Music and Geometry - part I: pitch class

Introduction

In this series of 4 articles I’m going to introduce you to what for many will be the most exotic piece of music theory you’ve ever come across. Even if you think you’re not good at theory, you may want to try following along and – who knows – perhaps pick up an insight or two.

Music shows deep inner structure on multiple levels that appeals to mathematically inclined people. Math professor Guerino Mazzola has written a book called the “Topos of Music”. It’s safe to assume that only a handful of people in the world really understand the finer details of what it describes. You literally need a PhD in mathematics (in category theory to be more precise) to even begin making sense of it. Professor Mazzola argues that all this complexity is unavoidable because the music it models is just as complex.

Luckily also other books showing neat connections between music and mathematics have been written. And luckily these books are accessible to a wider audience. One of the more intriguing ones, in my humble opinion, is “A Geometry Of Music” by professor Dmitry Tymoczko of Princeton University. In these articles we will dip our toes in some of the very basics of his ideas. To be perfectly clear: I’m not affiliated to professor Tymoczko or the book in any way, and all awesome ideas are his, whereas all mistakes in the explanation are mine.

Pitch line

If you just consider notes in a chromatic scale, starting with the very lowest note you can imagine, and rising to the very highest note you can imagine, it’s not so difficult to depict these notes as points on a line. Although the picture below highlights only a few special notes on the line, actually all conceivable pitches, including microtonal ones or pitches so high they can only be heard by dogs, have their place on the pitch line which has no beginning and no end. It’s an infinite line. In geometry, such a line is considered a 1-dimensional space.


Some simple composition operations can be done by reasoning on a pitch line directly. Take transposition, e.g., and observe how it corresponds to sliding along the line. Or consider inversion of a melody and observe how it corresponds to mirroring pitches around some other pitch on the pitch line.

Pitch class

Yet this pitch line doesn’t model all existing intuitions about pitches. One intuition that is not visible on this line, is the feeling that notes one octave apart in some sense sound the same. Who doesn’t remember the lyrics to the Do-Re-Mi song from the sound of music? When reaching the end of the song, it’s made very clear that “this will lead us back to do”, as if all do’s (C notes) are equal in some sense.

A more difficult way to say the same thing is to say that in some contexts it makes sense to abstract away the exact pitch, and instead reason on pitch class. In other words we don’t speak of a concrete C4 or D3 note anymore, but of a more abstract concept of a “C” note, or a “D” note, with no indication of exactly which octave it belongs to.

The question arises what introducing this “pitch class” abstraction does to our pitch line. How does incorporating pitch class change the pitch line?

The solution is straightforward, but not necessarily trivial to come up with by yourself. To model the concept of pitch class, we need to transform our pitch line so that all differences between octaves disappear. This can be done by curling up the pitch line in three dimensions, so that all equivalent notes (e.g. all “C” notes) perfectly line up. After this is done, we will squash the curled line until it’s flat again. If you think deeply about it, the lining up and especially the squashing of equivalent pitches indeed is a kind of geometrical equivalent to replacing concrete notes with pitch classes. Just like with the pitch line, the pitch class circle has plenty of room to host all microtonal pitches you can come up with. In case you’ve been wondering: the pitch class circle as such has nothing to do with the so-called “circle of fifths”. Other than the fact that both are represented on a circle, there’s no special relation between the two circles.

Please watch the first video animation to better understand how replacing concrete pitch with pitch class has the effect of curling up the pitch line into a pitch class circle.

Before going on to our next topic, let’s sum up a few quick facts about the concept of pitch class and related subjects:

• “Pitch” indicates a specific point on a pitch line.  C4, for example, is a different pitch than C5.  The higher the number, the higher the sound.
• “Pitch class” indicates an entire group of pitches, related by their “octave equivalence”.  The pitch class of C, for example, includes C4, C5, C9 and so on.
• “Pitch class” includes notes with enharmonic spellings.  The pitch class of “C”, for example, would include such pitches as C3, B#5, D double flat 6, and so forth.

Pitch classes are “octave equivalent”, which means that pitches in different octaves are still in the same class.   Pitch classes are also “spelling equivalent” which means that notes spelled differently but sounding the same, which we call enharmonic spellings, are in the same pitch class.

Here is an example of a piece which is built entirely on one pitch class, until the final few measures.

Fear of change and its influence on the practice of music composition

1 Introduction

In this blog entry I will formulate some thoughts about how fear of change can explain a number of principles in music composition. It's very well possible that all this has been written before, and much better explained than I will ever be able to do, but I'm in philosophical mood today, and perhaps you'll start to think differently about some things after reading this text. If you experience a feeling of skepticism while reading this article, and feel like it's written by an internet crackpot theorist, rest assured that this is in complete correspondence with what the article predicts will happen :)

2 Fear of change as an organizing principle in the universe

Fear of change, while sounding specific to human psychology, really permeates the universe. In Newtonian physics, any action will cause a counter-action that resists the original action. If you push your table top down, the table top pushes back and cancels out your intention to change it (until you hit it so hard that it breaks or deforms). Dynamic processes strive for equilibrium, that is, a state in which all changes canceled each other out perfectly and nothing happens anymore. Exactly why all things strive for minimal energy to the best of my knowledge is not known to anyone but it's an empirical observation that has held together science for a few centuries already and has been observed over and again in experiments and observations.

In psychology, “fear of change” is a well-known topic. When Copernicus found that the earth rotates around the sun, it caused massive resistance from the world population. When confronted with the implications of quantum theory (that he helped to establish), Einstein resisted the change in world view it would bring about and declared that "God doesn't throw dice". Announcements for big changes in an organization, e.g., are typically met with skepticism, and quickly resistance and conservatism will pop up to cancel out the announced change. Just google for “change management” to find a myriad of books explaining how to reorganize a corporation. As I will argue in this blog entry, this same mechanism of fear of change (or better "resistance to change") also permeates music theory. 

At the same time the universe doesn't appear to like complete rest. Quantum physics (a revolutionary theory that of course was met with a lot of skepticism at first!) predicts that there's no such thing as complete “rest”. The Heisenberg uncertainty relation necessitates that even at a temperature of 0 Kelvin (the lowest possible temperature in the universe) there must still be a small rest energy. In nature and technology we also observe constant evolution. Change is inevitable it seems. Similarly, in music, listening to a piece that consists of a single note without volume or rhythmic variation that lasts forever is not a pleasant experience. Ask anyone who's suffering from tinnitus what it's like...

Finally, I want to stress that this fear of change is not a bad thing per se. After all, it has helped us survive since the stone age. It was probably safer to eat the berries that your parents ate than to try new berries every day. And it continues until today, since not all big changes or revolutionary “new insights” really have the merit they claim they have (and that may well apply to this blog entry too!)

3 Fear of change in music

In this section I will list some places where I see fear of change in action in music composition. If you know about more examples, by all means, comment!

3.1 Music style

In modern classical music, certain experiments have been branded "interesting", whereas other experiments have proved to be wildly successful with wide audiences. 

The “12 tone” music style, that resolutely throws away the organizing principle of “sounding good” and replaces it with a different organizing principle of “using all available notes and treating them without differences”, as introduced by Schoenberg, results in music that has many leaps and bounds and, let's face it, has failed to attract a significant audience. On the other hand, “minimal music” with composers like Philip Glass, Steve Reich, Brian Eno, Michael Nyman, Terry Riley and a myriad of others, makes slowly evolving music and continues to be wildly successful with wide audiences. Compared to the 12-tone music, minimal music minimizes change. It also offers just enough changes to keep it from being boring. As such it avoids complete rest.

3.2 Writing melody

When writing melody, e.g. in the context of counterpoint, or in the context of a song, it is advised to avoid big leaps. The reason given by ancient theorists is that smaller leaps are easier to sing. What makes a bigger leap harder to sing accurately than a smaller leap? Is the larger change of pitch a cause for distress in our brains? At the same time, I also took an introduction to counterpoint class, in which I was warned to avoid “turbulence”, i.e. writing a flurry of notes that doesn't seem to go anywhere. Minimize the change, while avoiding complete lack of direction (lack of direction would be a form of equilibrium or rest).

3.3 Voice leading

When moving from chord to chord, voice leading is the principle that makes you do these movements while minimizing the changes in notes. Voice leading is an important topic in many courses on harmony and jazz theory. It is perhaps the most common principle that governs modern music styles (apart from those styles that avoid it on purpose, like the 12-tone music mentioned earlier). Voice leading is a direct application of minimizing change between chords. The fact that you move between chords and don't just stay on the same chord all the time, is a direct application of avoiding complete rest. 

Minimizing changes between chords historically probably also has a second reason: when playing chords on a keyboard it is easiest to play chords that are close together, i.e. where you minimize the changes in required hand and finger movements. Minimizing unneeded movements is absolutely required when learning to play an instrument at the level of a virtuoso. This synergy between physical minimization of change and pyschological minimization of change probably has led voice leading towards the huge role it plays in music composition.

3.4 Fugue construction

While constructing a fugue according to the classical rules, the composer first states the theme, then restates the theme a fifth away from the original theme (but without introducing new accidentals, a so-called modal transposition), then returns to the original theme. This is the so-called “exposition” part of the fugue. During the exposition, the listener is taught the theme that will return in all kinds of variations later on. The theme is taught three times (minimize change), but the second time a fifth away compared to the first and third time (no complete rest). Why a fifth away? At first sight, a fifth seems like a large jump. Why didn't the composer just write the theme a second higher?

There's again an application of the principle here and it requires some explanation. If you transpose all notes from the C major key a perfect fifth up, you get the notes from G major. If you compare the notes of C major and G major, you will notice that they share all the same notes, except for the note f (in C major) compared to a note f# (in G major). G major therefore represents a key that is as close as possible to C major (since it differs in only one accidental) while not being completely the same (since it differs in at least one accidental). A theme written in C major that is modally transposed from c to g will therefore sound maximally the same as the original theme (minimize change). Next time you wonder why moving along the circle of fifths is so popular, fear of change may be the answer you look for.

3.5 Modulation

Modulation is the art of moving from musical key to musical key. When you move from one key to another, you want to gently guide the listener towards this change. When you read about modulation, you will often be advised to modulate to “near” keys, that is, musical keys that do not differ in number of accidentals too much. This is a direct application of minimization of change. 

One can also modulate to more distant keys. In those cases a sudden change, known as direct modulation, is frown upon by composers and theorists. To modulate between keys, especially to distant keys, several advanced techniques have been invented and they involve clever voice leading, sometimes going as far as substituting chromatic notes for enharmonic equivalents, towards a cadence to confirm the new key. These techniques incrementally introduce small changes to the audience so they are guided from the old key into the new key without sudden changes.

3.6 Writing hit songs

Commercial pop music often reuses the same chord progressions. During the 1980-ies, these chords where typically I, IV, V (think: C F G). Nowadays, the new chord progression used in virtually all hit songs is (I, V, vi, IV) (think: C G Am F). Why is that? Why exactly those progressions? Why did I,V,iv,IV come after I,IV,V?

Look at I, IV, V. Remember from the section about fugue construction that transposing a theme a fifth up will maximally retain the existing melody notes. The same is true when transposing a theme a fifth down (note “c” transposed a fifth down gives an “f”. This can just as well be thought of as transposing it a fourth up). When transposing something a fifth up, you need an extra sharp (or one less flat) to completely preserve the same melody. Similarly, when transposing a theme a fifth down, or equivalently a fourth up, you need an extra flat (or one less sharp) to completely preserve the melody. This means that by playing with the chords I, IV, V you have minimized the changes in the set of notes that need to be recognized by an audience, and maximally preserved the possible melodies that can be written on top of these chords.

Complete rest is still not desirable, and so after 20 years of I, IV, V the time was ready for a new chord progression that finds a way to minimize change while avoiding complete rest. And this new chord progression appears to be I, V, vi, IV. It's an evolution from I,IV,V in that it introduces an extra chord. Because the audience is already very used to I, IV, V, this extra chord can inject a bit of much needed change into the music again. The new chord of course is not chosen arbitrarily. It's chosen in such a way that it minimizes change with respect to the old chord progression. 

As before, when going from I to V, you need only one extra accidental. When going from V to vi, you need one less accidental (since vi is the minor equivalent of major I). Then when going from vi to IV, you need one less accidental again, and finally when going back from the final IV to I to sing the next verse, you need a single extra accidental again, making the circle round. Changes have been minimized between every two consecutive chords, and total rest is avoided by moving between different chords.

I'm afraid we're still stuck with I, V, vi, IV for a while, but if you want to define the future, grab your chance, and design a new chord progression that minimizes change while avoiding complete rest :) Unfortunately, you may have trouble selling it to the music publishing companies, since they will probably resist these sudden changes you try to introduce ;) “Never change a winning team/theme!”

Stefaan Himpe - Fairy tale for piano solo

Fairy tale for piano solo

It's been a while but I finally found some time to write, practice and record a new piece. All things music have been on a break for a while now, and I'm glad that I can take a break from taking a break.

You can download the piece from my soundcloud account: https://soundcloud.com/stefaanhimpe/fairy-tale-for-piano-solo

So... what do we have here really? 

The piece started as a sad ending theme for a video game. It was never used for that purpose and so I decided to rework it into something longer. Maybe it's interesting if I tell you how I think about the music, bar by bar. (Then again, maybe it's not ;) Feel free to skip the explanations. Most of all, I'm also curious how much of this I will still recognize in a year or in 5 years. I may even come back to my description and update it as my insights change.)

Bars 1-4: The piece starts off quite melancholic (which was a requirement for the sad ending of the video game). This part is written in C minor key, a key that is traditionally regarded as the key in which to express a declaration of love with lament of unhappy love, sighing of the lovesick soul. Clearly someone is thinking back about something.

Bars 5-8: The beginning is repeated with a slight variation in the accompaniment to avoid making it sound the same twice. It is as if the accompaniment adds more detail to the memories as they are repeated.

Bars 9-12: Suddenly the melancholic fantasy is interrupted with a little waltz. A short distraction? Or perhaps a related memory, a sudden free association. 

Bars 13-20: The distracting thoughts are pushed away and make place again for the fantasy. As the fantasy repeats itself, the melody notes again add more details to the memories. By coloring the melody using notes outside the scale, the color of the memories changes. Little imperfections that keep the story alive as it were.

Bars 21-27: The fantasy continues. The fantasizing person remembers and overthinks some of the consequences that resulted from whatever happened in bars 13-20.

Bars 27-32: Stress level increases a bit. Temper gets a bit heated. 

Bars 33-34: Some soothing thoughts manage to calm down the person.

Bars 35-38: There's our distraction waltz again. It's barely interesting enough to keep our thoughts away from what happened.

Bars 39-48: The fantasy is resumed, but in bar 45 it suddenly takes a different path. The person has second thoughts about how everything really went back then. For a while, things are remembered as perhaps more festive than before (bars 47-48).

Bars 49-52: These finish the first fantasy and prepare the music to change into a different key of f minor. F minor is the key in which one traditionally expresses deep depression, funeral lament, groans of misery and longing for the grave. No doubt the passage that follows will have a sinister side to it.

Bars 52-85: A buildup of emotions, a waterfall of notes follows. In the left hand we have a sad but very static accompaniment with typical f minor notes. In the right hand, in bars 53-64, we also have very dissonant chords which work to create a somewhat uneasy feeling. In bars 65-76, the dissonant chords are now replaced with a locrian motif that further increases the uneasyness in the music. In bar 74, the deepest note of the piece (a very low "d") is reached. Then bars 77-85 repeat the same techniques but a fifth higher and with many more, and much faster notes, which adds even more drama to the already dramatic state of mind of our fantast. The fifth higher brings us in the key of Bb minor, which traditionally is used to express feelings of mocking God and the world, discontented with itself and everything, preparation of suicide. Quite restless and dramatic indeed :)

Bars 86-91: The deepest darkest memories subside and make place again for the sweeter earlier theme.

Bars 92-113: The earlier theme is repeated, but again the colors have changed to something more bitter-sweet. This time the chords in the right hand sound much more yearning than in the beginning of the piece. As if the darkest memories increased the feeling of having lost something valuable and make it more painful to think back about what was lost.

Bars 114-117: Eventually, the person snaps out of his trip through memory lane and we hear the waltz playing again.

Bar 118: The piece ends with some bitter sweet ending chords. A near happy end, and time to get back to work ;)

Musical key and mood

Key determines mood? Really?!

It is often argued that with the advent of equal temperament all differences between musical keys disappeared, i.e. C major doesn't sound fundamentally different from Eb major. Yet composers still attribute certain moods to certain musical keys. In part this is probably determined by tradition, and in part it may be influenced by the mechanics of playing an instrument: playing a piece in C major on a piano (using only white keys) is quite a different experience from playing that same piece in C# major (using many black and some white keys) because of the differences between physical location of black and white keys on a piano. Also, bowed instruments will often play an F# differently than a Gb, despite both being the same note on a piano.

So what are these traditional moods associated to certain keys?

This list is based on a list made up by Christian Schubart in his book Ideen Zu Einer Ästhetik Der Tonkunst...

Key	Mode	Mood
C	major	completely pure key; speaks of innocence, simplicity, naivety, child's talk
Db	major	key to bring out unusual feelings; can smile but not laugh; can grimace but not cry
D	major	key of triumph, hallelujah, war-cries and victory-rejoicing
Eb	major	key of love, devotion, intimate conversation with God
E	major	noisy shouts of love, laughing pleasure and not-yet-complete full delight
F	major	key of complaisance and calmness
Gb	major	key of triumph over difficulty, sigh of relief after difficulaties have been overcome
G	major	key that is rustic, idyllic and lyrical, calm and satisfied passion, any gentle and peaceful emotion of the heart
Ab	major	key of the grave, death, putrefaction, judgement, eternity
A	major	declaration of innocent love, satisfaction with one's state of affairs, hope of seeing one's beloved one again when departing, youthful cheerness and trust in God
Bb	major	cheerful love, clear conscience, hope, aspiration for a better world
B	major	strongly colored, announcing wild passions, anger, rage, jealousy, fury, despair and every emotion of the heart
C	minor	declaration of love with lament of unhappy love, sighing of the lovesick soul
C#	minor	penitential lamentation, intimate conversation with God, sighs of disappointed friendship and love
D	minor	melancholy, womanliness
Eb	minor	feelings of anxiety, the soul's deepest distress, brooding despair, blackest depression, most gloomy condition of the soul
E	minor	naive declaration of love, lament without grumbling, sighs accompanied by a few tears, desires to resolve into the pure happiness of C major
F	minor	deep depression, funeral lament, groans of misery and longing for the grave
F#	minor	a gloomy key, resentment and discontent, it languishes for the calm of A major or happiness of D major
G	minor	discontent, uneasiness, worry about a failed scheme, bad-tempered gnashing of teeth, dislike
G#	minor	grumbling, struggling with difficulty, heart squeezed until it suffocates
A	minor	pious womanliness, tenderness of character
Bb	minor	mocking God and the world, discontented with itself and everything, preparation of suicide
B	minor	key of patience, calm awaiting one's fate, submission to divine dispensation, mild lament without breaking out into offensive murmuring or whimpering,

How to write a Crab Canon

What's a crab canon?

According to wikipedia:

A crab canon—also known by the Latin form of the name, canon cancrizans—is an arrangement of two musical lines that are complementary and backward, similar to a palindrome. Originally it is a musical term for a kind of canon in which one line is reversed in time from the other (e.g. FABACEAE <=> EAECABAF).

Ever since I read about this in the Goedel, Escher, Bach book by Douglas Hofstadter I've had some fascination with this form of music. I've tried to write some but I usually got stuck after about three notes. In some previous articles I discussed how I created a 5-part canon and a 6-part invention using a technique I have invented (or more likely: rediscovered). Now I have extended this technique to create crab canons and palindrome canons et voila! A brand-new no-sweat, no-tears 4-voice palindrome crab-canon in the somewhat exotic meter of 11/4 appears... (why 11/4? Because I can ;) )

You can hear it on Youtube:

or SoundCloud:

The article explaining the full construction of this piece is available for download here. You may have trouble downloading it with some versions of internet explorer. In that case use chrome or firefox instead.

6-part Invention

What? Not another one??

In a previous post I explained what technique I used to write a Canon in 5 parts without much sweating. I've been experimenting a bit more with the technique and here's the result of one of those newer experiments: a 6-part invention.

Structure

As it goes with such pieces it's interesting to highlight the structure of the piece. The piece starts with a theme. The theme is repeated at different starting pitches (basic fugal treatment), and counter-subjects are introduced which are really just the theme in canon to itself. The canon is left to play in 3 voices at a few different starting pitches, and then gradually morphs into a different canon with a different theme. The new canon is also given some basic fugal treatment, and finally both canons sound simultaneously forming a 6-part super-canon with two themes playing simultaneously in 3 different time-delayed versions each. The piece rests in peace after a soothing final chord.

How did you make this?

The piece is generated using the techniques explained in my tutorial on writing canons, with some cool twists. I intend to fully explain the creation of this piece including the full score and algorithms in a new tutorial that will appear in a separate post. Stay tuned :)

Enough already... let me hear what this is all about

Ok.. ok... Here it is:

And here it is as well:

Swan song

Near-tonality

Something that fascinates me is the concept of near-tonality. I'm not certain if this is existing terminology, and if it means what I would like it to mean in the context of this blog post. I use it to denote music that somehow "almost" sounds tonal. A typical example would be a soloist playing something "out-of-key", while the accompaniment remains "in-key". The notes sound wrong in the context, but if you keep playing out-of-key systematically something funny happens: once you get used to the new musical language, it all starts to sound right again. You start to hear how different types of dissonances can evoke different moods, and what first sounded all weird and wrong suddenly sounds beautiful, comforting and soothing.

Music please?

As you may have guessed, all this was but a long introduction towards my most recent somewhat experimental LMMS composition: Swan song!

After 2 minutes of tonal introduction, I dip my toes into the rich world of near-tonality and atonality.

• +1 @ you if you can listen to it once without questioning my mental sanity
• +2 @ you if you can listen to it three times in a row without it getting stuck inside your head for the next two days ;)

Modes

Using modes in composition

I have some fascination with composing based on modes. It can make music sound so refreshing. For the longest time i've been very confused about modes. Here's an attempt at clarifying some aspects related to modes. The explanation assumes you are already familiar with key signatures in major and minor scales (i.e. number of sharps and flats required to make a key like D major sound like D major).

A typical explanation about modes is something as follows. First you start with a C major scale

c d e f g a b c

Now you play the same notes, but you start on the note "d"

d e f g a b c d

and you have created a dorian mode. Similarly, starting on the note "e": "e f g a b c d e" results in a phrygian mode, starting on "f": "f g a b c d e f" results in a lydian mode, starting on "g": "g a b c d e f g" results in a mixolydian mode, starting on "a": "a b c d e f g a" results in a aeolian mode, and starting on the "b": "b c d e f g a b" creates a locrian mode.

If you're like me, immediately a few questions arise (which are never answered by most basic tutorials)

1. Is "d e f g a b c d" is a dorian mode of the C major scale? or of the D major scale?
2. Do different modes have a specific sound to them? some mood or character?
3. How do you quickly construct a dorian mode (or any other mode) of - say - the E major scale without memorizing the notes for each possible combination of (scale, mode)?
4. What difference does it make if you play "c d e f g a b c" or "d e f g a b c d", it's all the same notes anyway?

Is "d e f g a b c d" a dorian mode of the c major or of d major ?

The rule is simple: if it starts on a "d" it's derived from a d-scale

What different modes exist? Do they have a specific sound to them? some mood or character?

Some modes lend themselves more naturally to achieving specific moods in your music, but you can by no means generalize. Some very sad and melancholic music has been written in major keys (I'm thinking e.g. of Scriabin's prelude op 17, no 6, or the Plainte by Caix d'Hervelois) Despite all those over-generalizations, the following list seems to work well (these are all modes derived from a major scale; you could derive even more modes by starting from other scales, say a harmonic or melodic minor scale, too):

• c ionian "c d e f g a b c": happy music. This is also called the c major key.
• c dorian "c d es f g a bes c": irish folk-like
• c phrygian "c des es f g aes bes c": spanish sounding
• c lydian "c d e fis g a b c": happy, playful, somewhat comical effect, attention raising (think: the simpsons theme)
• c mixolydian "c d e f g a bes c": uniting pleasure and sadness; creating a effect of yearning for something or someone
• c aeolian "c d es f g aes bes c": sad music. This is also called the c natural minor key.
• c locrian "c des es f ges aes bes c": (this is not often used as it sounds weird to most western ears)

Some nice effects can be achieved e.g. if a melody is written in one mode, and later on is echoed in a different mode.

How do you quickly construct the dorian mode of - say - the E major scale ?

This may not work for you, but it works for me. It requires that you don't have to think about key signatures for major keys (i.e. you do know those by heart), and you also don't have to think twice about the names of the simplest modes built on the major keys (i.e. ionian "c d e f g a b c", dorian "d e f g a b c d", phrygian "e f g a b c d e", lydian "f g a b c d e f", mixolydian "g a b c d e f g", aeolian "a b c d e f g a", locrian "b c d e f g a b")

• First I remember that the simplest dorian mode is "d e f g a b c d"
• Then I remember that D major really needs two sharps (fis and cis) to sound like D major.
• From those two memories I can quickly construct the following rule: "the dorian mode is like the major scale, but with two less sharps (or two extra flats, or one less sharp and one extra flat)." This rule can be applied to any scale.
• Example: E major requires four sharps. If I want to have the dorian mode I just need to drop two of those sharps, so I end up with "e fis g a b cis d e". Second example: now I want the E lydian mode. Simplest lydian mode is "f g a b c d e f". Compared to F major, this has one less flat (bes became b), or equivalently: one extra sharp. E lydian therefore requires 5 sharps instead of 4, i.e. fis, cis, gis, dis, ais.

What difference does it make if you play c ionian "c d e f g a b c" or d dorian "d e f g a b c d", it's all the same notes anyway?

• One difference is in which notes you emphasize on the strong beats (typically first, third, fifth notes of the mode) also the notes you chose to return to at the end of a musical fragment (often something like the first note of the mode).
• If you're writing a melody in a mode, you will want to use the notes that make it sound different from a major scale more prominently to emphasize that you're not working in a major/minor scale. So a melody in G mixolydian should feature the natural f, and a melody in d dorian should feature the natural f and the natural c. It's exactly those altered notes (compared to the major scale) that lend your melody its extra qualities/mood.