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This now is music theory.
In the overtones of any note, lets take C as example, appear some useful intervals. For note C they are represented by E, G, D and B. (D appears later than E and G, because it is less consonant.) We can create some chords with these notes, these are:
C = C-E-G
CM7 = C-E-G-B
Csus2 = C-D-G
Cadd2 = C-D-E-G
What gives? When we reverse the chords (equivalent to consider undertones instead of overtones) and then shifting the new root back to C, we get
C reversed = Cm, C-E♭-G.
CM7 reversed = stays a CM7. An M7 (or maj7) chord is quite common.
Csus2 reversed = Csus4, C-F-G. A sus4 chord appears quite often.
Cadd2 reversed = Cmadd4, C-E♭-F-G.
Now, is "minor add 4" an actual chord in the sense that is used?
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Overtone scale use is not an exact science at all. When you start getting to those less common chords, context is as important as anything in their use. We can define these things in a strict manner, but relation to music is the real issue. In the practice of music, these exactitudes generally have little bearing. They become interesting rather than relevant, beautiful, enjoyable, etc.
A m(add4) chord doesn't really have much meaning by itself.
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I know that context is of great importance, for example to tell an inverted minor chord from a sixth chord. I also know that the harmonic series (which the overtone series follows) only provides us some intervals, while the chromatic scale consists of many notes which do not appear in overtones.
Currently I think that the harmonic series or overtone series can be used to establish basic harmony. For example it is possible to explain why a just major triad is extremely consonant and why we perceive the root note as root indeed. While a minor chord is also consonant, even though a bit less, it is not possible to explain it directly with overtones. So I am trying out if the reversing of the intervals of consonant chords (so consonant that they can be explained using overtones) also leads to a, maybe somewhat less, but still harmonic chord.
If this hypothesis is correct, an m(add4) should be a bit less consonant than an add2 chord.
edit: I just tried it. m(add4) is suprisingly harmonic – compared to add2. m(add4) is a bit less consonant, but still creates a colorful sound with no sharp dissonance.
edit2: Of course there are a lot of useful chords which use way more tones than appear in the overtones of the root note. I am not trying to creating a complete chord theory here.
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Consonance and dissonance are relative to the listener as well. While there is some degree of science related to the topic, I would be very wary of giving much value to anything beyond the basic and common chord structures. I think it's good to go through that overtone system in an organized fashion if it helps you understand your "ear" better. Something you might be interested in is general psycho-acoustic effects. Specifically, check out writings on Steve Reich's "Music for 18 Musicians." That stuff is fascinating.
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I listened through a piece available on Reich's webpage. http://www.stevereich.com/mp3.html Strange, yet interesting. I felt taken on a journey. It was summer, but the sun was low and the wind already a bit chilly. But I did feel great. It somewhat reminded me of earlier works of Mike Oldfield.
It takes of course a lot of work and creativity to write and to perform good music. I view notes and chords similar to colors. The artist still have to paint the picture and I will not be able to explain the picture when I just describe the wavelength of the color red or the chemical composition of the red paint he used.
I am not always theorizing with numbers. I have a keyboard, a glockenspiel, and a flute. Even though I play neither instrument good enough for an actual song, I like to have this experience of structured sound. I heard the effect of the dominant or the leading tone long before I read about such concepts. A great part of my theoretical effort is to find a satisfying explanation why I do feel tension, or resolution. There is surprisingly little literature about those topics. But I am now satisfied with the explanations I found why the bass note of a perfect fifth interval appears as root: Overtones and combination tones both show, that the bass note if amplified and contains the higher note, but the higher note does not contain and therefore does not amplify the bass note.
Most disquisitions about the minor triads I read however are just playing with numbers or geometric figure. The "explanations" are so elaborate that I feel they are not really an explanation but rather a mathematical coincident.
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Well said, sharkeyanti, and I'll go one further.
Modern music theory is not concerned with natural acoustics (except psychoacoustics), even for the "simplest" triads, but rather with theorizing musical behaviors and patterns. For example, the enharmonic respellings that crop up in Haydn and Mozart are understood as having functional structural connotations -- i.e., double meanings, modulatory significance -- rather than as pointing to some abstract desire for a "pure" natural triad. While there have always been critics of the acoustical basis for a music theory, the main reason for this almost universal change of approach is that harmonics does a poor job of explaining even the most "basic" products of Western musical culture (and a terrible job for many other cultures).
Recent experiments (e.g., Neil McLachlan's "Consonance and Pitch" from this year, which also gives a historical overview) show that harmonics can't even account for the basic experience of consonance and dissonance. It might be useful, then, when conceptualizing the various "sus" chords, to consider how they got their names and what sorts of expectations they carry in a conventional tonal context.
That being said, there's always a place for discussion of historical ideas about ratios and harmonics, the old musica theorica. Zarlino in the late sixteenth century devised a theoretical tuning based on the syntonic diatonic, using ratios going up to the number 6 (the senario). Rameau tried to base his theory on "natural" properties of sound and even ingeniously argued that the supposed acoustical basis for a chord root (the "fundamental") also explained the behavior of chords in a musical context. He ran into the same problem as [F_]aths with the minor third, which (along with its inverted "shadow," the major sixth) does not occur over the fundamental. As Joel Lester writes in Compositional Theory in the Eighteenth Century:
+ Show Spoiler +Rameau never abandoned the search for an origin of the minor third. He was not alone in this search -- theorists from antiquity to the present have sought the sources of their musical system in divisions of the string or [numerically identical] harmonic resonance...Ultimately, the reason a solution has never been found is the fallacious basis of the search: an ethnocentric belief that music, a cultural phenomenon, is a Natural phenomenon... (pp. 102-104)
Lester points out some assumptions this kind of theorizing requires. My two favorites: (1) octave equivalence, which is useful in early music pedagogy but deeply flawed in both practice and in mathematical reality. (2) the assumption that simple ratios (or string divisions) represent actual musical intervals that one hears in various tunings.
I'm really glad the OP has opened up basic Western musical concepts to these forums. You don't have to be a great performer to develop subtle listening skills; likewise, conceptual development helps performance and listening alike, regardless of genre.
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Thank you for the extensive criticism, gorbonic.
Those are actually my current two big issues: Not being able to order the intervals of the chromatic scale by consonance, and not being able to explain the major sixth. Treating it as an octave complement of the minor third is the best I can do, and the minor third only appears as fifth complement of the major third in the first place.
+ Show Spoiler [Books I read about music harmony] +
I read the entire Harmonielehre vom Diether de la Motte. He does not go deep into the theory, he rather focuses on harmony concepts used in practice. I also read most of “13 Tonstufen in der Duodezime” from Heinz Bohlen. He got deep into the theory and proposes to use 13 notes spanning a twelfth. The “Unterweisung im Tonsatz”, The Craft of Musical Composition by Paul Hindemith does not seem to be available as book, nor online, so I depend on Wikipedia here. Hindemith tried to offer a new explanation for our 12 step chromatic scale, but while many of his ideas seem to be good, he does not argue in a consistent way.
Still I think, the consonance of the minor third is explainable:
A) The minor third appears not in overtones, but as an interval in major triads. We only slightly modify the major triad by a minute change of the inner interval. It gets shifted by just a chromatic semitone, to get its relation of the middle note from being 1/4 higher than the root note, to 1/5 higher than the root. The gestalt of a triad is still there. While the minor third does not support the root note with overtones or combination tones, the perfect fifth still does. We still use both types of thirds, which are not that different anyway. We still see that it is a triad.
B) In music practice, we use the concept of scales. If we construct a scale from a string of three major triads (lets say the triads for F, C and G) that scale will also include minor triads. This has to happen. Let us connect F-A-C, C-E-G and G-B-D.
F-(maj)-A-(min)-C-(maj)-E-(min)-G-(maj)-B-(min)-D
We get A-minor and E-minor. Many tuning systems also allow for an acceptable D-minor. Only the root note determines if we perceive the scale as major or minor, the white-key notes allow for as many major as minor triads.
On June 07 2013 04:31 gorbonic wrote:Lester points out some assumptions this kind of theorizing requires. My two favorites: (1) octave equivalence, which is useful in early music pedagogy but deeply flawed in both practice and in mathematical reality. (2) the assumption that simple ratios (or string divisions) represent actual musical intervals that one hears in various tunings.
I think the mathematical properties of the octave justify its use. Your second point is in my opinion the great advantage of the western chromatic scale: One can construct a pentatonic, or diatonic, or chromatic scale only using octaves and fifths. A diatonic scale can also be constructed with three triads. To get the five missing steps for a chromatic scale, we calculate the octave inversions of the other intervals. While the tuning is a bit different (when we begin with C, the notes A and E-flat have to be adjusted by the syntonic comma) we can use the context to underline if we use distances by fifths for the relationship, or simple ratios.
I consider equal temperament the height of western music. While it sacrifices the purity of thirds, it allows to use all notes in different contexts and thus provides a rich palette with only 12 steps. From any note, it includes the ratios 1:2, 2:3, 3:4 with extremely good, and 4:5 and 5:6 with acceptable precision. It also provides a circle of (slightly tempered) fifths.
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I've played music my entire life, and this post has taught me more than I ever knew xD
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hello mates, I'm pro musician, saxophonist, I have never seen this link, mmm I'll read it and if I could help let me know PM 
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"Natural minor has a lowered 6th and 7th scale degree and its overall pattern is W-H-W-W-H-W-W."
I have thought long and hard about this. I am now confident to say that this description is not correct even though the outcome is the same. Western scales are not constructed by tone steps. The scale is constructed by strings of fifths or by triads (leading to different tunings) and then arranged into a chromatic scale of half-tone steps.
No-one understands why the minor scale is "w-h-w-w-h-w-w". But one can understand it as a string of three minor triads. Or as one out of seven modal scales (this approach takes a bit longer to explain, but extends the major-minor-dualism to seven modes, which is useful to explain diatonic scale functions. I see some issues in common diatonic function descriptions as well.)
Why are the half-tone steps of a minor mode there and not somewhere else? One cannot explain it unless one explains how major works with major triads and that one have to flatten the third to get a minor triad. Since three triads are involved, three notes have to be flattened. Since the triad in the middle is the tonic one, one can also compute on which position the flattening occurs.
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OK, I finally got around to updating this thread since the site move, so now all the images have returned to their normal place. And I did want to address something you mentioned previously Faths:
Why are they arranged WHWWHWW in that specific order? This is explained very simply by understanding that the natural minor scale is simply the aeolian mode. Period. While the definition from the text is being written from a post-analysis perspective, it's not wrong in the slightest.
Music has always been more practical than scientific, and that trying to ascribe purely scientific principles to post-describe what happened in practical terms is somewhat problematic. It's interesting to look at this way for sure, but it's not indicative of how music evolved.
Let me give you an example. In the text A History of Western Music, Peter Burkholder writes"Aristoxenus (a pupil of Aristotle) distinguishes between continuous movement of the voice, gliding up and down as in speech, and diastematic movement, in which the voice moves between sustained pitches separated by discrete intervals. A melody consists of a series of notes, each on a single pitch (in ascending or descending order)." Additionally, in ancient Babylonia/Mesopotamia and Greece, tetrachords which were the common unit or space for four-note combinations (to which there are three most common iterations), span a 4th and not a 5th. There's a lot more, but a lot of what we know of reinforces the idea that melodic content was formed around speech patterns and scalar melodic tendencies rather than the harmonic series (which ofc was only 1st proofed in the 14th century). Of course I don't know 100% because I'm not a musicologist with expertise in ancient theoretical models, but I'm going off a lot of what I've learned, intuit, and have in books.
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will Part II and Part III be posted anytime soon? i'm quite interested in this
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On November 22 2014 01:10 kaleidoscope wrote:
will Part II and Part III be posted anytime soon? i'm quite interested in this
As a "playing what feels right" guitarist trying to learn music theory, I'm also quite interested in this. I also have a mathematics degree so that makes me doubly interested in what the exact science is behind music.
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On November 26 2014 10:04 mierin wrote:Show nested quote +On November 22 2014 01:10 kaleidoscope wrote:
will Part II and Part III be posted anytime soon? i'm quite interested in this As a "playing what feels right" guitarist trying to learn music theory, I'm also quite interested in this. I also have a mathematics degree so that makes me doubly interested in what the exact science is behind music.
It is not exact science in the sense of tuning. If you try to compute pitches with the different approaches I describe here, you get slightly different pitches. Guitars and piano use a tuning which does not get either approach exactly right, but which offers a good approximation.
This tuning leads to C# and Db (and other enharmonic notes) having the same pitch, while the perfectly tuned frequency would be slightly different. But I focus on white-key notes for the basics.
Older penatonic scales, like C-D-E-G-A offer quite nice ratios to note C:
C = 1:1
D = 1/8 higher than C
E = 1/4 higher than C
G = 1/2 higher than C
A = 2/3 higher than C
C' = 2x as high as C. 2x or 1/2x the pitch fits extremely well, so that we use the same note name.
Simple ratios are easy to discern and to understand, and easy to sing along with. C-G, ratio 3:2, is especially nice. Not only the ratios to note C, the ratio of any note to any note is a more or less harmonic, "consonant" interval.
There is another principle at work here, the already mentioned 3:2 ratio G-C, the "perfect fifth" (since it is the fifth degree in our today's 7-tone, diatonic scale.)
Lets view fifths.
C - G - D' - A' - E''
We have a chain of fifths. And getting the notes of the pentatonic scale. Some notes are in different octaves, but they don't matter for the note name.
Lets view the common diatonic scale C-D-E-F-G-A-B in fifths:
F-C-G-D-A-E-B
Again, we have a chain of fifths. There is another principle at work at the same time, important for tonality (for establishing a note as tonal center.) Lets look at a major triad, C-E-F. The pitch ratio of a just intoned major triad is 4:5:6.
4:5:6 is a part of the overtone spectrum. (Every sound has its base pitch, as well as its double, triple, quadruple and so on frequency at the same time.) That means, each sound contains its own major triad! Therefore the major triad is a very natural sound.
Lets us have a look at the C-major tonic triad: C-E-G. (Note a triad also spans the fifth, C-G.)
So if we use a chain of three triads, the subdominant F-A-C, a fifth below the tonic triad C-E-G, and the dominant triad a fifth above the tonic, G-B-D, we again get all seven diatonic notes.
These three triads are important to create a tonic center, in this case "C".
The diatonic scale includes some disharmonious, "dissonant" intervals like F-B or B-C. This expands not only the melodic possibilities, these intervals can be used to generate tension and to create the desire for resolution.
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That was the best intro to music theory I've ever read. Was Part 2 ever published? Sorry to bump but it's so good and I do hope there's a sequel somewhere.
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EPT
