to his ideas, instead of just considering the "just-ness" of an interval.

This may be a bit much to chew on after that, but the diminished seventh chord with its minor thirds has a close affinity to tonality if one lowers any tone; thus F-Ab-B-D becomes G7 if we lower the Ab to G (F-G-B-D).

This is true if we "manipulate" the intervals and lower a note. Music is a moving, changing thing, not static. Thus, BabyGiraffe's ideas are "too static;" they deal only with static, vertical interval relations, not movement of notes.


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## millionrainbows

BabyGiraffe said:


> You can't have clarity, because there is no clarity - chords don't have inherent function (meaning) in isolation, *only in relation with each other in a real composition)...*


No,* in relation to a single tonic,* which changes everything: thus, the interval (and chord derived from it) is seen in relation to *an octave, with a tonic.* Thus, a chord is never truly "in isolation" in this context. But you probably want to do away with octaves, right? You need to consider tonality.


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## BabyGiraffe

millionrainbows said:


> From my studies of tonality and intervals, the 12 ET system is best for tonality because the fifths are almost perfect, only 2 cents off, and this encourages stable triads and root movement by fifths, not repeating the entire chromatic until (fifths=7 semitones) 7 X 12= 84, or the other direction (fourths=5 semitones) 5 X 12=60. These intervals are "built for travel" outside the octave into new areas.
> 
> The other intervals (M2, m3, M3, tritone)are smaller and recursive within the octave, not having to venture outside the octave to repeat. This also means they are more limited tonally, since none of them (except the minor second) can generate the entire chromatic when multiplied; only fourths, fifths. and m2s can do that. The smaller intervals are "built for chromaticism" and staying-put within the octave, in localized areas.
> 
> So these smaller intervals are more suited for chromatic tonalities (as Bartok knew), not our tonality of fifths-relations. BabyGiraffe should figure this in to his ideas, instead of just considering the "just-ness" of an interval.


Figure what? Each system has its own advantages and disadvantages- with the right timbres even 5, 7 and 9 (or even 8 and 10, someone built such inharmonic pianos - as proof of concept) equal can sound good.

12 equal is the best for augmented in (5-limit tuning) and diminished tonality in 5 and 7-limit, that's it. (Not for meantone, which is the assumed Western music tuning and notation..) So, it is certainly fine for late 19th- 20th century music, but not for anything before this (excluding some lute music etc; it can also work for pseudo medieval music, but good HIP groups train in something like 53 equal in practice).


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## BabyGiraffe

millionrainbows said:


> No,* in relation to a single tonic,* which changes everything: thus, the interval (and chord derived from it) is seen in relation to *an octave, with a tonic.* Thus, a chord is never truly "in isolation" in this context. But you probably want to do away with octaves, right? You need to consider tonality.


You can get close to have a single tonic only in just intonation and even then there you can modulate in most cases, depends on the size of pitch gamut.

With 12 notes gamut, it's trivial to have like 8 usable major or minor keys in tune even in JI, so single tonic theory fails. 
You can have like single tonic when you are using something like arabic scale with only 1 good fifth and major third maybe? And the rest of notes - various 1/4 tones inflected intervals that will never be heard as tonic.


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## millionrainbows

BabyGiraffe said:


> Figure what? Each system has its own advantages and disadvantages- with the right timbres even 5, 7 and 9 (or even 8 and 10, someone built such inharmonic pianos - as proof of concept) equal can sound good.
> 
> 12 equal is the best for augmented in (5-limit tuning) and diminished tonality in 5 and 7-limit, that's it. (Not for meantone, which is the assumed Western music tuning and notation..) So, it is certainly fine for late 19th- 20th century music, but not for anything before this (excluding some lute music etc; it can also work for pseudo medieval music, but good HIP groups train in something like 53 equal in practice).


You seem to be ignoring the limitations on meantone temperament; it only works for a limited area of tonality. C-G-D-A E going in sharps or C-F-Bb-Eb-Ab going the other direction. And some of these sound worse than others.

I think you need to also reconsider the importance you seem to place on major thirds. Like the guy with the Marshall that Wes mentioned, I find flat sevenths and ninths to be more compelling. Of course, I'm a pagan...


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## millionrainbows

BabyGiraffe said:


> You can get close to have a single tonic only in just intonation and even then there you can modulate in most cases, depends on the size of pitch gamut.
> 
> With 12 notes gamut, it's trivial to have like 8 usable major or minor keys in tune even in JI, so single tonic theory fails.
> You can have like single tonic when you are using something like arabic scale with only 1 good fifth and major third maybe? And the rest of notes - various 1/4 tones inflected intervals that will never be heard as tonic.


Recognize the practical primacy of the octave. You can sing them. _"Close to having a single tonic only in just intonation?"_ Who cares. The octave recognizes no particular intonation. The primacy of the ear, and of the 2/1-1/2 relation is most important of all; otherwise, why mess with meantone and other tempering systems? _Tempering is for octave preservation, not just for better thirds.
_


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## BabyGiraffe

millionrainbows said:


> You seem to be ignoring the limitations on meantone temperament; it only works for a limited area of tonality. C-G-D-A E going in sharps or C-F-Bb-Eb-Ab going the other direction. And some of these sound worse than others.
> 
> I think you need to also reconsider the importance you seem to place on major thirds. Like the guy with the Marshall that Wes mentioned, I find flat sevenths and ninths to be more compelling. Of course, I'm a pagan...


So, in 12 notes meantone we have:
8 major thirds, 9 minor thirds, 11 fifths. 1 major and 1 minor third are in the same key with the mistuned fifth, so->

7 major keys and 8 minor keys at least.

Btw, since late 15th century many keyboards and organs had 14 (or more!) notes per octave, not 12, because of split black keys.


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## BabyGiraffe

millionrainbows said:


> Recognize the practical primacy of the octave. You can sing them. _"Close to having a single tonic only in just intonation?"_ Who cares. The octave recognizes no particular intonation. The primacy of the ear, and of the 2/1-1/2 relation is most important of all; otherwise, why mess with meantone and other tempering systems? _Tempering is for octave preservation, not just for better thirds.
> _


Aren't all pianos with stretched octaves????
Even digital modelling pianos do this to sound more natural.
Tempering is to have enharmonic equivalences, so your music works without annoying intervals that are potentially not needed. You can temper the octave without problems, too, if it is not too radical.

Here is an example, try tuning a synthesizer to this - I give you the wolf on C.

0: 0.000000 
1: 117.273616
2: 234.547232
3: 310.703874
4: 427.977490
5: 504.134131
6: 621.407747
7: 738.681364
8: 814.838005
9: 932.111621
10: 1008.268263
11: 1125.541879
12: 1201.698520

M3 is 386.86052 cents, P5 are 697.56439 cents, min3 are 310.70387 cents.


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## BabyGiraffe

Btw, the actual 12 notes meantone scale as generated by fifths, has 66 unique permutations of the scale steps (like there is neapolitan major, diatonic and melodic minor sharing the same intervals, but ordered in different ways), so there is plenty of choices for ideal intervals in a composition even with 12 keys in "meantone".


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## millionrainbows

BabyGiraffe said:


> Btw, the actual 12 notes *meantone scale *as generated by fifths,has 66 unique permutations of the scale steps (like there is neapolitan major, diatonic and melodic minor sharing the same intervals, but ordered in different ways), so there is plenty of choices for ideal intervals in a composition even with 12 keys in "meantone".


I don't get you, because there is no single meantone "scale". Stacking fifths doesn't close the octave. You need to explain what you're talking about.
Meantone *tuning* is a temperament of an octave.

If you look up "meantone scale" in WIK, it doesn't exist as a term.

That's because "meantone temperament" is a *musical temperament, that is a tuning system, obtained by slightly compromising the fifths in order to improve the thirds.*

Meantone temperaments are constructed the same way as Pythagorean tuning, as a stack of equal fifths, but in meantone each fifth is narrow compared to the perfect fifth of ratio 3:2.

You need to start using these terms in a more precise way.


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## BabyGiraffe

millionrainbows said:


> I don't get you, because there is no single meantone "scale". Stacking fifths doesn't close the octave. You need to explain what you're talking about.
> Meantone *tuning* is a temperament of an octave.
> 
> If you look up "meantone scale" in WIK, it doesn't exist as a term.
> 
> That's because "meantone temperament" is a *musical temperament, that is a tuning system, obtained by slightly compromising the fifths in order to improve the thirds.*
> 
> Meantone temperaments are constructed the same way as Pythagorean tuning, as a stack of equal fifths, but in meantone each fifth is narrow compared to the perfect fifth of ratio 3:2.
> 
> You need to start using these terms in a more precise way.


Meantone means MEAN TONE - 10/9 is equivalent to 9/8. This is mean tone. The generator can be anything that is tempered in meantone, so there is no syntonic comma!!!!!!!!!!!!!!!!!!!!!!!!!!!! Fifths are the canonical generator, but it doesn't have to be a fifth (Vicentino was the first that suggested a scale, generated by neutral third, splitting a fifth that is tempered in meantone, this is again a meantone)!
There is an infinite number of equal temperaments that do this. "Classical" meantone tunings are equivalent to them. 1/4 syntonic comma is equivalent to 31 equal, 1/6 comma is 55 equal, 1/5 is 43 equal.
1/11 comma is 12 equal (almost, actually fifths have to 699.something cents), 1/3 comma is 19 equal, Zarlino's 2/7 meantone is 50 equal.

http://www.tonalsoft.com/enc/number/2-7cmt.aspx

About scale vs temperament -> the scale was usually a chain of fifths, but like I said, the existing intervals of this 12 notes scale can be re-arranged into other scales that cannot be generated with stacked fifths (like you can't generated non-diatonic scales with this method). 
Btw, we can take random exotic notes from equal temperament that supports meantone and say that this is also meantone, but this is hardly what most people think about meantone.

I saw you linked books by Barbour and Partch, but it seem don't remember much of what you have read (if you did that) in these books.


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## millionrainbows

BabyGiraffe said:


> Meantone means MEAN TONE - 10/9 is equivalent to 9/8. This is mean tone.


I looked up "mean tone" and there is nothing. Even if I've read Partch, I still expect a good usage of terms.


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## millionrainbows

BabyGiraffe said:


> There is an infinite number of equal temperaments that do this. "Classical" meantone tunings are equivalent to them. 1/4 syntonic comma is equivalent to 31 equal, 1/6 comma is 55 equal, 1/5 is 43 equal.
> 1/11 comma is 12 equal (almost, actually fifths have to 699.something cents), 1/3 comma is 19 equal, Zarlino's 2/7 meantone is 50 equal.


I thought we'd discussed this before. The "equivalences" you cite are not mathematically precise, so this is some sort of "ball park" thinking you have developed. Where did this come from? Can you cite a source?


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## BabyGiraffe

millionrainbows said:


> I thought we'd discussed this before. The "equivalences" you cite are not mathematically precise, so this is some sort of "ball park" thinking you have developed. Where did this come from? Can you cite a source?


Dude, read the article I linked. 
"Using vector addition again to compare the 2/7-comma meantone "5th" with the 50-EDO "5th", we get a difference between the two of: ~0.189653305 cent = ~1/5 or ~11/58 cent"

Read also wikipedia articles on meantone.

"https://en.wikipedia.org/wiki/Quarter-comma_meantone"

"hTe perfect fifth of quarter-comma meantone, expressed as a fraction of an octave, is 1/4 log2 5. This number is irrational and in fact transcendental; hence a chain of meantone fifths, like a chain of pure 3:2 fifths, never closes (i.e. never equals a chain of octaves). However, the continued fraction approximations to this irrational fraction number allow us to find equal divisions of the octave which do close; the denominators of these are 1, 2, 5, 7, 12, 19, 31, 174, 205, 789 ... From this we find that 31 quarter-comma meantone fifths come close to closing, and conversely 31 equal temperament represents a good approximation to quarter-comma meantone"

You can compare intervals found in in 31 equal and 1/4 comma meantone and again it is 0.something cents. For all practical purposes it's 1/4 comma unless you want 205 or whatever equal.

Another article:

https://en.wikipedia.org/wiki/Meantone_temperament

Btw, there are precise mathematical definitions and tuning methods on xenharmonic wiki along code for algorithms, but these are way over the head of any normal person.


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## Wes Lachot

BabyGiraffe said:


> You can't have clarity, because there is no clarity - chords don't have inherent function (meaning) in isolation, only in relation with each other in a real composition employing some kind of systematic language (like sticking to certain patterns in rhythm, melody or harmony, not playing random sequences as heard in pseudo-randomly generated computer music or serialism)...


Hmmm. Clarity to me is more of a quest, a journey, than a destination. I agree that chords only have meaning in context--that's one thing I think we all can agree on. I do think that some languages are clearer than others, and my quest goes on. The quest for what supposedly "can't be had" is pretty much exactly what I have in mind, so thanks for the reminder.


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