Scale Degrees

The harmonization of any scale into triads always happens in the same way.
If you start from the major diatonic scale (Ionian mode), you will have the 7 degrees with their relative intervals:
I (tone) II (tone) III (semitone) IV (tone) V (tone) VI (tone) VII (tone) I ...
each of which is harmonized by a triad according to a formula like:

Agx=gx+g3x+g5x
with x ranging from 1 to 7

where Agx is the triad built on degree x (for example, if x=1, then Agx is the triad of the first degree), 3x is the degree following the degree following degree x, 5x is the degree following the degree following degree 3x (if, for example, x is degree V, then 3x is degree VII, and 5x is degree II).

Now let's take, for example, the C Ionian scale (the one you indicated): the triad built on degree VII will be, according to the previous formula:
AVII= VII+II+IV, which means B+D+F = B diminished chord

The definition by which a triad is called major, minor, diminished, or augmented—I assume you already know it, but anyway:
triads are chords formed by the superposition of two harmonic intervals of a third and are called:
major if the lower interval is a major third and the higher interval is a minor third (0,4,7)
minor if the lower interval is a minor third and the higher interval is a major third (0,3,7)
diminished if both intervals are a minor third (0,3,6)
augmented if both intervals are a major third (0,4,8)

Perhaps stated this way it will seem a bit complicated, but it serves to help you understand that in this manner, not only the major scale is harmonized, but any scale of any known mode. And not just for triads, but also for sevenths, etc... For example, even the Aeolian scale (that is, the natural minor scale), which is nothing other than the sixth mode of the Ionian scale. Known and unknown, I would add: if you want to create your own scale, you can follow this scheme to harmonize it.

PS: regarding your other post, be careful because Anglo-Saxons call I IV V a "chord progression" regardless, whereas here we generally mean by (harmonic) progression an ascending or descending transposition of a sequence (harmonic)... but as usual, it is a matter of definitions.

Those chords are formed using only the notes belonging to the scale.

That is, if you take a C major scale (C-D-E-F-G-A-B), on each degree you can build "diatonic" or "natural" chords, meaning they are built using only the notes present in the scale. From which: CM - Dm - Em - F - G - Am - Bdim.

Obviously, all of this makes sense within the tonal system...

In fact, things get complicated when we talk about minor modes, where the influences of "modality" are greater. In fact, on some degrees there is not just one diatonic chord, given the presence of alterable degrees (as I imagine you know, in minor scales there is the possibility of altering the sixth and seventh degrees in an ascending direction).

That being said, for Schoenberg all this discussion would make no sense, and in reality, in musical literature, it has "little" [relevance]. There are some chords, traditionally considered altered, that actually appear very frequently. The chord on the second degree of the major scale is very often Major, as it functions as a dominant of the dominant (meaning it works as a tonicization, a momentary modulation that increases the sense of tension toward the destination). If you are in C major and want to do an I - II - V - I sequence, you can decide to make the D chord major (II+) and, in this way, act as if the II+-V progression in C is a V-I in G (that is, for example, DM - GM).

In the same way, the problem of natural chords arises with seventh chords. The dominant seventh chord should be considered an alien chord to the key, since it has an altered note (the seventh, precisely, which is minor). But talking about it in these terms means nothing, given that we are talking about the most used seventh chord in musical literature.

This is to say that tonality is more of a set of forces than a list of right and wrong notes.

Thank you so much Bianca and Thallo!

So if I understood correctly, the schematic formula that Bianca is talking about applies to all major scales, so I will have:

C Major

CM - Dm - Em - FM - GM - Am - Bdim - CM

C# Major

C#M - D#m - E#m - F#M - G#M - A#m - Cdim - C#M

D Major

DM - Em - F#m - GM - AM - Bm - C#dim - DM

D# Major

D#M - E#m - G#m - G#M - A#M - Cm - Ddim - D#M

E Major

EM - F#m - G#m - AM - Bm - C#m - D#dim - EM

E# Major

E#M - G#m - Am - A#M - CM - Dm - Ddim - E#M

F Major

FM - Gm - Am - BbM - CM - Dm - Ddim - FM

F# Major

F#M - G#m - A#m - Bm - C#M - D#m - Fdim - F#M

G Major

GM - Am - Bm - CM - DM - Em - F#dim - GM

G# Major

G#M - A#m - Cm - C#M - D#M - E#m - Gdim - G#M

A Major

AM - Bm - C#m - DM - EM - F#m - G#dim - AM

A# Major

A#M - Cm - Dm - D#M - EM - Gm - Adim - A#M

B Major

BM - C#m - D#m - EM - F#M - G#m - A#dim - BM

So Thallo, you are saying that if I am in C major and I want to do a I - II - V - I sequence, I can decide to make the D chord major, without necessarily following the formula that

If in the I II V I sequence in C major you use D major instead of D minor... the degrees change. It is called substitution, so: I V/V V I. Therefore tonic area, dominant, tonic.

Sorry, I'm not very clear on this thing, the harmonic sequence I II V I always follows the Tonic - Subdominant - Dominant succession

since in the subdominant group there is the II degree... I can't quite understand it well

If you were truly on a II of C major, you should find a D minor. Since you have a D major, you must relate the chord to the next one, which is G major (V of C major). If you look for the D major chord in G major, you will discover that you are on a V. Therefore, you are tonicizing, using a secondary dominant, by making a substitution. Thus I V/V (FIFTH OF THE FIFTH) V I.

but V/V (fifth of the fifth) is always D Major? which would be the dominant of G, right?

writing I - V/V - V - I will I have C - Dm - G - C?

Fox wrote:

Thank you so much Bianca and Thallo!

So if I understood correctly, the schematic formula that Bianca is talking about applies to all major order scales, so I will have:

It applies to all scales of all modes. Just be careful that if you take, for example, the melodic minor scale, you must distinguish the ascending part from the descending one and therefore there will be two different harmonizations (note that the ascending melodic minor is a fundamental mode -in current modal theory-, while the natural minor (aeolian) is the sixth mode of the major (ionian), which however has nothing to do with harmonization). Try to perform a quick check yourself by harmonizing some modes (you can find a lot of material online).

Furthermore, note that it is also possible to harmonize scales not only with the principle of thirds, but also with the principle of seconds or fourths (always using triads, sevenths, etc...), still following the same logic, but that was not your initial question, just as the possibility of inserting chords into a sequence that do not belong to the harmonized scale you started from was not either, which seems obvious to me. The techniques with which to choose chords outside the scale are varied, but in the end, it is always up to your imagination and your skill to choose which ones.

Thank you very much everyone, you are very knowledgeable and kind but I continue not to understand one thing

If the C major sequence is the following: CM - Dm - Em - FM - GM - Am - Bdim - Cm

what is the A minor one?

There are things I don't agree with...

Bianca is always a bit "naïve" in her approach, and together with Marzapane, she mixes modal and tonal terminologies.

Let's clarify one thing: harmony, in all its elements, follows a historicized path. This means that that set of phenomena which we today consider "the rules of harmony" had a precise development, which we cannot analyze according to rules invented on the spot. Bianca's scheme regarding the rules for harmonizing scales is not only complicated but is incorrect, simply because it does not consider what the TRUE development of tonal harmonization was, which exists and is a studied and traceable historical phenomenon.

You cannot talk about it in a single post, but the summary is this: the rules of modal polyphony (which have nothing to do with modal scales, such as Aeolian or Ionian modes, which are inventions of Glareanus later adopted by jazz) identified certain movements called cadences. First cadences are born, and then chords. Chords gradually become autonomous entities, just like tonalities, but in her approach to the discourse, it is Schoenberg who got it most right: every tonality has a very high number of "belonging" chords. In a tonal piece written in the classical period, I can find Neapolitan triads, augmented sixths, alternation between major and minor chords, augmented triads, diminished sevenths, and an entire series of chords that have nothing natural about them. Identifying a limited number of chords that can be built on a scale is a purely didactic matter.

Even in another recent discussion of yours, Fox, I found answers that spoke extensively about modal theory. At this point, I want to understand if you were the one who asked for references of that type, because otherwise I am surprised that it is "straying" there. Maybe the number of users who like jazz is simply increasing, I don't know :-)

If you are a student, as you seem, I think it is right to plant a seed of doubt in your mind: Gregorian modality, Renaissance modality, jazz modality, and classical tonality are all 4 DIFFERENT things. The idea, for example, that you can "harmonize" in seconds or fourths, as Bianca said, is nonsense if we are talking about tonality, and it is absolutely normal if we are talking about jazz and, sometimes, ancient polyphony.

Yes, I am a classical harmony student; I don't know jazz and I have no background in the subject, I only study harmony at high school but I have many gaps.

So, if I understood correctly, harmonizing with seconds and fourths is normal in jazz but not in classical tonality.

What I have understood is that the harmonic sequence I IV V I is very ancient and expresses:

I rest

IV momentum towards tension

V tension

I Rest

then I understood that there are groups or areas, namely:

Tonic

Subdominant

Dominant

the tonic group includes the I and VI chords of the scale

The subdominant group includes the II and the IV

The dominant group includes the V and the VII

and I understood that I can create different combinations, using the degrees belonging to each group and following that specific sequence (tonic-subdominant-dominant)

Then I understood that major scales are formed by the distance of: W - W - H - W - W - W - H

C Major
CM - Dm - Em - F - G - Am - Bdim - CM

I understood that D is minor because if it were major there would be two alterations, namely F# and C#, which are not present in the C major scale; the same applies to the other degrees, (CM - Dm - Em - F - G - Am - Bdim - CM) I hope I understood correctly... because this thing is still not very clear to me

On the other hand, I haven't understood how a minor scale (melodic and harmonic) is constructed.

For example, I know that the melodic minor scale is formed by the distance of: W - H - W - W - W - W - H (ascending) W - H - W - W - W - W - H (descending)

but I don't know how the degrees are formed, I would say it like this: Am - Bdim - CM - Dm - Em - F - G#dim - Am (ascending)
Am - G#dim - F - Em - Dm - CM - Bdim - Am (descending)

but maybe I've made a mess, I don't know, I'm confused!

thanks anyway for the help!!

What I have understood is that the harmonic sequence I IV V I is very ancient and expresses:

I rest

IV momentum towards tension

V tension

I Rest

Let's set aside the fact that it is ancient...

A cadence, in the tonal system, is a sequence of chords that marks an 'articulation'. It is as if it were a punctuation mark, a comma, or a semicolon, or a period.

There are many types of cadences; the V-I cadence (with both chords in root position) is called a perfect cadence. When you add other chords, like IV-V-I, it is called a compound cadence. But in every music theory book, you will find even better definitions than these.

The "meanings" you have given to the chords make sense but should be treated with caution. They are not always part of "classical" tonal theory, but they are often the result of reflections by specific theorists.

Functional theory, for example, considers chords (and also scale degrees, and thus individual notes) as "functions," meaning as elements of a very precise discourse. There are three functions: tonic, subdominant, and dominant. EVERY possible chord can be indicated as a function, even if ambiguous chords exist... functional theory becomes interesting precisely when it deals with complicated situations, especially analytical ones, in which it interprets functions within complex compositions.

At a "basic" level, the chord on the first degree has a tonic function, as does the one on the third. The dominant chord and the leading-tone chord have a dominant function. Chords on the second, fourth, and sixth degrees have a subdominant function. In reality, things are much more complicated than this, and functional theory interprets many other chords, including those I was talking about (the "foreign" chords).

The rest is substantially correct.

Regarding minor scales, there are essentially three types: the natural minor scale, the harmonic minor, and the melodic minor.

The natural minor scale has a minor third and the other natural degrees, i.e., following the list of notes given by the key signature. For example, a natural D minor will have only Bb as an altered note. The other notes will be natural, as indicated by the key signature.

The harmonic minor scale alters the seventh degree to allow for the formation of a major dominant chord. You will notice, in fact, that in natural D minor, the diatonic dominant chord formed with the scale degrees is A-C-E, which is a minor triad. To allow for the presence of a major dominant chord (i.e., the presence of leading-tone tension), the C must be altered to make it C#. Thus, harmonic D minor will have D E F G A Bb C# D.

The melodic minor scale resolves the problem of the augmented second in the harmonic minor scale. Between Bb and C#, there is, in fact, an augmented second, a very dissonant leap. To avoid this, the Bb is also altered to make it B natural. But since this would completely eliminate the key signature, this alteration is maintained only in the ascending scale. The ascending melodic minor scale will, therefore, be D E F G A B C# D, and the descending one D C Bb A G F E D.

Infinite thanks thallo, this comparison was truly useful to me!!

thallo wrote:

Let's make one thing clear: harmony, in all its elements, follows a historicized path. This means that that set of phenomena which we today consider "the rules of harmony" had a precise development, which we cannot analyze according to rules invented on the spot. Bianca's scheme regarding the rules of scale harmonization is not only complicated but is incorrect, simply because it does not consider what the TRUE development of tonal harmonization has been, which exists and is a studied and traceable historical phenomenon.

it is not incorrect, precisely because it works. if you look at any harmonized scale today, you have proof of it

and it isn't even complicated, it is just expressed in a synthetic and generalizable form, not to say "put a finger on the first note, then skip one," etc.

the fact that it historically developed in a different way is certain, but that is not what fox was asking.

It simply seems to me that he wanted to understand why the harmonized scales he sees are made this way and not otherwise. Harmonic functions or ancient modality are of no use to him in answering his question, so much so that he asks "but then how is the A scale harmonized..."

Just as it is obvious that harmonization by seconds etc., is not used to study tonal harmony, but only to understand that even that is a convention...

Yes, I am interested in both aspects, the one related to tonal music and the harmonic functions that determine its purposes (which I have now understood)

also the formula that establishes its origin. Initially, I had not understood either one, but now everything is much clearer!

I was also unaware of the existence of modal music, which I will certainly delve into in the future because you have made me so curious, but first I must study tonal music in depth..

thanks!

Bianca, the Pastafarian church claims that the hole in the ozone layer was caused by the decrease of pirates in the world. And all historical data says fewer pirates = more pollution.

Despite this, there are no scientific links between the two things, only a cause-effect link invented and supported by public opinion :-)

I believe there are infinite possible models to decipher tonal harmony, to formalize it. But tonal harmony HAS its own formalization, and when I say "its own" I mean a historically established formalization. It is true that the issue is not so clear-cut, that there are different opinions, different weights. But seeing harmony as a synchronic subject, as arithmetic could be, is WRONG. Harmony is a diachronic subject, which means its rules change over time, and it is wrong to experience it as a set of always identical rules.

Then, if you want to create a little scheme for your own use because it's easier for you or I don't know, you can do it, but you must be aware that there EXISTS a historical reality regarding the development of harmony.

I mean, guys, out there are still plenty of people convinced that the Italian augmented sixth and the German augmented sixth are built on the augmented fourth!!??

P.S. for Fox

I write nonsense too :-) rereading I saw that I told you the VI degree has a subdominant function. Not quite... I mean, it is one of those ambiguous degrees, it can also have a tonic function. I don't remember if there is a discussion on functionalism on the forum, there surely will be. But with these arguments, one can easily get indigestion :-)

thallo wrote:

I believe there are infinite possible models to decipher tonal harmony, to formalize it. But tonal harmony HAS its own formalization, and when I say "its own" I mean a historically established formalization. It is true that the issue is not so clear-cut, that there are different opinions, different weights. But seeing harmony as a synchronic subject, as arithmetic could be, is WRONG. Harmony is a diachronic subject, which means its rules change over time, and it is wrong to experience it as a set of always identical rules.

Then, if you want to create a little scheme for your own use because it's easier for you or something, you can do it, but you must be aware that there EXISTS a historical reality regarding the development of harmony.

I mean, guys, out there there are still plenty of people convinced that the Italian augmented sixth and the German augmented sixth are built on the augmented fourth!!??

Thallo, but I agree with what you say regarding historical development, and if you read it, I've already said it (parenthetically, studying arithmetic as a "synchronic" subject leads to not understanding arithmetic well). The harmonization by triads of a scale, which you find left and right, doesn't tell you much about harmony, yet it is used everywhere (and not just in "light" music). And the way it is constructed is that (and it certainly isn't naive), in the sense that even if you don't know a thing about history, you will still build scales following that scheme. It is not a universal law obviously, but a practical scheme used by musicians, not musicologists.

Then one can argue as much as one wants, but in fact that is what it is, regardless of whether harmony developed this way or that. For someone approaching the subject (say, a guitarist? someone studying improvisation? it seems more than legitimate to ask how one practically harmonizes a scale without starting from the Vedas).

Another thing is asking about the reason for a cadence or a sequence, a philologically much more interesting discussion, and another thing is studying harmony.

Fox seems interested in both aspects; for the first, he already has an answer (though he will need a lot of practice and exercise), for the second, he will need a bit more time.

I mean, is this formula

Agx=gx+g3x+g5x
with x ranging from 1 to 7

more immediate for a guitarist???

Further down you put numbers in parentheses (without explaining them, for that matter), and that is the notation of Allen Forte's Set Theory. Set Theory has NOTHING to do with the "classical" morphology of tonal chords; it is a pretentious way to complicate the situation. And to complicate it a lot, because (0, 2, 6, 9), which is the formula for the dominant seventh, according to Set Theory is given in third inversion, because that is the most compressed one. You didn't want to talk about Set Theory? You were unlucky to use that formalization, then, because someone before you in the history of harmony has already used it, giving it a totally different connotation.

I would reflect on the fact, for example, that for Pitch-class Set Theory, the major triad is equal to the minor one

Anyway, it is a merely descriptive analysis