Formula for harmonized triads of the minor scale

Analyses of individual works for Classical Guitar and general discussions on analysis. Normal forum copyright rules apply.
Tonit
Posts: 751
Joined: Tue May 22, 2018 1:44 am

Re: Formula for harmonized triads of the minor scale

Post by Tonit » Wed Sep 11, 2019 10:13 pm

Hi VasquezBob,
VasquezBob wrote:
Wed Sep 11, 2019 9:12 pm
Tonit wrote:
Wed Sep 11, 2019 12:22 am
VasquezBob wrote:
Tue Sep 10, 2019 11:32 pm


Thank you for the response and explanation. I still don't know why any minor scale should be based on a major scale that has the same tonic, namely, "C" in the case above. Minor scales are extensions of related major scales and they carry the same accidentals (where appropriate); added accidentals being included for harmonic and melodic minor scales, q.v. That's why I thought that the approach might be for a different type of scales. Anyway, thanks again.
That in fact is a good question.

There appears to be more than one path to get at the answer. To address the question of where do all the flats (b's) come from, one just needs to go from "C minor" to its relative major which is "E" but, which E? The E major on the sharp side or the flat side of the signature chart? Well, if E is the tonic and F is the supertonic, then, E must be flat; if we are to apply the whole tone, whole tone, half tone to the upper and lower tetrachords. Now, one can either look-up Eb major or write it out and one will find that "E, A, and B" are flat and that's from where all the flats descend (pardon the play on words). Just an aside: sometimes it's easier to "see" the answer when one applies the musical alphabet rather than using Roman numerals, q.v., but, to each his/her own.
Well there is a little confusion going on.
if E is the tonic and F is the supertonic, then, E must be flat, if we are to apply the whole tone, whole tone, half tone to the upper and lower tetrachords.
No, supertonics can be 1/2 step above the tonic, like Phrygian or Locrian supertonics.
But I suppose you are talking about Eb major scale. But here you mixed up the "tetracord" idea I noted.
It is not for this purpose I introduced it to you.

I might have explaiend, but what you should look at is "conjuncture" and "disjuncture" natures of major and minor modes when you see the both upper/lower tetrahords of maj/min are balanced up.

And like I have noted, it is for the convenience purpose that one same key signature for a major and its relative minor is used, and except that we may have the "relative" key relation in case of actual transposition from a major to its relative minor or vice versa in a composition, there is no relation being sugested by one key signature.

Also, what you should not be confused is that, the sharps and flats used for "scale degrees" only apply when we discuss the scale degrees, and not applied when we discuss about the diatonic chords in classical theory, because, as per noted, we do not have to specify if "modal interchange" (or "borrowed chord") is not involved. Accordingly, you DO have to apply sharp(s) and flat(s) to roman numerals in case of Jazz analysis as Jazz involves many "modal interchange" chords.

Dirck Nagy
Posts: 976
Joined: Sat Sep 28, 2013 4:47 pm
Location: Wisconsin, USA

Re: Formula for harmonized triads of the minor scale

Post by Dirck Nagy » Thu Sep 12, 2019 12:13 am

Mark Clifton-Gaultier wrote:
Sat Sep 07, 2019 7:41 am
The pointless and apparently random introduction of unrelated terminology is particularly unhelpful within a discussion particpated in by those seeking clarity.
.
+1

Tonit
Posts: 751
Joined: Tue May 22, 2018 1:44 am

Re: Formula for harmonized triads of the minor scale

Post by Tonit » Thu Sep 12, 2019 1:49 am

Dirck Nagy wrote:
Thu Sep 12, 2019 12:13 am
Mark Clifton-Gaultier wrote:
Sat Sep 07, 2019 7:41 am
The pointless and apparently random introduction of unrelated terminology is particularly unhelpful within a discussion particpated in by those seeking clarity.
.
+1
OK sorry if I have introduced "Tetrachord" to confuse.
I will try to share the most concise explanation.

Also, it was hard to locate, but I have found the journal article I read when I was learning so I would like to share here for your further consideration and reference: www.jstor.org/stable/909364?read-now=1& ... b_contents

The title of the paper is The Origin of the Major and Minor Modes (Concluded) published in 1917.

Hopefully going over it would make it easier to understand my explanation below (NB. the following is NOT the point made by the original article)

----------

The major scale is preferred to the minor scale as the standard, because the former is more "singable" than latter and the western music has made progress with the "most singable" scale available. And this "most singable" in the sense of the western music encompasses two key aspects: stability and balance, because of its history.

As has been taught, due to the harmonic reason, "harmonic minor" scale was introduced to have the same tritone resolution as in the mejor scale, but it is very hard to sing, because of the augmented 2nd interval between ^6 (scale degere 6) and ^7, thus "melodic minor" was introduced to cancel this issue, which backs up this "most singable" scale preference, resulting in three different minor scales, and each of them can be found in very specific circumstances, so that we have to use them accordingly.

And even though the problems have been seemingly resolved by having three different scales, the resultant scales are turned out to be "ill balanced". "ill balanced" here means, when we divide the octave into two, the two parts are none similar to each other, making it difficult to sing in two part chants, for example (this would be yet another long post so please check the history of chants). The divided 1/2 segments are called "tetracords", comprising "lower tetrachord" and "upper tetracord".

When we divide a major scale with the tonic 8va (ie 8 notes) in half, the resulting tetrachords are identical.

In case of C major, it will be divided into: Lower (C-D-E/F) and upper (G-A-B/C) where "-" means a step and "/" means half step. As you see, the upper and lower tetrachords have the same intervalic structure, while if we divide A natural minor scale in the same manner, it would be: Lower (A-B/C-D) and upper (E/F-G-A) where the tetrachords are not the same.

However so, when we have a duplicated note shared in both upper/lower tetrachords (ie 7notes with one duplicate), A natural minor scale can also be subdivided into two tetrachords of the same structure: Lower (A-B/C-D) and upper (D-E/F-G). This way of division is referred to as "conjunct" compared to "disjunct" that has no shared note.

While both conjunct and disjunct are reasonably equally subdividing the scales as it seems, the conjunct division is not exactly in the center but is at a poin a little closer to the lower tonic, or making the division a little "top heavy", compared to disjunct that divides between ^4 and ^5 that is exactly in the middle, only that we do not have any note there.

So this imbalanced tetrachords and variations in the upper tetrachord was not facilitating the singing, compared to more stable and well-balanced major scale.

This is why major scale has been preferred over minor scale as the standard.

However so, this is only true in the most part of the tonal western music historically, and also there are many kinds of music which feature the augmented 2nds in their melodies, such as flamenco or some other middle-eastern musics. It just turned out, the western music has developed relying on this well-balanced major scale.

I hope the point is a little clearer now.
Last edited by Tonit on Thu Sep 12, 2019 2:43 am, edited 1 time in total.

VasquezBob
Posts: 198
Joined: Mon Oct 15, 2018 10:54 pm

Re: Formula for harmonized triads of the minor scale

Post by VasquezBob » Thu Sep 12, 2019 2:28 am

Tonit wrote:
Wed Sep 11, 2019 10:13 pm
Hi VasquezBob,
VasquezBob wrote:
Wed Sep 11, 2019 9:12 pm
Tonit wrote:
Wed Sep 11, 2019 12:22 am


That in fact is a good question.

There appears to be more than one path to get at the answer. To address the question of where do all the flats (b's) come from, one just needs to go from "C minor" to its relative major which is "E" but, which E? The E major on the sharp side or the flat side of the signature chart? Well, if E is the tonic and F is the supertonic, then, E must be flat; if we are to apply the whole tone, whole tone, half tone to the upper and lower tetrachords. Now, one can either look-up Eb major or write it out and one will find that "E, A, and B" are flat and that's from where all the flats descend (pardon the play on words). Just an aside: sometimes it's easier to "see" the answer when one applies the musical alphabet rather than using Roman numerals, q.v., but, to each his/her own.
Well there is a little confusion going on.
if E is the tonic and F is the supertonic, then, E must be flat, if we are to apply the whole tone, whole tone, half tone to the upper and lower tetrachords.
No, supertonics can be 1/2 step above the tonic, like Phrygian or Locrian supertonics.
But I suppose you are talking about Eb major scale. But here you mixed up the "tetracord" idea I noted.
It is not for this purpose I introduced it to you.

I might have explaiend, but what you should look at is "conjuncture" and "disjuncture" natures of major and minor modes when you see the both upper/lower tetrahords of maj/min are balanced up.

And like I have noted, it is for the convenience purpose that one same key signature for a major and its relative minor is used, and except that we may have the "relative" key relation in case of actual transposition from a major to its relative minor or vice versa in a composition, there is no relation being sugested by one key signature.

Also, what you should not be confused is that, the sharps and flats used for "scale degrees" only apply when we discuss the scale degrees, and not applied when we discuss about the diatonic chords in classical theory, because, as per noted, we do not have to specify if "modal interchange" (or "borrowed chord") is not involved. Accordingly, you DO have to apply sharp(s) and flat(s) to roman numerals in case of Jazz analysis as Jazz involves many "modal interchange" chords.
Okay, I'm getting it (I think): The thread is talking about medieval modes, jazz, classical, diatonic all in the same discussion. So, I guess my very first question should have been "what musical system is this thread talking about?" Or, "are several musical systems being intertwined; and, if so, which ones are they?" With that info, I may be able to follow the discussion and, hopefully, participate in a meaningful way. Many thanks for all you patience.

Tonit
Posts: 751
Joined: Tue May 22, 2018 1:44 am

Re: Formula for harmonized triads of the minor scale

Post by Tonit » Fri Sep 13, 2019 2:18 pm

Hi VasquezBob,
That's right.
But in the first place, the chord analysis in the phot included by the OP mixes up these classical and jazz ways that I hope you understand.

Aside from that, I would like to suggest one video resource that would help us understand the challenges of the musicians before maj/min modes from renaissance and on:



In fact the entire channel content is a great well of resources for earlier musics that I found in the last few days, as I have encountered something I didn't know quite so often in there. Obviously I am not any specialist in that.

I hope this further helps you.

User avatar
Mark Featherstone
Posts: 544
Joined: Tue May 27, 2014 12:26 pm
Location: Alameda, CA

Re: Formula for harmonized triads of the minor scale

Post by Mark Featherstone » Fri Sep 13, 2019 11:58 pm

...the chord analysis in the [post] included by the OP mixes up these classical and jazz ways...
Well, as I said, the harmonized minor scale that I presented does NOT mix up classical and jazz approaches. Rather, it provides the harmonized minor scale by reference to the parallel major scale.

Some further clarifications:

(1) The harmonized minor scale that I posted originally is generic for all harmonized natural minor scales; that is to say, it does not represent only C minor in relation C major, but rather is applicable to each and every harmonized natural minor scale when compared to its corresponding parallel major scale.

(2) The inclusion of the b (flat) accidental in the harmonized minor scale pattern is meant to indicate flatted with respect to the parallel major scale, not flat per se. I hope the diagram below helps.
Parallel minor scales and flatted notes.jpg
And now I realize for the first time that the b (flat) has nothing to do with how the triads are harmonized but rather the status of each root wrt the parallel major. Oy vey. So much work to understand a system that is so unhelpful. Still, I understand major vs minor scales much better now! :)

PS: I should add that in some instances, the parallel minor scale will have flatted notes as a result of flatting the 3rd, 6th and 7th notes of the parallel major scale. So, for example, in the scale of C minor, E, A and B (3, 6 and 7) are actually flat as a result of flatting the corresponding natural notes in C major.
You do not have the required permissions to view the files attached to this post.
Francisco Navarro Concert Classical, cedar top, 630 mm scale, 50 mm nut

"The trouble with normal is it always gets worse."
Bruce Cockburn

Tonit
Posts: 751
Joined: Tue May 22, 2018 1:44 am

Re: Formula for harmonized triads of the minor scale

Post by Tonit » Sat Sep 14, 2019 2:28 am

Hi Mark,
Mark Featherstone wrote:
Fri Sep 13, 2019 11:58 pm
...the chord analysis in the [post] included by the OP mixes up these classical and jazz ways...
Well, as I said, the harmonized minor scale that I presented does NOT mix up classical and jazz approaches. Rather, it provides the harmonized minor scale by reference to the parallel major scale.

Some further clarifications:

(1) The harmonized minor scale that I posted originally is generic for all harmonized natural minor scales; that is to say, it does not represent only C minor in relation C major, but rather is applicable to each and every harmonized natural minor scale when compared to its corresponding parallel major scale.

(2) The inclusion of the b (flat) accidental in the harmonized minor scale pattern is meant to indicate flatted with respect to the parallel major scale, not flat per se. I hope the diagram below helps.

Parallel minor scales and flatted notes.jpg

And now I realize for the first time that the b (flat) has nothing to do with how the triads are harmonized but rather the status of each root wrt the parallel major. Oy vey. So much work to understand a system that is so unhelpful. Still, I understand major vs minor scales much better now! :)

PS: I should add that in some instances, the parallel minor scale will have flatted notes as a result of flatting the 3rd, 6th and 7th notes of the parallel major scale. So, for example, in the scale of C minor, E, A and B (3, 6 and 7) are actually flat as a result of flatting the corresponding natural notes in C major.
I understand what you are explaining.

In so far as I was taught in my university, all the classical instructors of "traditional studies" curricula used no accidentals for roman analysis.

So in the classrooms today the figure musikai shared:
Image

This has been the case all the time.

In the meantime, all the jazz instructors of jazz "harmony" classes there used upper case roman numerals plus accidentals as required, as per my explanation:



So up to here have been discussed already.

Also, I quickly checked on Wikipedia and surprisingly found exactly the same as you posted:

https://commons.wikimedia.org/wiki/File ... _minor.png

To my further surprise, the figure is cited from Tonal Harmony With an Introduction to Twentieth-Century Music (Kostka, Payne, 1995; ISBN 0-07-035874-5)

Now THIS was THE textbook taught in my traditional classes, and still I have it being shipped from US to Japan to Europe.

I do not recall anything about the cited figure out of 700 pages, wherein vast majority (if not all) of roman analyses were made without any such prefixed accidental as you are trying to explain.

So and so, I have to take it back and agree with you that your example figure exists, most likely for the clarification, but only for clarification most possibly.

Standing corrected, though, in practical teaching of the classical roman analysis, we use no prefixed accidental, as also cited on the Wikipedia page.

And please also take note that all those prefixed accidentals to roman analysis numerals are most often found in Jazz analyses, only with upper case roman numerals, for the purpose of indicating modal interchanges and part of other non-diatonic chords very often found in jazz/pop/folkloric musics.

P.S. Tonal Harmony With an Introduction to Twentieth-Century Music is a great book to know about the most of the questions you encounter to know the music before getting to the era where the rules became meaningless, and it was my favorite read on and off campus in the earlier years at school, and one of the only three books I shipped all the way back from the US: One is this book, and the other one is Persichetti's 20th Century Harmony, and the Real Book (but mine is a 1/4 "Small Book" so called).
Last edited by Tonit on Sat Sep 14, 2019 9:19 pm, edited 1 time in total.

User avatar
Mark Featherstone
Posts: 544
Joined: Tue May 27, 2014 12:26 pm
Location: Alameda, CA

Re: Formula for harmonized triads of the minor scale

Post by Mark Featherstone » Sat Sep 14, 2019 5:45 am

Thanks for the detailed response! I am very interested in these books on music theory/harmony that you list, but I notice you did not mention Harmony by Walter Piston and Mark DeVoto. I thought that this was THE book on harmony. Do you prefer the books you mention? I am pretty new to music theory, so the Piston is really my only lengthy book on the topic.

Mark
Francisco Navarro Concert Classical, cedar top, 630 mm scale, 50 mm nut

"The trouble with normal is it always gets worse."
Bruce Cockburn

musikai
Posts: 511
Joined: Fri Jun 22, 2007 10:32 am
Location: Augsburg, Germany

Re: Formula for harmonized triads of the minor scale

Post by musikai » Sat Sep 14, 2019 6:26 pm

@Tonit

I had to correct an important error in my illustration and have updated it in my original post.
I accidentally used german words but found out that this is utterly wrong in english.
(german "Mollparallele" isn't "parallel minor" but "relative minor" in english. The english "parallel minor" is "Mollvariante" in german)

If you can please change the picture in your posts too.
Major-Minor Scale_0001.png
You do not have the required permissions to view the files attached to this post.
Free Sagreras Gitarrenschule PDF
Free Project: LibreOffice Songbook Architect (LOSA)
See website link

User avatar
Mark Featherstone
Posts: 544
Joined: Tue May 27, 2014 12:26 pm
Location: Alameda, CA

Re: Formula for harmonized triads of the minor scale

Post by Mark Featherstone » Sat Sep 14, 2019 7:40 pm

Thanks, musicai. That's a useful figure.

Ah, yes, these deceptive changes in word meaning between languages. I have a reasonable command of French, but can always get tripped up by these "faux amis" as they are called. My most hilarious instance was when I walked into a lingerie shop in France and said to the young saleswomen that I would like to buy a "camisole". They burst out laughing and it took some time to realize that the term in French means "straightjacket"! This was back in the 80's and the meaning may have changed since then as I see that Google Translate translates English "camisole" to French "camisole". But elsewhere a google search indicates that an older meaning of camisole is indeed straightjacket.
Francisco Navarro Concert Classical, cedar top, 630 mm scale, 50 mm nut

"The trouble with normal is it always gets worse."
Bruce Cockburn

Tonit
Posts: 751
Joined: Tue May 22, 2018 1:44 am

Re: Formula for harmonized triads of the minor scale

Post by Tonit » Sat Sep 14, 2019 9:20 pm

musikai wrote:
Sat Sep 14, 2019 6:26 pm
@Tonit
If you can please change the picture in your posts too.
Done!

musikai
Posts: 511
Joined: Fri Jun 22, 2007 10:32 am
Location: Augsburg, Germany

Re: Formula for harmonized triads of the minor scale

Post by musikai » Sun Sep 15, 2019 9:08 pm

@Tonit
Thank you!!
Mark Featherstone wrote:
Sat Sep 14, 2019 7:40 pm
I have a reasonable command of French, but can always get tripped up by these "faux amis" as they are called. My most hilarious instance was when I walked into a lingerie shop in France and said to the young saleswomen that I would like to buy a "camisole". They burst out laughing and it took some time to realize that the term in French means "straightjacket"! This was back in the 80's and the meaning may have changed since then as I see that Google Translate translates English "camisole" to French "camisole". But elsewhere a google search indicates that an older meaning of camisole is indeed straightjacket.
Haha, thanks for that! :)
As a young boy on holiday in Italy I often went to the beach and it was a Friday when I met this nice girl and we liked each other. Although we couldn't speak each other's language well we decided to meet again. She showed 3 fingers, pointed to the bridge we were standing on and told me "domenica". I immediately understood we would meet us at 3 o'clock here in this same place the next day ("domenica"="demain"="tomorrow"). So on Saturday I waited and waited the whole day but she didn't appear and a friend already began to comfort me.
But on Sunday ("domenica") all went ok!

In India in a village at the mountain side I stood at the reception of an Inn and was asked for my name:
"I need your name"
"Kai" I said.
"Please tell me your name!"
"Kai" I replied and 3 Indians sitting on a couch at the window began to giggle slightly. :lol:
"Can you please tell me your name!"
"Yes, Kai"
giggle-giggle-giggle from the couch :lol: :lol:
"Name. Your name!"
"It's Kaaaiiii"
laughter gets louder :lol: :lol: :lol:
"Please, just give me your name!
"Mh, my name is Kaaeeeyyy" and this time I pronounced it english.
"Ahhh, haha, I see!"

"Kai" = "what?" in Marathi
Free Sagreras Gitarrenschule PDF
Free Project: LibreOffice Songbook Architect (LOSA)
See website link

Tonit
Posts: 751
Joined: Tue May 22, 2018 1:44 am

Re: Formula for harmonized triads of the minor scale

Post by Tonit » Sun Sep 15, 2019 10:48 pm

Hi Mark,
Mark Featherstone wrote:
Sat Sep 14, 2019 5:45 am
Thanks for the detailed response! I am very interested in these books on music theory/harmony that you list, but I notice you did not mention Harmony by Walter Piston and Mark DeVoto. I thought that this was THE book on harmony. Do you prefer the books you mention? I am pretty new to music theory, so the Piston is really my only lengthy book on the topic.

Mark
I also had it :D but somehow I liked the other one better that I bought.
Piston's was given out to me for free to use in some other classes like music history or something I don't remember, but both books were first to give away among students :lol: They were generally more into playing in the bands. I was into reading more so.

Given the staggering status difference, my harmony instructor was also teaching at Harvard and compiled his own material handed out to us, so it wasn't required at all to go over those books. But of course once also there was Igor the great who I can only imagine to have a class with.

Anyhow both are good books for traditional studies without a doubt.

For me, I had already completed Jazz harmony classes by the time I went into the traditional studies, where I learned it mainly by contrasting it with Jazz, and the white book was more convenient for this than the brown one.

User avatar
Mark Featherstone
Posts: 544
Joined: Tue May 27, 2014 12:26 pm
Location: Alameda, CA

Re: Formula for harmonized triads of the minor scale

Post by Mark Featherstone » Mon Sep 16, 2019 4:39 am

musikai wrote:
Sun Sep 15, 2019 9:08 pm
"Kai" = "what?" in Marathi
Haha, that's great! :D
Francisco Navarro Concert Classical, cedar top, 630 mm scale, 50 mm nut

"The trouble with normal is it always gets worse."
Bruce Cockburn

Return to “Analysis of Classical Guitar Works”