How to stack intervals when building a chord?

Good hour good folks! This is a very very crucial question for me, so thank you in advance.

I went ahead and tested the interval ranking (list of intervals from most consonant to dissonant in form of ratios) by building a chord with the 3 most dissonant intervals within the ranking which are

•         Tritone - ratio (23:16)
•         Minor 2nd - ratio (15:16)
•         Minor 7th - ratio (9:16)

Image Description (Root to tritone - root to minor 2nd - root to minor 7th.)

As seen in the image above, according to the interval ranking there is no chord more dissonant than this.

Now, the testing..

In the image above, if i were to take the Minor 7th interval (A#) and reposition it to the note (D), the chord will have now become significantly more dissonant than in it's original position. This would disprove that my use of the interval ranking was correct, as the chord is now more dissonant.

Maybe.. i positioned the most dissonant intervals all relative to the root as opposed to placing them relative to the note below each successive interval, so lets visualize this updated structure below.

Image Description (Root to tritone - tritone to minor 2nd - minor 2nd to minor 7th.)

As seen in the image above, the intervals are now structured not relative to the root but by consecutive notes.

Again, the testing..

In the image above, If i again, take the minor 7th (F) and reposition it to (G#), the chord is now more dissonant than in its original "supposedly max dissonance" chord structure shown above.

----------------------------------------------------------------------------------------------------------------------------------------------------------------------------

"Question Section"

•     Question 1. In my second image example, did i utilize the interval ranking adequately? When stacking intervals, do you use intervals based off the note before the successor note or do you stack intervals all relative solely to the root?

•     Question 2. In my second image example, i used the most dissonant intervals available within the interval ranking but yet when i shifted the minor 7th (F) and re-positioned it to (G#), the chord was more dissonant. So my 2nd approach to stacking intervals still doesn't prove to work effectively. What is the process behind this and how should intervals be stacked in a chord?

•     Question 3. If we reference the second image and and focus on the F# and G notes right beside each other, they're pretty dissonant. If i push the G up or down an octave and leave the F# where it is, now they aren't as dissonant.

Is the reason for this because when the two notes are beside each other in the same octave, their waves are both traveling at a similar rate but when one note is brought up an octave, because the frequency of that note has been doubled, now the two notes waves line up more as the higher octave note is now traveling twice as fast as the note an octave below?

• Question 4. If i stack intervals as explained in my second image example, which is by selecting my desired interval ratios relative to the note previous to it in my chord as opposed to the root, do the successive notes after my new note play a direct role "interval relationship wise" with the previous notes or does it just boil down to the note before the new note? Let me elaborate..

If i have C and E (Major 3rd) and then add a G on top of that, the C to E is a Major 3rd and the E to G is Minor 3rd. But in that chord structure the C to G could be considered a Perfect 5th and now the G is no longer a Minor 3rd interval within the chord. Which is right and which is wrong? Especially when more notes are added, the intervals can have multiple perspectives. So the questions boils down to, do we only focus intervals relative to the note before the new note added? If not, it would be a mess.

Thank you very much guys!

21 minutes ago, Seery said:

I went ahead and tested the interval ranking (list of intervals from most consonant to dissonant in form of ratios) by building a chord with the 3 most dissonant intervals within the ranking

Why did you stop at three? 

22 minutes ago, Seery said:

In the image above, if i were to take the Minor 7th interval (A#) and reposition it to the note (D), the chord will have now become significantly more dissonant than in it's original position.

D is the inversion of A# when working in equal-tempered pitch classes. In other words, both notes are a whole step away from C.

24 minutes ago, Seery said:

When stacking intervals, do you use intervals based off the note before the successor note or do you stack intervals all relative solely to the root?

It depends on the type of music you're writing. In jazz, you'd base things on the primary or secondary roots. In early atonal music, it could be either. In Renaissance music, both matter.

30 minutes ago, Seery said:

So the questions boils down to, do we only focus intervals relative to the note before the new note added?

Both, technically. When you play a major triad (C E G) the ratio is 4:5:6. Major third ratio is 5:4 and minor third ratio is 6:5. 6:4 is a perfect fifth. 
Major seventh chord (C E G B), you get 8:10:12:15. 8:10 = 4:5. 10:12 = 5:6. 12:15 = 4:5. The major seventh interval from bottom to top is 8:15. All of the ratios matter, that's what makes them ratios.

1 hour ago, Monarcheon said:

Why did you stop at three? 

I could have continued but i decided that 3 intervals would suffice for the purpose of the demonstration. Of course, adding additional intervals would have added further dissonance but at the same time, i was able to maximize the dissonance for my first displayed chord simply with those 3 intervals.

1 hour ago, Monarcheon said:

D is the inversion of A# when working in equal-tempered pitch classes. In other words, both notes are a whole step away from C.

Granted, but the question I'm getting at is, how come even though i utilized the 3 most dissonant intervals within one chord, i can produce further dissonance with other intervals? That answer would demonstrate my general question which is what is the correct scientific approach to stacking intervals? The answer could possibly revolve around how pitches interact with each other.

1 hour ago, Monarcheon said:

It depends on the type of music you're writing. In jazz, you'd base things on the primary or secondary roots. In early atonal music, it could be either. In Renaissance music, both matter.

If we reference my two piano roll images, they're using the same intervals with the only difference being image 1 has stacked it's intervals all relative to the root and image 2 has stacked it's intervals relative to the previous note to which the successive note was placed. Referencing the interval ranking, it did not deliver in producing the most dissonant chord available to the intervals as it should have in theory. This would prove that neither of my two approaches which you have stated in the above quote, are the correct approach to stacking intervals evidently as the two tests were inaccurate. So that is my question, what is the actual formula?

Tell me what you think of this possibility.. Maybe the correct formula for stacking intervals in accordance with ratios, is not in fact neither of the two approaches we have mentioned. We demonstrated this by using the 3 most dissonant intervals without success. But what if we approached producing the most dissonant chord by stacking duplicates of an interval as opposed to selecting multiple intervals? 

If we stacked minor 2nd intervals repeatedly (lets say 4 times in a chord) we would have C, C#, D and D#. Would that chord be more dissonant than what a Tritone, minor 2nd and minor 7th chord could produce? It would seem so, which could lead us to believe the scientific approach to stacking intervals would be as such?

50 minutes ago, Seery said:

I could have continued but i decided that 3 intervals would suffice for the purpose of the demonstration. Of course, adding additional intervals would have added further dissonance but at the same time, i was able to maximize the dissonance for my first displayed chord simply with those 3 intervals.

No, you didn't. Those three intervals are not the most dissonant if you wanted to make the most dissonant tetrachord. It would be Forte: 4-1, (0123). I just don't see why you, in your premise, think that (0137) is going to be more dissonant, especially considering it has an even interval class vector, while (0123) has a more left-heavy vector. In other words, I don't agree with your premise that the most dissonant tetrachord is one that has multiple different types of dissonances in it. You'll notice that (0137) is just the major triad (037) with an added minor second (1), so of course it isn't going to be the most dissonant, even when all bunched up. 

50 minutes ago, Seery said:

Tell me what you think of this possibility.. Maybe the correct formula for stacking intervals in accordance with ratios, is not in fact neither of the two approaches we have mentioned.

This whole ratio business gets really useless if you're considering all of this in equal temperament. In equal temperament, the perfect fifth is not 3:2.

50 minutes ago, Seery said:

We demonstrated this by using the 3 most dissonant intervals without success. But what if we approached producing the most dissonant chord by stacking duplicates of an interval as opposed to selecting multiple intervals? 

If we stacked minor 2nd intervals repeatedly (lets say 4 times in a chord) we would have C, C#, D and D#. Would that chord be more dissonant than what a Tritone, minor 2nd and minor 7th chord could produce?

That's what I'm saying. I don't see why in your initial argument you claimed you had found the most dissonant chord which is why I was calling you out for stopping at 4 notes. 

17 minutes ago, Monarcheon said:

It would be Forte: 4-1, (0123)

I'm not familiar with this term. Does it represent the chord with the pitches C,C#,D and D#?

20 minutes ago, Monarcheon said:

This whole ratio business gets really useless if you're considering all of this in equal temperament. In equal temperament, the perfect fifth is not 3:2.

From my previous research, i understand that Just Intonation is in ratios and 12TET in cents but i was also told that although the two tuning systems are not exact, they're good enough to utilize as reference. That's what you mean right when you say a perfect 5th is not a 3:2 but 700 cents?

22 minutes ago, Monarcheon said:

That's what I'm saying. I don't see why in your initial argument you claimed you had found the most dissonant chord which is why I was calling you out for stopping at 4 notes. 

Understood!

Now if i stack the intervals Major 3rd and Perfect 5th relative to the root, that is a Major Chord but if i stack the intervals Major 3rd and i stack the Perfect 5th relative to the E, now that's a Major 7th with the 5th omitted. Same intervals, two different chords. You say that intervals can be stacked through both methods but it seems a mess really as there's two alternatives to one process.

Which is the more scientific method if our focus is on harmonic series/overtone relationships?

12 minutes ago, Seery said:

I'm not familiar with this term. Does it represent the chord with the pitches C,C#,D and D#?

Basically. It's a little more nuanced, but in this case it works. [0123], which is C, C#, D, D# and (0123) are slightly different.

13 minutes ago, Seery said:

From my previous research, i understand that Just Intonation is in ratios and 12TET in cents but i was also told that although the two tuning systems are not exact, they're good enough to utilize as reference. That's what you mean right when you say a perfect 5th is not a 3:2 but 700 cents?

Cents are even more weird to think about it. In pitch class space only, just think about intervals between notes, ratios for simple relationships, if you must. Makes it simpler unless you're utilizing the harmonic series for compositional effect.

14 minutes ago, Seery said:

Now if i stack the intervals Major 3rd and Perfect 5th relative to the root, that is a Major Chord but if i stack the intervals Major 3rd and i stack the Perfect 5th relative to the E, now that's a Major 7th with the 5th omitted. Same intervals, two different chords. You say that intervals can be stacked through both methods but it seems a mess really as there's two alternatives to one process.

What? I'm saying the major seventh chord's ratios are 8:10:12:15. In any given ratio between two of those numbers you get the correct ratio for that interval specifically. i.e. 10:15 reduces to 3:2 as does 12:8. They're both perfect 5ths, which makes sense, because both E to be and C to G are perfect fifths. That's not two processes, it's just one concise ratio, despite having multiple internal relationships. 

10 minutes ago, Monarcheon said:

What? I'm saying the major seventh chord's ratios are 8:10:12:15. In any given ratio between two of those numbers you get the correct ratio for that interval specifically. i.e. 10:15 reduces to 3:2 as does 12:8. They're both perfect 5ths, which makes sense, because both E to be and C to G are perfect fifths. That's not two processes, it's just one concise ratio, despite having multiple internal relationships. 

You've wonderfully cleared up all my other points which i appreciate. This last one about stacking intervals all from the root or a note to note basis, is not becoming clear to me. I'll ponder on your response along with additional research. If i have any more questions regarding this post, ill hit you up. Thank you for this.

@Seery, it's both. Let's take a look at the ∆7 chord. 

8:10:12:15 = C E G B

All of the intervals related to C make sense.
10:8 = 5:4 which is a major third. That checks out because C to E is a major third.
12:8 = 3:2 which is a perfect fifth. That checks out because C to G is a perfect fifth.
15:8 = 15:8 which is a major seventh. That checks out because C to B is a major seventh.

HOWEVER:

All of the inner intervals also make sense. 

12:10 = 6:5, which is a minor third. That makes sense because E to G is a minor third.
15:10 = 3:2, which is a perfect fifth. That makes sense because E to B is a perfect fifth.
15:12 = 5:4, which is a major third. That makes sense because G to B is a major third.

So when you play a chord (C∆7, specifically), you hear ALL of those relationships at the same time. It is both all notes relative to the root and a note-to-note basis at the same time.

@Monarcheon

So basically,

Within the Major 7th chord exist these following intervals, no more and no less? Major 3rd x 2, Perfect 5th x2, Minor 3rd and Major 7th?

If this is the case then the ratios within the Major 7th chord are no more and no less than 4:5 x 2, 2:3 x 2, 5:6 and 8:15. which constitutes it's exact level of dissonance.

and because of the duplicates in the Major 7th chord of 4:5 x 2 and 2:3 x 2, this duplication of ratios would make this chord more dissonant than a chord that would not have ratio duplicates or at least less than 4 unlike Major 7th?

5 minutes ago, Seery said:

and because of the duplicates in the Major 7th chord of 4:5 x 2 and 2:3 x 2, this duplication of ratios would make this chord more dissonant than a chord that would not have ratio duplicates or at least less than 4 unlike Major 7th?

I don't know about that. My point was whenever a chord is played all of the ratios found in it (much like finding an interval vector) are heard and applied to your ear, the consonance and dissonance don't much matter. Same goes with inverted, chords– the ratios would just be different, but you'd still hear all of them. 

I guess if you have a different chord to that of Major 7th which maybe has a duplicate of a minor 2nd and a single tritone, although it has only one duplicate, the more dissonant choice of intervals would sum up to a harsher chord. 

That can be further analysed if you look at it from a harmonic series/overtone perspective. In fact your explanation revolves around the overtone relationships, just that its stated through way of ratios. So seeing it that way makes much sense to me. Thank you so much and I'll ask here if any other difficulties arise. You've been tremendous help!

I think you have to consider the background. In no tonal music, tritones are considered neutral. If the background is consonant it's dissonante and vice versa.

I also thing the most dissonant tetracord is C-C#-D-Eb

Yes I think we would agree C-C#-D and Eb is the maximum dissonance within 4 notes possible 

If by dissonant, you mean most ambiguous/unstable, that designation goes to those harmonies which divide the octave evenly: the tritone, the augmented triad, whatever you call a chord structured like C-D#-F#-A, etc. If by dissonant, you mean most harsh-sounding, that will depend largely on the timbre. With pure sine tones, C5-C#6 doesn't sound harsh: it produces no sensory dissonance. Play that same harmony on a piano, and there will be plenty of sensory dissonance. If by dissonant, you mean ugly-sounding, that's highly subjective.

In short, I don't really think the process you used is getting the results you intended. Also, I don't know why you would want to build chords in this particular fashion.