What exactly are interval class vectors?

Good hour,

I posted my first question here recently regarding interval ratios and their relationships. A user here provided me with great responses which within the array of knowledge, she mentioned "interval class vector". I went ahead and did some research on it and it's really begun to intrigue me because i think it can play a significant role in my understanding of music.

So far all i understand about it is that it holds 6 interval classes including their inversions..

• 1. Minor 2nd/Major 7th
• 2. Major 2nd/Minor 7th
• 3. Minor 3rd/Major 6th
• 4. Major 3rd/Minor 6th
• 5. Perfect 4th/Perfect 5th
• 6. Tritone

That is absolutely all i know about interval class vectors. I don't understand how it works, what its exact purpose is nor what does it mean.

So my question basically is could you kindly and in layman terms explain to me

1. The purpose
2. How it functions
3. and what does it mean for musical composition

Thanks guys!

Can't really help - that's a good start, innit? because I can't see how the term "vector" fits in (which traditionally is an attribute/movement having magnitude and specific direction). It seems to have been borrowed by all kinds of things though to mean something different and lend them an air of authority. I've been composing music for some time, well aware of intervals and their inversions (how can one not be?!) but never had to classify them or consider more of a trajectory except in movement to the next (musical) event: chord? note? 

But there's a department somewhere that comes up with this sort of minutiae and no doubt there'll be students of same who can answer your question and explain exactly why "vector" to me. 

(I believe I need enough theory to do what I have to do compositionally, no more!)

Yes, the interval vector, represented between this marks  <   > tells you how many intervals of each class there are in a given PC set.

Let's say you have a PC set with this interval vector:  <320102>

That means that in your PC set you have 3 minor second, 2 major second, 0 minor 3, 1 major 3, 0 fourthfs/fifths, 2 tritones.

This gives you a quick information that your PC set is highly dissonant (having seconds and tritones, and less o zero thirds...).

The interval vector gives also information for composing: an interval such as <223232> is very "plain", there are no intervals predominating. Instead, an interval vector such as <400301> has much more "personality" (more contrasting) regarding the more frequent intervals.

Another useful issue about intervals vectors for composition is that it tells you how many notes the original PC set has in common with its transpositions. This is important because if you want a smooth transition from a set to a transposition, you have to choose sets with common tones. If you want a "hard" transitions, use transpositions with no common tones.

In the example, the original set of three notes (018) has been transposed into its six possible sets. The interval vectos <100110> tells you that the original set has 1 note in common with the first transposition, 0 in common with the second and third transposition, 1 note in common with the fourth and fifth, and 0 notes in common with the last transposition.

@Quinn It's a vector because it displays varying magnitudes (in this case, intervals in a set) on a singular matrix structure.
 

17 hours ago, Quinn said:

(I believe I need enough theory to do what I have to do compositionally, no more!)

You have to realize: music theory, effectively, is a way for musicians to be lazy. We don't have time to count individual pitches that sound good. Having a system in place to describe them easily helps.
There's a second point in music theory, where it becomes mostly analytical, while still attempting to create systems. Composers often leave this behind (though some don't, like Carter and Babbit), but that's why it's a field unto itself. You can use the concepts for composition, but they're a means of understanding something beyond performance practice or orchestration. Simply speaking, it's great you have what you need to know and like many people stop when it no longer becomes compositionally functional, but there are people who take it further as means of further insight into existing works. Both viewpoints are valid. 

Many systems that were born in the 20th century are different ways to organize music material.

We are accustomed to tonal hierarchy and that's the system almost everybody uses by default. But there are other systems.

Atonal dodecaphony takes the row as a unit to rely on.

PC Sets take the interval relationship between a series of pitches as the base. That relationship is fixed along the use of the set. I'm aware many people reject these kind of systems because they don't sound, logically, as the tonal hierarchy. Interval relationship provides a different feeling. I think that comparing PC sets, or whatever, with tonal music is mistake. It's as if you compare realistic painting with cubism.

PC Set musicians and theorist developed a corpus of operations for composing. I agree with the opinion that theory goes, sometimes, too far. At some point it's useless (for me, at least) such complex operations. I don't see how they can be translated into music composition. But that happens also with tonal music. At some point, it seems that theory takes its own path. Which is good, as theory itself.

Contemporary music brought many new systems to organize music, not only from a harmonic point of view, but also in terms of new forms.

In the 80's another system was devised, it's called Tone Clock System. "It encourages the composer to generate all their harmonic and melodic material from a limited number of intervallic configurations"...

All sounds a bit like articles published in Die Reihe, a contemporary music periodical published between roughly 1957 and 1968 (English version). All too often the composers' commentaries were far more interesting to read than the music was to 'listen' to if a performance was available. One wondered what those composers were aiming to achieve if listening was expected. 

EDIT: It got me wondering what purpose music serves. A listener is concerned with the final product and matters of taste and preference come into its reception. However, for composers using algorithmic means or aleatoric (hoping you get what I mean - some formulaic system of their invention), they derive satisfaction from the process of composing (presumably in the knowledge that any communication commonality between composer and listener would be coincidental). A big problem I had with systems like serialism was that too much of a composition was predetermined at the point of setting up the basic thematic material, the row, whatever you want to call it.

On 9/25/2019 at 7:37 AM, Seery said:

she mentioned "interval class vector"

Did she now?

God where do people come up with these goddamn names? I'm serious, there's stuff like "Trimodular block" and so on, I don't know. It's like people want to seem like they're talking about spaceships and lasers rather than some boringass cadence some dead guy wrote centuries ago. But hey, whatever, at least it sounds exciting.

11 minutes ago, SSC said:

God where do people come up with these goddamn names? I'm serious, there's stuff like "Trimodular block" and so on, I don't know. It's like people want to seem like they're talking about spaceships and lasers rather than some boringass cadence some dead guy wrote centuries ago. But hey, whatever, at least it sounds exciting.

lol, well i'm certain that you do know that we just borrowed a lot of terms from math. Maybe your point is why bother using math to describe music, and I feel like it's just be concise about what you want to say about it. It definitely can be heavy-handed, though, I agree.

2 minutes ago, Monarcheon said:

lol, well i'm certain that you do know that we just borrowed a lot of terms from math. Maybe your point is why bother using math to describe music, and I feel like it's just be concise about what you want to say about it. It definitely can be heavy-handed, though, I agree.

 

My point is mostly, I've never found myself in a situation where I would need to even think of something like that. Additionally, this, I think, is something that happens almost exclusively in English, since in Spanish, Portuguese and German I have absolutely no examples of anything even resembling "Interval Class Vectors," or whatever. I would go as far as to say that in my experience, talking about music theory in English is probably one of my least favorite things to do since the jargon is fancypants for no real reason, lol.

11 minutes ago, SSC said:

My point is mostly, I've never found myself in a situation where I would need to even think of something like that. Additionally, this, I think, is something that happens almost exclusively in English, since in Spanish, Portuguese and German I have absolutely no examples of anything even resembling "Interval Class Vectors," or whatever. I would go as far as to say that in my experience, talking about music theory in English is probably one of my least favorite things to do since the jargon is fancypants for no real reason, lol.

 

Interesting! I'm pretty proficient in Spanish, but never got so specific about music. The German class we're required to take for the theory masters is going to focus on music treatises so I'm curious to see how much extra jargon there is to translate. 

Se say vector interválico.

Many times translaciones are tricky. For example triad and triada is not the same.

20 minutes ago, Luis Hernández said:

For example triad and triada is not the same.

What? They're not?

Tríade -> Tríada -> Dreiklang -> Triad...?

They all mean a 3 note chord, unless there's some secret I'm not aware of. Also, the only references I can find for "vector interválico" are Spanish translations of stuff that's originally in English, lol.

@SSC   

No. Not the same.

Triad is a three note chords BY THIRDS (major or minor)

But in Spanish (my language) Tríada is a chord of three notes (whatever interval among them). That's confusing if you don't know it. It happened to me when I spoke here of triads (assuming a chord by fourths or whatever) and people told me I was wrong.

On the other hand, vector interválico is tha valid name for that concept. Vector and interválico are words in Spanish, they're not neologisms. In fact, they come from Latin.

The reason this doesn't show up usually is because even using the word "triad" no matter the language it's almost always to talk about, well, a triad lol. I don't think I've used "tríada" or whatever to talk about random 3 note chords that aren't primary or secondary triads to begin with.

Though, according to spanish wikipedia:

Quote

Una triada es un acorde de 3 notas, usualmente en intervalos de terceras; fundamental, tercera y quinta.

Existen cuatro tipos de triadas principales, determinadas por las relaciones interválicas entre las notas que forman el acorde. Son las denominadas triadas comunes:

https://es.wikipedia.org/wiki/Tríada_(música)

For contrast, grove:

Quote

A chord consisting of three notes which can be arranged to form two superimposed 3rds. If the lower 3rd is major and the upper 3rd is minor, the triad is said to be major (C–E–G); if the lower 3rd is minor and the upper 3rd is major, the triad is minor (C–E−G). If both 3rds are major the triad is augmented (C–E–G), and if both 3rds are minor the triad is diminished (C–E−G).

I don't really see the difference, except on the wording. In the spanish article it's said it's usually in thirds, but then both proceed to list the same 4 types of triads...

Hell, in fact:

Quote

Als Dreiklang wird in der Musik ein dreitöniger Akkord bezeichnet, der im einfachsten Fall aus zwei übereinandergeschichteten Terz-Intervallen besteht.

https://de.wikipedia.org/wiki/Dreiklang

In German it's the same thing. It says it's the easiest case where it's 3rds, but I guess it doesn't say it's ONLY that. ... But then again, I've not seen an english definition that specifically says it can only be that either. The Grove definition says they CAN be arranged that way.

I'm not seeing where's the difference, honestly, between the languages.

Quote

Some twentieth-century theorists, notably Howard Hanson^[2] and Carlton Gamer,^[3] expand the term to refer to any combination of three different pitches, regardless of the intervals amongst them. The word used by other theorists for this more general concept is "trichord".^[4] Others, notably Allen Forte, use the term to refer to combinations apparently stacked of other intervals, as in "quartal triad".^[5]

On the english wikipedia this shows up. Go figure people don't agree on what things are called haha.

A chord is any combination of pitches (no matter the intervals involved).

A triad is a three note chord.

A triage unit is the emergency room (ER) found in your local hospital.

Seriously though, why stress over these types of topics. Just write your music. Leave the analysis and over-analysis to the theorists. Let them make sense of what you write and how it works -that's what they get paid for (?). If a combination of notes sounds interesting to you, explore it. 

4 hours ago, jawoodruff said:

A chord is any combination of pitches (no matter the intervals involved).

A triad is a three note chord.

A triage unit is the emergency room (ER) found in your local hospital.

Seriously though, why stress over these types of topics. Just write your music. Leave the analysis and over-analysis to the theorists. Let them make sense of what you write and how it works -that's what they get paid for (?). If a combination of notes sounds interesting to you, explore it. 

 

Lemme just guzzle that tea you're pouring.

I actually madly respect anyone who studies music enough to reach such advanced levels of knowledge and analysis but I seriously don't think I have the temperament for that sort of thing.

1 minute ago, KJthesleepdeprived said:

I actually madly respect anyone who studies music enough to reach such advanced levels of knowledge and analysis but I seriously don't think I have the temperament for that sort of thing.

 

It's one thing to know the material (and you should) but it's another to stress over it like it's a golden grail. Hell, most of the CP period was spent with composer's doing everything they could to buck the prevailing system of music theory. Yes, Mozart knew it... but his chromaticism (at times) slapped CP theory in the face!