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I'm working on writing some sad sounding songs. Every time I'm jamming by myself on the guitar, I can play a few chords and sing or whistle what the leads could be. Leads come to me easily, and I can create vocal melodies and guitar melodies that seem pretty interesting to me.

But my rhythm guitar chord progressions invariably end up as some combination of Em, C, D (sometimes Em, C, G, D) or Am, F, G. So this sounds pretty sad, and it's easy to write some leads over it, and the leads definitely make things more interesting, but I need to be more creative. Every song can't have that same chord progression. (or a transposition of it)

I've tried adding in more chords, but it doesn't sound right. And I've tried doing key changes too, it also doesn't sound right. While the key changes can make things sound "evil" it doesn't make it sound sad, in fact the sadness seems to be lessened by adding in extra chords.

One idea I have is trying sus chords, as those can definitely add a nice flavor and can still fit in with the theme of a sad song.

I also like building up several major chords before going back to a minor chords as the root, as it sort of has the effect of building up "false hope" in the listener before crashing back down to a minor sound...basically I think why Em C D works so well.

And of course I can play around with variation of the chords, like arpeggios, extra notes, etc, but that doesn't change the basic chord progression for the rhythm guitar, and it's getting into lead guitar territory anyway (ie avoiding the problem I am having rather than trying to solve it)

And it's also easy to play around with patterns, timing, etc, etc, but again that doesn't get around the fact that the rhythm guitar is still playing the same three damn chords.

Voicing is two guitars and vocals.

Happy to keep experimenting but what exactly should I investigate now? What aspect of music theory should I study? I don't know anything about modes, is that an area worth study and perhaps is applicable here? Or perhaps there are new chord shapes I can look at? (I'm really just doing major and minor chords at the moment)

Or what else would you recommend?

tl;dr: How can I keep the rhythm guitar parts sounding interesting but still keep the emotional content that I want? (And what aspects of music theory should I study and what books could I look at?)

Thanks!

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    When working with a major tonic, you can modulate down a whole step for a sad sound. For example, here's a progression that might work in the tonal center of CMaj: Emin - Amin - Emin - CMaj - GMaj - FMaj - GMaj - FMaj. Or you could just do: CMaj - CMaj - BbMaj - BbMaj – jdjazz Jul 12 '17 at 15:02
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    It means your song is in the key of CMaj--that's the "home base" and the place where the chords tend to resolve/want to move to. It's the tonal center for the entire song. – jdjazz Jul 12 '17 at 22:17
  • Is a "tonic" different from the "root"? – cat pants Jul 13 '17 at 0:18
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    The term "root" can apply to any chord, whereas the term "tonic" refers to the single tonal center of a song. Many songs have many different chords each with a different root, yet only one tonic chord. – jdjazz Jul 13 '17 at 0:29
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What aspect of music theory should I study?

The domain of music that you're working in and want to study is composition. I think it would be extremely beneficial for you to study the elements of composition, because that's how you'll develop tools for (a) understanding what you like in a song, (b) knowing why a song feels sad, and (c) experimenting with new chords. Finding a good book on modern music composition probably won't address the question of "what makes a song sad?" but it will give you the skills to answer this question yourself.

I don't know anything about modes, is that an area worth study and perhaps is applicable here?

You're right here--modes will be very important, because they form a family of chords that work well together. Here's a chart that shows this. Each of row of the chart shows a particular family of chords:

http://www.playpiano.com/101-tips/images/family-Chords-chart-2.jpg

The first row applies to the keys/tonal centers of C maj and A min, the second row applies to the keys of D♭ and B♭m, etc. So the first row tells you what chords you have at your disposal to use when constructing a song in C maj or A min. Above the top row, you'll see column headings with Roman numerals (I, ii, iii, IV, etc.). These letters are assigned to the degrees of the scale, and they are "key-agnostic" and apply to every key. It's valuable to refer to the Roman numerals instead of referring to the letters themselves. For example, when we use Roman numerals, we can see that C-Dm-Em and G-Am-Bm are the same progression: they're both I-ii-iii, but in two different keys. That's the benefit of thinking about Roman numerals--they allow us to generalize chord progressions so that we don't have to learn each progression in every single key. Thinking about songs using Roman numerals gives a broader way of conceiving of chord progressions, which will help you vastly expand your understanding of the different progressions. When using Roman numerals, we sometimes use lowercase numerals for minor chords and uppercase numerals for major chords.

The family of chords you see above is a great starting point, and there are tons of great songs written exclusively within the family. However, it's worth knowing that this family of chords isn't prescriptive. For one, chords in this family can be modified. A common example is changing the iii chord (e.g., Em in the tonic of C) to a III7 chord (e.g., E7). Second, there are plenty of songs that use chords outside of this family altogether. For example, Coltrane Changes represent a chord progression that cycles through three different tonics/tonal centers.

Or perhaps there are new chord shapes I can look at? (I'm really just doing major and minor chords at the moment)

Playing a chord with a different shape/"voicing" will change how the sound comes across. This can sometimes relate to whether a song feels happy or sad, but generally it will have a much smaller impact than other things because the chord quality (major vs. minor) remains the same when changing only the voicing/shape.

On a related note, adding "higher extensions" refers to adding notes into the chord that are above the 5th. For example, instead of simply playing Cmaj as C-E-G (or 1-3-5), you could try playing Cmaj9 (C-E-G-B-D or 1-3-5-7-9). As with all chords, the particular notes can be moved around into different orders, and some notes can be omitted. There are many different voicings/shapes, and their impact on the happiness/sadness of a song will probably be small.

For the purpose of composing sad-sounding songs, I don't think the things under this heading (different voicings/shapes and higher extensions) are too important to focus on, at least initially. As you see from the picture above, none of the chords in the family are required to have higher extensions. I think the question you've asked can be studied successfully without worrying about either of these things.

What else would you recommend?

Analyze any songs you like and think are great. Determine the tonic/tonal center, and write out the chord progression using Roman numerals. The chord progression is crucial, but there are other things to note, including tempo and instrumentation. (The theme song from Disney's movie "Up" is an outstanding example of how tempo can be used to affect how happy or sad a song sounds. Also note that the song is sadder when it switches to solo piano in a higher register.) Melody can have a big impact too--does the contour of the melody go up? Does it go down? Does the melody consist of a single tone being held for a long time while the chords change beneath it? Does the melody consist of many notes being played quickly?

When you do this analysis, it can be beneficial to study not only sad songs, but also happy songs. Consider the ways in which a sad song and a happy song might share certain elements, and then find the specific qualities that distinguish one as happy and the other as sad. Choose a variety of sad songs that you like, and potentially select songs from different genres (e.g., The Two Lonely People by Bill Evans, True Love Waits by Radiohead, Adagio for Strings by Samuel Barber). Google can be useful here for finding works to analyze. I found the Adagio for Strings by searching "sad song classical music."

I think you're taking a great approach by studying these elements of composition and music theory through examples and through your own composition. In that same vein, I encourage you to write songs as you study these elements of composition. When you uncover a technique you think is interesting, write your own song with the same chords, or with the same tempo adjustment, or with the same melodic counter, etc. This will promote deeper internalizing of the technique and will give you a chance to apply it in a new context. Maybe keep a notebook of the techniques you've analyzed. Writing them out in your own words is a great way to learn, just as you've already done:

I also like building up several major chords before going back to a minor chords as the root, as it sort of has the effect of building up "false hope" in the listener before crashing back down to a minor sound.

Building up your own set of rules and writing them out in a way that you find intuitive is an outstanding way to learn anything. As you dig deeper into composition/music theory books, you'll find formal theory rules that codify some of the things you've discovered, which is really cool. And there will be other cases where you write things down that don't appear in music theory/composition books, but they'll still be extremely useful to you. I remember having the realization that Brad Mehldau's tune called Song-Song uses a melodic theme over both a minor chord and then later over a major chord. This was a revelation to me, and while I haven't come across the idea in a formal composition book (maybe I haven't read enough books!), I still have used it to compose. In fact, the first time I came across this technique, I was wrong--I articulated it in a way that didn't correctly describe Mehldau's "Song-Song." Despite the fact that I was incorrectly describing Mehldau's tune, I still had a tool that was useful to me and helped me write a new song. You'll come up with tools/techniques like this too, and they can be really valuable to you in your composing.

  • "" the first row tells you what chords you have at your disposal to use when constructing a song in C maj or A min""- Please explain this.i write songs in C major/A major with each notes in scale with specified meter (4/4,3/4...) without considering the chords.once song is finished then for instrumental i apply Chords matcheing with each bar.Hence iam confused how you construct songs using chords? – robert winsly Jul 19 '17 at 16:22
  • I think there are multiple ways to compose, including (1) composing melody first and chords second, (2) composing chords first and melody second, and (3) composing chords and melody simultaneously. (My answer describes method 2, and your comment describes method 1.) I think all three are valid. – jdjazz Jul 19 '17 at 16:56
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Which artists inspire you?

If you like sad music, I'd recommend listening to Chopin. He can sound sad even when his composition is written in a major key.

If you'd like, here's a recording of me playing a Chopin Nocture - it's in minor, but the chord progressions are fascinating:

Perhaps in the rock genre you could check out Neil Young or Elliot Smith. Both songwriters were simplistic but chose a careful progression of chords.

Do you like Radiohead? "Exit Music For A Film" has a fascinating minor chord progression that is reminiscent of Chopin.

Maybe you would like harmonic minor chords.

e.g. ii-V-i - Dmin7b5 (D, F, Ab, C) - Gdim7 (F, Ab, B, D) - Cmin9 (C, Eb, G, B)

or e.g. VI-V-i - Ab (C, Eb, Ab) - G (B, D, G) - Cmin (C, Eb, G)

The chords are sort of arbitrary. I think it's best to find music that inspires you and try to follow some of the chords you hear. Along the way, you will learn new progressions that you can use for your own music.

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Happy to keep experimenting but what exactly should I investigate now? What aspect of music theory should I study? I don't know anything about modes, is that an area worth study and perhaps is applicable here? Or perhaps there are new chord shapes I can look at? (I'm really just doing major and minor chords at the moment)

Or what else would you recommend?

I recommend going back to learning mode (instead of composing mode) and adding more "tools" to your composing "toolbox". You're trying to get a certain sound, and you can't get it, which means you don't have the knowledge in your mental library to go to. You have to add that knowledge in so it's there when you need it.

Since you find yourself using the same chords over and over again, it means to me that most of the songs you know how to play use those chords. Learning to play songs written by someone else is a very important part of learning music. By studying where people have gone before you, you learn both how to follow in their footsteps as well as how to go off on your own.

One great songwriter to study to understand how to create music that is sad all the way through, from individual notes to entire textures, is Robert Smith of The Cure. The Cure album Disintegration has some of the saddest music I've ever heard on it - so sad I don't even like this album, but Robert Smith's writing and guitar playing on Disintegration are both a big influence on me.

"The Same Deep Water As You" is basically sad songs 101 through 423 - it's an entire undergraduate degree in sad songs and it's only three chords. After you graduate from that song, there's "Pictures of You" to give you another song to unpack.

Beyond that album, there are plenty of sad songs to learn and study, like "Sound of Silence" by Paul Simon, "Black" by Pearl Jam, "Mad World" by Tears for Fears - really I could go on and on, but you should learn the songs that you think are sad.

Having learned a lot of songs and having studied a lot of music theory, I'm compelled to mention that theory has done virtually nothing to help me write, but learning other's songs has been the most important thing in my writing ability. It's like you're fueling up your brain with raw materials to make songs with. Or going to the grocery store and buying all kinds of ingredients that you can use to make a stew or a soup or a roast or whatever.

You should be learning songs that inspire you all the time. By learning songs, you'll not only learn more chords but also different ways to use those chords. You'll start to see patterns on the guitar neck. With only one chord to start with, you'll be able to hear and see so many options of what other chords to use, and you'll know whether those options will make the song funny, sad, danceable, or whatever.

If you don't think you've learned at least 100 - 200 songs so far (even if you've forgotten all but 3 of them), then that's probably holding you back.

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I'd agree that you would be benefited by adding some variety to your musical vocabulary. I see from your question that you are taking a somewhat typical approach to sad music, using minor chords/keys. It's definitely good to sprinkle in the major chords to have that juxtaposition and make your minor chords more sad by comparison.

The thing that I think you should consider is that a lot of what makes a sad song sad is how it is written and performed. There are plenty of sad songs that are in major keys and/or consist mostly of major chords. Tempo can have a big impact as well. A sad song is typically at a slower tempo, which I think has a lot to do with faster music inspiring movement and sadness often puts us in a state of dormancy.

You might want to try out some Major 7 chords. While an emotional interpretation of art is entirely subjective, it seems to be pretty commonly accepted that a major 7 chord typically doesn't hold the same level of happiness. I tend to find that they have a more sentimental sort of feel to them and a sense of longing, which can easily be used to convey a loss of love or life.

Melodically, I would pay attention to your non-chord tones and the tension they can create. Benjamin Zander’s TED Talk does a good job of talking about how Chopin uses the melodic direction and tension to convey sadness. You don't necessarily need to learn a bunch of theory to do this either, it's really just about hearing how the notes that aren't in the chord feel when played against it and manipulating the tension it creates.

It's a little unclear whether or not you are writing lyrics but the lyrical content can really transform a song. Paul Simon's song "You're Kind" is a good example of a song that doesn't sound inherently sad but definitely tugs at my heart strings a little. The song really sounds like a sort of happy love song that subtly transitions to the fact that the relationship is ending. I'd generally recommend listening to Paul Simon's music for ideas. He really encompasses a wide emotional range with some songs that sounds happy but are sad, some that are happy but sound sad, as well as those that sound as they are. Still Crazy After All These Years is a great album from his early catalog that does well to convey the full emotional spectrum.

I'm a big advocate of theory in general but it's definitely not something you absolutely have to focus on. It could be helpful as far as determining what it is about a given song that is giving you the feeling of sadness but there is no theory that explicitly covers how and why something is emotive since the emotional response is entirely subjective. Many theorists speak to the emotive response within a piece that they are analyzing but there is no checklist for what to include if you want someone to cry when they listen to your tune. I'd recommend listening to and learning some more songs that give you the emotion you're going for and trying to figure out what those pieces have in common. Make sure to look outside your preferred genres as well, particularly at instrumental pieces from the Jazz and Classical realm, as these are genres with a very long history that have lots of examples for you. Instrumental music can give you the focus on the actual musical aspects of what makes the piece sad without paying attention to the lyrics. You can also try listening to some sad songs in foreign languages. This would allow you to hear the emotional expression of the vocals without focusing on the lyrics. I'd also recommend listening to some movie/tv soundtracks but be careful to notice that the visual content can influence the emotive response drastically.

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Sounds like you're locked into your repertoire of stock chords, ones you know and can describe with simple chord symbols. Put your guitar down. Grab a keyboard. Play the tune you've come up with. What single notes (not stock chord shapes) go well against it?

If you MUST work on guitar, take a different approach. What chords could go under melody note C? C major, Dbmaj7, Dm7, E+, F, F#dim, Gsus, Ab, Am, Bb9, B7b9 - and that's only a start! If you don't know all these chords, start learning.

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One of the things I love music is the ability to convey emotion other people. Obviously, this is pretty personal, and people will interpret a piece in different ways. I mention this, because a chord progression that sounds sad to one person may sound trite and insincere to another. I'm not trying to pour cold water on your question, but to frame my answer. My ideas may not work for you.

My first suggestion would be to analyse songs that are 'sad'. Remember, this is going to be different for everyone, so pick ones that feel sad to you. What makes them feel sad? What chords do they use? Don't just trust a chord chard that you found online; play along, and write out the chords yourself. It's a really good way to learn.

What else could you look at? Well, let's talk about the chord progression you've mentioned. Both Em C D and Am F G are the same progression. I would call it a vi IV V in G Major and C Major respectively. You may observe that the tonic chord doesn't make an appearance. I think this ambiguity is part of the reason this progression feels sad. If you're not familiar with this part of music theory, perhaps start researching chord structures and notation.

You've already brought up suspended chords. Here's some sounds you should explore:

  • Suspended Fourths and Seconds. Don't feel like you have to resolve them.
  • Major Sevenths
  • The name varies, but try suspended sharp fourths. For example, the chord G-C#-D in the key of D Major.
  • Also, you can just add a fourth/second/both to a major or minor chord. As in, C-E-F-G in C Major.

These chords all have varying degrees of tension. However, they avoid intervals like dominant sevenths. That's because I don't really like that sound. You may have a completely different opinion.

I also like borrowing chords from parallel keys. Take Am Fmaj7 G D. I borrowed the D major chord from A Major, rather than my home key of A Minor. Or is it G Major? It's a little ambiguous. Perhaps that's why it works. E Major would be another chord I'd consider borrowing.

There's many other techniques; I would start by transcribing some recordings. Then, try to figure out why they work, and see what ideas you can replicate.

Here's some further reading that may be of interest:

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Here is an answer to you question and give you a different perspective on the problem.

Let's look at a jam with Am - G - F - F

for now, in std tuning (EADGBE). Playing the roots on the 6th string, you'll play using the CAGED chordal system. That's great. Track one guitar doing that, and track a second guitar playing inversions.

  • Am = A C E: Root, minor third, fifth

  • G = G B D: Root, third, fifth

  • F = F A C: Root, third, fifth

The minor third and major thirds in these three chords give the tonal movement you're after. Typically, these notes will be buried inside the 'standard' chord shapes on the guitar, but if you play inversions and let them be the highest notes of the chords, you'll really accentuate the movement in the melody.

Harmonic Minor & Chromatic notes

One of my favourite techniques.

(Quick background) In the key of C, the 5th degree is G Major chord. G Major has a B in it, which is one semi-tone away from C and thus provides a bit of suspense/ tension. Playing a G Major followed by a C Major gives a feeling of resolution. In the minor key, you don't get that resolution by moving from the Em to Am for instance. The way to get that tension, is to play E Major in place of the E minor.

(Resume) So, now that you're playing some nice CAGED chords, and some inversions (playing them up near the 12th fret for a 'pretty' sound), you can cycle through Am - G - F and occasionally drop down to E Major.

Am - G - F - F

Am - G - F - F

Am - G - F - F

Am - G - F - E

What makes this sound interesting is the notes played in order, C - B - A - Ab. It's got a nice chromatic movement going on which will really catch the ear, you won't find that movement inside the diatonic scale. When you resolve from the E Major, to Am, it is quite powerful.

Depression

Pink Floyd's Comfortably Numb starts out with a few bars of B minor chord, layered with a B- power chord. It then proceeds to play B min - A Maj - G Maj - E min chords. After a chorus, B min is replaced by B sus 2. Later in the song, I think it returns to B min.

Chord review:

  • major = Happy
  • sus4 = even more joyful
  • minor = sad
  • sus 2 = depressing,

Mixed feelings

Pink Floyd's twenty something minute song Echoes, from the B-side of Meddle, has a very interesting approach with chords. In the verse it uses C# min, G# min, F# min, G#7. In the chorus it changes to really interesting C#, G#, F#m, G#7 (you won't see a movement like this using the diatonic system). Then there is the famous bit C#, C, B, A#, A.....A, A#, B, C, C#..... which, so I understand, Pink Floyd took Andrew Lloyd Webster to court over for copywrite infringement (Phantom of the Opera). Echos takes the listener on a journey, invoking a number of mixed emotions along the way.

Sentinel

Hawkwind's 2010 album Blood Of the Earth features a song called Sentinel. It follows the progression, Em, C, Bm, C. The chorus then plays G, D#, Dm, D# - not in the same diatonic scale as the verse. When I heard it for the first time, I was very impressed, and I still am.

Arpeggios & Delicacy

Strumming can be good fun, it can be used to convey gentle or aggressive sentiments. Playing single note lines from chords (listen to the Hawkwind example), can convey a delicate sound. For sad music, delicate sounds can be useful. The Hawkwind example plays two ascending notes from the 'bottom' of the chord, then the descending notes from the 'top' of a chord.

Let there be silence

I heard a quote from Roger Walters a number of years ago. To paraphrase, "Don't play stuff for the sake of it. Sometimes, it's better to let there be gaps between parts, room for the music to breath.

According to Spinal Tap

Dm is the saddest key.

Power chords

Sometimes, a lack of major or minor tonality can be just as useful by employing 5th chords. You'll see this technique used in a number of Pink Floyd songs, notably songs from The Wall album.

Dynamics & The Sound War

Industry production of music entered the business into The Sound War, an arms race of volume. Loudness tends to be a big factor in the success of a song. People tend to like louder volume music. Queen famously had their engineer remove the limiter from a Live Aid gig during their set and replace it with their own. According to the story, they were MUCH louder than everyone else, and their performance has gone down in history as one of the best. Loudness and record sales have shown positive correlation. Mix engineers try to maximise the volume of music played over the radio by using compression and limiting. When you listen to a song next, watch the volume meter jump up and down. If its a song from the last 20 years, you'll probably see it jumping around the yellow and red section of the meter. Nowadays, the sound war is starting to fade with most music being streamed online rather than purchased on physical media or consumed on the radio. YouTube is leading the way, by normalising the music (turning it down). This allows the engineers to make use of dynamics when mixing songs! Dynamics doesn't mean play quietly. It means play quietly when it works for the song, and play loudly as well when it works for the song. Classical music makes heavy use of dynamics. Try to incorporate dynamics into your arrangement. Judas Priest and Black Sabbath use dynamics, and they are Heavy Metal. They have some great songs. Judas Priest's Beyond the Realms of Death is a sad song. The band has two guitar players and they really can play well. Notice how the song uses an arpeggied minor chord with a descending bass note during the clean/acoustic guitar sections, then goes into raging power chords for a powerful effect during the chorus. Black Sabbath's Wheels of Confusion is an interesting song to listen to. The introduction is quite sad, then the verse kicks in and it's more upbeat. When you get about 5 minutes 10 seconds into the song, the tone changes completely and turns very dark.

Finding different places to play chords on the neck

Basic chord theory:

  • Major chords are comprised of a Root, Major Third and a Fifth note.
  • Minor chords are comprised of a Root, Minor Third and a Fifth note.
  • Sus 2 chords are comprised of a Root, Major Second and a Fifth note.
  • Sus 4 chords are comprised of a Root, Perfect Fourth and a Fifth note.

There are some very informative tutorials on the major scale, the minor scale and chord construction. I learnt from the Dorling Kindersly Complete Guitarist. It's a wide topic, but its very rewarding to learn even the basics. With this knowledge and the knowledge of which note is played at each fret of each string on your guitar (half a days learning), you will be able to pick any cluster of corresponding notes to make a chord or an inversion of a chord. There is an A minor triad shaped like part of an A Major 7th chord, which is played half way up the neck. When played in conjunction with an open A minor chord, I think it's really pleasing.

Seventh Chords

Major and Dominant Seventh Chords can be used to add flavours to your chords. For each degree, the major (capitals I, IV,V), minor (lower case ii, iii, vi) and diminished (vii) chords have next to them, listed the type of diatonic 7th chords available.

  • I - Maj 7
  • ii - Dom 7
  • iii - Dom 7
  • IV - Maj 7
  • V - Dom 7
  • vi - Dom 7
  • vii - Dom 7b5

Sometimes they add embellishments which are great. Sometimes, they really detract from a passage. An A minor 7 chord (A C E G) has C E G in it which are the notes of a C major chord. If you have a C chord in a passage and want some variation, try replacing your C with Am7. Play it in a jam such as

Em, D, C, C

Em, D, Am7, Am7

Chord substitutions

Wikipedia has an introduction article to chord substitutions. There are some very interesting uses of substitutions which you can hear in music by The Beatles.

Summary: How to use your guitar to make a song sad

  • Have a story in mind to tell, and use your instrument to tell it by applying dynamics, chromatic movements, arpeggios and other techniques.
  • Variation is a big part of keeping the attention of listeners. We humans are remarkably good at blocking out repetitive things in order to dedicate more 'brain-power' to other tasks. Add variation to your song, and you'll maintain attention. Variation also provides you with the opportunity to give mixed emotions during different parts of a song. If you want an example, queue up 16th note hi-hats in a DAW and play 4 bars worth. You'll see it grates on the ear. now apply randomised velocities and some randomised positioning, (in logic pro the humanise function does this), and notice the difference.
  • Use dynamics
  • Don't play notes/ chords for the sake of it. Let the music breath
  • As a musician, you're job is to manipulate and guide the emotions of the audience. Find chord progressions, strumming/arpeggio patterns and tempos which allow you to do so.
  • Listen to other people's music which you like and re-create their songs. You'll learn and remember more technique by re-creating a handful of songs than you will by reading
  • C-B-A-G# is more accurate. And what the heck is a minor dominant seventh. You surely mean a minor seventh. – Tim Jul 16 '17 at 17:38
  • @Tim yes, minor seventh. Got caught up in writing dominant and major to provide distinction – DWD Jul 16 '17 at 17:42
  • "Dm is the saddest key." In ET all minor keys are the same by definition. – user1803551 Jul 17 '17 at 4:20
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    @ user1803551 A bit of humour to intersperse a long answer – DWD Jul 17 '17 at 7:41

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