# Thoughts on Hypomodes



## Mr. Big Noodles (Jul 8, 2014)

I&#8217;ve been fooling around with some ideas regarding melody, harmony, voice leading and tonality lately and I wanted to share something that I came up with. I&#8217;ll probably lose everybody, and I&#8217;m kind of rambling, but what the hell.

Some of you may have seen my posts on this forum regarding the use of tetrachords to conceptualize a scale. This is perhaps an oversimplification of my views on harmony, since I really could not care any less about scales and modes. Rather, I would like to posit that tonality must emanate from a tonic, from the &#8220;1&#8221;, that every pitch bears a relationship and a harmonic function relative to the tonic, and that there is a tonal gravity that pulls every pitch toward the tonic. Here is what a scale looks like if you look it up in a book:

1234567

Here is what the tetrachordal scales look like if you follow the link I posted above:

5671234

See that? I put the &#8220;1&#8221; in the middle. Getting away from the numbers, this is what the tendency of harmony looks like:

>>>1<<<

The emphasis is on the 1, but coming from both directions. This is how the guitarist and pretty much every other musician in the world learns and practices scales:

1234567123456712345671765432176543217654321

Three octaves up and then down, right? It&#8217;s a good technical exercise, but don&#8217;t mistaken it for music theory. I&#8217;m not talking about learning and playing scales, but rather learning the concept of the tonic. In tonal music, it needs to be the most important pitch, and we must understand how to approach it. See the tonic as the tone from which all other tones emanate, the center of a tonal cosmos.

If the tonic is at the center, then what is at the edge? I consider the dominant to be the boundary of the pitch space.

5>>1<<<5

You can also bounce back towards the dominant, to prepare for the return to the tonic:

5<<1>>>5

While I am not well-versed in the history of the subject, I believe that this is how the Greek hypomodes were conceived: you stick the tonic at the center, then give yourself a little bit of room to go above the tonic as well as below. The dominant octave, because of its harmonic strength, gives the ear a nice signpost. Hanging onto the dominant is the next best thing to being resolved at the tonic.

Given this definition of the hypomode, the two solid foundations are the tonic and dominant, and anything can be built around that. You can stick with a single pitch collection, or use a pitch collection that is not built of static pitches. The melodic minor is one of those that does not have static pitches: the lower tetrachord (1 2 &#9837;3 4) never changes, but the upper tetrachord has two forms - an ascending version (5 6 7 1) and a descending version (1 &#9837;7 &#9837;6 5).

I have also discussed Bartók&#8217;s lydian/phrygian polymodal chromaticism in the past. That is another system that uses different ascending and descending pitches. It is notable in that the textbook use of the technique results in the use of all twelve chromatic pitches, but the tonic and dominant are constant throughout.

1 2 3 #4 5 6 7 1 &#9837;7 &#9837;6 5 4 &#9837;3 &#9837;2 1

This confirms that every chromatic pitch is viable at any given time, provided that the tonic-dominant framework is still there. With this in mind, our modal palette really opens up. Try playing around with hybrid modes: use the lower tetrachord of the phrygian mode and the upper tetrachord of the dorian mode:

5 6 &#9837;7 1 &#9837;2 &#9837;3 4

Or maybe use the tetrachords of those modes interchangeably as an ascending/descending set. Or create a hybrid of numerous modes:







So far, we&#8217;re doing little more than building scales. However, I think there is a way to extend harmonic function using the hypomode as justification. First, we&#8217;ll start with the hypomode of A minor.






I have both the natural and melodic minor here. We&#8217;re going to go with the melodic minor, because it allows us to alter the dominant degree. Since I have two dominants in the hypomode, I can alter the dominant in two ways: an ascending form (#5) and a descending form (&#9837;5). Maybe ascending/descending is not the best terminology, since I am thinking that the two dominants are more of a boundary than something you simply pass by.






Some of you might have noticed that this is the octatonic scale. This was a coincidence for me, but noticing this fact caused me to think for a while. We often hear about the octatonic scale in discussions about symmetric scales and chord-scale mumbo-jumbo (where it is analogous to a diminished seventh chord, or a dominant seventh chord with a bunch of alterations in the rotated form), but to my ear I am finding that as long as I hear that E# and E&#9837; as some sort of &#8220;5&#8221;, the tonic is still very clear (which is not usually the case with symmetric scales). This opens up harmonic and melodic possibilities. You can swap the placement of the two altered 5&#8217;s too, which leads to some interesting melodic lines:






I started thinking of how this would affect harmony, since harmony can be derived from a pitch collection, and I&#8217;ve gotten some gnarly chords by turning the regular dominant seventh chord into a dominant seventh chord with a doubly altered root.

5 7 2 4 > &#9837;5 7 2 4 #5

The #5 will come across as &#9837;6, which is a minor ninth in the traditional dominant chord. However, 6 can be added for extra crunchiness, though it is difficult to do so with just one guitar.

This is the bread and butter 7(&#9837;9) chord:


```
E7(&#9837;9)

e-
B-6
G-7
D-6
A-7
E-
```
And here are a couple of voicings of this split-altered-root dominant seventh chord:


```
E&#9837; in the bass

e-
B-6
G-7
D-6
A-6
E-
```


```
E# in the bass

e-
B-4
G-7
D-6
A-8
E-
```
I don&#8217;t know how one would begin to name those. To me, they're still E7... with two roots, neither of which are E. I run into this problem from time to time. Standard notation for the win.

These are surprisingly delicate chords. They are difficult to resolve into plain old Am triad satisfactorily because they don&#8217;t have the normal &#8220;dominant seventh&#8221; dissonance, but if you want a spicier tonic chord, I think the added harmonic richness works well with added and altered tones on the tonic chord. Might be good for somebody looking for new harmonies. Build chords from the tones that extend outward from the tonal center rather than stack a bunch of stuff on top of a root that is predetermined by a scale.

More to come if I feel like it.


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## Solodini (Jul 8, 2014)

How do you see yourself applying this stuff, then?


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## Mr. Big Noodles (Jul 8, 2014)

In a way that gets me into girls' pants.


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## MrPepperoniNipples (Jul 8, 2014)

Mr. Big Noodles said:


> In a way that gets me into girls' pants.



that's a good way


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## Mr. Big Noodles (Jul 8, 2014)

In all honesty though, this is a big part of how I choose pitches.


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## Solodini (Jul 8, 2014)

Hypomodes or getting into girls' pants?


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## AugmentedFourth (Jul 8, 2014)

I think that potentially you could make those chords resolve better by somehow reintroducing the regular old 5 (E&#9838. Like maybe if you did


```
e||--------------------------|
B||-----------4--------------|
G||--------7-----7-----------|
D||-----6-----------6--------|
A||--8-----------------8--7--|
E||--------------------------|


---------------------------||
---------------------------||
-----------5--------------*||
--------7-----7-----------*||
-----7-----------7-----7---||
--5-----------------5------||
```

That kind of constructs a simple V - I loop with a split root on the V. The other chord with the root as E&#9837; is a bit more obtuse but may work the same way. Oddly enough though I think the one with E&#9837; as the bass sounds more like E7 than the E# one does, which is kind of in imminent danger of sounding like a borrowed Fm13. Or maybe I have no idea what I'm doing. 

I'll try to mess around with this later, see what I come up with.


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## Mr. Big Noodles (Jul 8, 2014)

Solodini said:


> Hypomodes or getting into girls' pants?



Ah, got me there, you tricky jackal you!

Contrary motion and tri/tetra/pentachords inevitably mean that you're going to run into hypomodes at some point. I think of it as surrounding a pitch on both sides.



AugmentedFourth said:


> I think that potentially you could make those chords resolve better by somehow reintroducing the regular old 5 (E&#9838. Like maybe if you did
> 
> 
> ```
> ...



I don't think it's necessary to reintroduce E during the dominant chord. The smoothest resolution to a tonic triad is this:


```
e-
B-4-5
G-7-5
D-6-7
A-8-7
E-
```
Root position works too.


```
e-
B-4-5
G-7-5
D-6-7
A-8-0
E-
```



> The other chord with the root as E&#9837; is a bit more obtuse but may work the same way. Oddly enough though I think the one with E&#9837; as the bass sounds more like E7 than the E# one does, which is kind of in imminent danger of sounding like a borrowed Fm13.


I agree with you. Having the &#9837;9 above the &#9837;1 makes it sound kinda lame. Still, it's a tight voicing that's not a pain in the ass to finger. Given the correct context and proper resolution, it works.


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## OmegaSlayer (Jul 8, 2014)

Mr. Big Noodles said:


> In a way that gets me into girls' pants.





Mr. Big Noodles said:


> In all honesty though, this is a big part of how I choose pitches.



Are you sure you didn't mistyped pitches and put a p instead of a b?


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## Solodini (Jul 8, 2014)

Mr. Big Noodles said:


> Still, it's a tight voicing that's not a pain in the ass to finger. Given the correct context and proper resolution, it works.


 
Are we still talking about hypomodes?


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## OmegaSlayer (Jul 8, 2014)




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## Cyn__Theia (Jul 8, 2014)

Way too elevated for this thread right now. 

Hahaha. 

Seriously, though, thanks for sharing, Mr. Big. Good insights and all that. A bit more technical than what I've been grounding myself with and applying as of late, but still really relevant to what I've been looking into and learning recently.

_"See the tonic as the tone from which all other tones emanate, *the center of a tonal cosmos*." _

This is nearly a spiritual sentiment for me.


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## Mr. Big Noodles (Jul 8, 2014)

Now you know how I feel every time I hear a note. Each one, so precious and unique, like a tiny little snowflake. Tears well in my eyes, the beauty is so difficult to bear.


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## viesczy (Jul 8, 2014)

Mr. Big Noodles said:


> In a way that gets me into girls' pants.



Good luck!

Every farking show, girls be all on the drummer and the singer, the two least talented people in the band.

Me, I'd get the guys wanting to talk gear or theory or technique! Get lost, but leave your GF here! 

Derek


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## Mr. Big Noodles (Jul 8, 2014)

viesczy said:


> Good luck!
> 
> Every farking show, girls be all on the drummer and the singer, the two least talented people in the band.
> 
> ...



This makes me lament that there aren't more gearheads who are also female.



Also, what are you talking about? My alt chords were singlehandedly responsible for the spike in Fruit of the Loom's third quarter earnings in 2009.


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## ElRay (Jul 8, 2014)

Mr. Big Noodles said:


> In all honesty though, this is a big part of how I choose pitches.



Make sure you use vibrato. Pitches love vibrato.

Ray


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## ElRay (Jul 8, 2014)

On topic:

I will have to digest this a bit, but is seems similar to Carter's interest in All-Interval-Tetrachords (AITs). I've noodled with them on the piano, but I haven't really found "What I'm Looking For" on the guitar. They can be arpeggiated, but the chords are too dense for my brain/fingers to find what I have this feeling "should be there".

Ray


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## Hollowway (Jul 9, 2014)

Dang it, Member Formerly Known As SchecterWhore, I love this stuff! I don't know that I can do much with it, but conceptually it's very cool. And are you/did you get a PhD in theory or something?


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## Mr. Big Noodles (Jul 9, 2014)

Working on it. All glory to the Hypomode.


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## Albionic (Jul 9, 2014)

i find 3 chords on an acoustic plus singing is more likely to get you into girls pants than any kind of scale


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## Mr. Big Noodles (Jul 9, 2014)

You and Orpheus, my friend. Man, he has a smooth voice.



ElRay said:


> On topic:
> 
> I will have to digest this a bit, but is seems similar to Carter's interest in All-Interval-Tetrachords (AITs). I've noodled with them on the piano, but I haven't really found "What I'm Looking For" on the guitar. They can be arpeggiated, but the chords are too dense for my brain/fingers to find what I have this feeling "should be there".
> 
> Ray



I understand the technical aspect of the all-intervals tetrachords, but I don't know enough about Elliott Carter to comment on their musical use, outside of "this set contains all six interval classes." My thoughts on hypomodes relate directly to pitch centricty, melodic tendency, and harmonic cadence. Perhaps there is a link, but I fail to see it. Care to help me out here?


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## Alex Kenivel (Jul 9, 2014)

Enlightening. 

Now I know what a tetrachord is. 

Thank you.


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## Mr. Big Noodles (Jul 9, 2014)

Welcome.


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## ElRay (Jul 9, 2014)

Mr. Big Noodles said:


> ... Perhaps there is a link, but I fail to see it. Care to help me out here?



That's what I'm trying to figure out. The AIT's are all inversions of [0,1,4,6] or [0,1,3,7], but I think I'm messing-up the conversion of that to subsets of 1,b2,2,b3,3,4,b5,5,b6,6,b7,7 which is the notation you're using. I'm missing something, but I can't tell if it's a connecting piece, or the piece that says I'm off on a tangent.

I also went back to look at the main Carter piece I was thinking of (Shard) and it seems he was using the AITs as a bit of a serialism constraint. The overall "structure" is two big overlapping polymeters that go in and out of phase. I'll have to mess around some more to see if he was going all Ron Jarzombek and pulling implied dominant (or other) chords out of something atonal/12-tone.


Ray


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## ElRay (Jul 9, 2014)

Hollowway said:


> ... Member Formerly Known As SchecterWhore ...



I might go Prince & SW-like and become "The member formerly known as ElRay" in favor of "The Dumpling Guy". I think Mr. Big Noodles needs a crime fighting side kick hero support person.

Ray


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## Given To Fly (Jul 10, 2014)

ElRay said:


> That's what I'm trying to figure out. The AIT's are all inversions of [0,1,4,6] or [0,1,3,7], but I think I'm messing-up the conversion of that to subsets of 1,b2,2,b3,3,4,b5,5,b6,6,b7,7 which is the notation you're using. I'm missing something, but I can't tell if it's a connecting piece, or the piece that says I'm off on a tangent.
> 
> I also went back to look at the main Carter piece I was thinking of (Shard) and it seems he was using the AITs as a bit of a serialism constraint. The overall "structure" is two big overlapping polymeters that go in and out of phase. I'll have to mess around some more to see if he was going all Ron Jarzombek and pulling implied dominant (or other) chords out of something atonal/12-tone.
> 
> ...



90% of _Shard_ is made up of two AIT's and the tempo relationship(s) can basically 108 : 144 or 3:4:3:4 with some metric modulations between the two groups of 3:4. There are further tempo/pitch relationships present when the accented notes are played accurately (which is no easy feat). I've played _Shard_ for awhile now yet still learn new things about his music even after all this time. I did an analysis of the rhythmic and pitch material for my Post-Tonal Theory class and its the one paper I'm actually proud of. The score analysis is color coded showing how everything relates to each other. There were only a few trichords I could not account for but they are towards the end and likely have some relationship to _Luimein_. 

Wow! The nerd part of me really needed that!


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## Poltergeist (Jul 15, 2014)

To hell with getting into girls' pants, I just want to be a theory wizard like Mr. Big Noodles.


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## octatoan (Jul 16, 2014)

Amen!


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## Mr. Big Noodles (Jul 17, 2014)

"Women want him, men want to be him..."

Does anybody want to discuss the topic?


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## ElRay (Jul 17, 2014)

Given To Fly said:


> ... I did an analysis of the rhythmic and pitch material for my Post-Tonal Theory class and its the one paper I'm actually proud of. ...



1) Would you be willing to share?



Given To Fly said:


> ... The nerd part of me really needed that! ...



2) Can we start calling you "Master Little Noodles"?


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## OmegaSlayer (Jul 17, 2014)

Mr. Big Noodles said:


> "Women want him, men want to be him..."



Those in between want to experience THROUGH him...


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## Given To Fly (Jul 17, 2014)

ElRay said:


> 1) Would you be willing to share?
> 
> 
> 
> 2) Can we start calling you "Master Little Noodles"?



I'll see what I can do. I used highlighters on the score and just made copies for the class, so I've never tried to scan it. If I can get it scanned I'll post it or send you and individual copy. 

No, you may not call me "Master Little Noodles."  I'll be impressed if you figure out where my user name comes from though.


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## octatoan (Jul 17, 2014)

Pearl Jam. 
Google 

(May I have a copy of the PDF too if you don't post it? )

EDIT: That is ONE nice song!


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## Given To Fly (Jul 17, 2014)

And some positive rep. is in order!


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## EcoliUVA (Jul 17, 2014)

WHAT HAVE YOU DONE WITH SCHECTERWHORE?!? Brain transplant?

My life is a lie.

But this post is cool. 10/10 would shecterwhore.


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## ElRay (Jul 17, 2014)

Given To Fly said:


> ... I'll be impressed if you figure out where my user name comes from though.



Sorry. I tuned out. It's a bad time, nothing can save me. I'm alone in a corridor, waiting, locked out.


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## Gregory Frus (Jul 20, 2014)

Wow! awesome stuff. Cool way of looking at tonality. I finally found a place where I can wear my "I love music theory" T-shirt without getting punked for my lunch money


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## Mr. Big Noodles (Jul 21, 2014)

... You have lunch money?



One thing I noticed is that the chord that I posted in the OP has a sort of hybrid function as an augmented sixth chord as well as that of its resolving dominant. A bit of a hybrid, or a compression of the normal chord progression.


```
e-
B-[COLOR=Red]4[/COLOR]-5
G-7-5
D-6-7
A-[COLOR=Red]8[/COLOR]-7
E-
```
E&#9837;
D
G#
E#

These substitute for E, the dominant, but could easily be contextualized in a progression such as this:

D#-E--E
C--D---C
A--G#-A
F--E---A

Ger+6 V7 i, for those who are savvy with the functional labels.
Double-subbing the root of the V7 chord, in effect, combines the predominant +6 function with the dominant function.


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## octatoan (Jul 21, 2014)

I'm not savvy with the functional labels 
What's Ger+6 V7 i mean?
What's Ger mean, in the first place?


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## Given To Fly (Jul 23, 2014)

Awesoham said:


> I'm not savvy with the functional labels
> What's Ger+6 V7 i mean?
> What's Ger mean, in the first place?



German


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## octatoan (Jul 23, 2014)

Y U NO DE


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## Konfyouzd (Jul 23, 2014)

My sax teacher talks ab stuff like this.... But also actively avoids playing the tonic hoping to just imply it by surrounding it. Not sure if that's quite the same idea...?


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## Mr. Big Noodles (Jul 27, 2014)

Awesoham said:


> I'm not savvy with the functional labels
> What's Ger+6 V7 i mean?
> What's Ger mean, in the first place?



This might get confusing.

When we write scales, you can think of each tone corresponding to a number. As an example, the scale of C major, C D E F G A B C, corresponds to 1 2 3 4 5 6 7 1, where every C is 1, every D is 2, every E is 3, and so on.

We can build a chord off of every degree of the scale by stacking other notes from the scale on top of that degree. We name them according to the intervals in the chord. If the chord contains a major third and a perfect fifth, it's major. If the chord contains a minor third and a perfect fifth, it's minor. If the chord contains a minor third and a diminished fifth, it's diminished. If the chord contains a major third and an augmented fifth, it's augmented.

We then combine the information of the scale degrees (1 2 3 4 5 6 7) with the information of the chord quality (major, minor, diminished, or augmented) to make a more complete picture of how the music is working. If we're in C major, the first chord is a C major triad (simply labeled "C"), which would be "1," but we call it by a Roman numeral instead, "I". The second chord in the scale is a D minor triad, ("Dm"), which corresponds to "2," but we use the Roman numeral "ii". Notice that this one is the lowercase form of the numeral. If the numeral is majuscule (such as I, IV, or V), then the chord is major. If the numeral is miniscule (such as ii, iii, vi), then the chord is minor. This continues with the scale up until we get to the seventh degree, which is a B diminished triad ("B°"), indicated by a miniscule numeral with an added symbol: vii°







This makes it easier to talk about chord progressions. If we had C Dm G, we could describe the chord progression as I ii V.

We can do the same for every other scale. Here is the natural minor scale:






When compared to C major, some of the notes are flattened. I account for this by placing a &#9837; in front of the relevant notes and chords. Next is the harmonic minor scale:






This one has the augmented triad, E&#9837;+ (&#9837;III+). More importantly, it has V and vii°, allowing us to use the progression V i (G Cm) or vii° i (B° Cm). We can also add the seventh onto the V chord (G B D) to make V7 (G B D F)

This gets us closer to the original question. You now know what V7 i means. I also mention Ger+6, which is a kind of chromatic chord. "Chromatic" means that it exists outside of the scale. Chromatic chords break down into several different types. One of the types of chromatic chords is the "augmented sixth" family. These chords are built using the interval of the augmented sixth. The role of the augmented sixth is to embellish or emphasize an octave by expanding chromatically in both directions. Here are two examples of the augmented sixth expanding to an octave:






We're going to use the second one to build chords, so that we can have enough space to put other notes between the augmented sixth.

Traditionally, we use augmented sixth chords to precede the V chord. There are three types of traditional augmented sixth chords: the Italian augmented sixth (It+6), the German augmented sixth (Ger+6), and the French augmented sixth (Fr+6). The names don't really mean anything, it's just how they're called.






Don't feel bad if you don't understand all of this: I just gave you three semesters of music theory in one post.



Konfyouzd said:


> My sax teacher talks ab stuff like this.... But also actively avoids playing the tonic hoping to just imply it by surrounding it. Not sure if that's quite the same idea...?



You're learning sax? Awesome.

It's not 100% the same thing, but naturally there is some relation. In harmony, we use chord progressions to set up the expectation of a tonic. Theoretically, by hearing a dominant chord, we should be able to tell which note the tonic is. That is to say that if you hear D7 (D F# A C), you should be able to predict that G is the corresponding tonic. This is the basis for much ear training.


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## iamjosan (Aug 4, 2014)

This is an interesting way to approach harmony. The E7 with the D# below it resolves nicely to an Amaj7add6 going from the F in the dominant chord to F# in the tonic on the top voicing.

The Amaj7add6 would be played like this: A-G#-C#-F#


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## octatoan (Aug 4, 2014)

Mr. Big Noodles said:


> This might get confusing.
> 
> When we write scales, you can think of each tone corresponding to a number. As an example, the scale of C major, C D E F G A B C, corresponds to 1 2 3 4 5 6 7 1, where every C is 1, every D is 2, every E is 3, and so on.
> 
> ...


I feel really, really good, you know? I already knew everything you said until the part about using aug6s to move to octaves, and I understood the rest.


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## AugmentedFourth (Aug 4, 2014)

So can you use hypomodes to justify moving the tonic note? I'm not entirely sure if this makes sense, but take for example how you used the ascending melodic minor scale to create doubly altered 5's. Well you could do a similar thing with natural minor, except move the tonic note and keep the 5's there.


```
E F G A B C D E
5>>>>>1<<<<<<<5

A &#8594; A&#9837;

 E F G A&#9837; B C D E
&#9839;5>>>>>1 <<<<<<&#9839;5

From the tonic:

1 &#9839;2 3 &#9839;4 &#9839;5 6 7

(augmented Lydian garbage)

From the 5:

1 &#9837;2 &#9837;3 &#9837;4 5 &#9837;6 &#9837;7

(Phrygian &#9837;4?)
```
Potentially this could be the basis for some interesting harmonic material. Or, more likely, it's hogwash. 

EDIT: This is a bit off topic but I didn't want to start a new thread just for this.
Does anyone have recommendations for books/resources on more "advanced" or "modern" music theory? Like, I've seen the Wikipedia articles on like musical set theory, Schenker analysis, and other obscure stuff that I really am clueless about. But I'm not sure if there's any accessable materials out there for what I guess is late undergraduate/masters-level concepts. Obviously this sounds pointed at MBN but I'd like to hear from anyone else who has experience there.


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## Given To Fly (Aug 13, 2014)

AugmentedFourth said:


> EDIT: This is a bit off topic but I didn't want to start a new thread just for this.
> Does anyone have recommendations for books/resources on more "advanced" or "modern" music theory? Like, I've seen the Wikipedia articles on like musical set theory, Schenker analysis, and other obscure stuff that I really am clueless about. But I'm not sure if there's any accessable materials out there for what I guess is late undergraduate/masters-level concepts. Obviously this sounds pointed at MBN but I'd like to hear from anyone else who has experience there.



The standard undergraduate textbook is Materials and Techniques of 20th Century Music (3rd Edition): Stefan Kostka: 9780131930803: Amazon.com: Books .

The standard master/doctoral degree textbook is Introduction to Post-Tonal Theory (3rd Edition): Joseph N. Straus: 9780131898905: Amazon.com: Books . 

The Kostka book might be considered accessible. It covers a lot of information in a very general way. If you can find a used copy at a used copy for a decent price it might be worth picking up. 

The Straus book is truly aimed at doctoral theory students and is not for the theory enthusiast. The word "introduction" is slightly misleading because it starts with the assumption the reader has already been exposed to atonality, set theory, and serial techniques along with the composers who used them such as Pierre Boulez, Elliott Carter, Milton Babbitt, etc. 

I want to encourage learning, but I also want to be responsible about it. Start with music first, then start to approach the theory behind it.


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## celticelk (Aug 13, 2014)

There's a newer edition of Kostka under a different title: Materials and Techniques of Post-Tonal Music (4th Edition): Stefan Kostka: 9780205794553: Amazon.com: Books


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## AugmentedFourth (Aug 13, 2014)

Cool beans. Thanks for reminding me about that book. I had read like the first 5 or 6 chapters a while ago but never finished it. I should probably get on that...


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## Given To Fly (Aug 13, 2014)

Another great source is Perspectives of New Music. Perspectives of New Music Home Page
You can buy individual articles, individual volumes, or get a full subscription.


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