Compos

Reprises

NordMix

test :

Cantaloupe Island – Herbie Hancock

This note focus on the drum break in the intro of the tune “Move By Yourself” by Donovan Frankenreiter.

In the above video :
- The original audio excerpt is aligned under the tempo.
- Each line of the grid is a 16th note (double croche). The classical time signature is 4/4.
- It’s usual to count and verbalize each measure, by saying 1 syllable per 16th note, like this : “1 i & a 2 i & a 3 i & a 4 i & a “,
- For the drum break, the 2 snare hits are on “a 3″, followed by keyboard+bass+guitar hits on “& i 1″.

Here is the staff notation view for the drum break :

Here is a short video of the staff notation software playing it :

Here is a warmup and maintenance procedure for intermediate keyboard players.

Intended use :

For a quick warmup before a gig, as an everyday exercise to keep fingers and head ready to play, or for education purpose to learn some essential things about harmony, around the cycle of fourths.

Procedure :

- Switch on a rythm or a metronome, set it at a speed you can play everything in time
- For each of the following key : C, Db, A, Eb, using the major diatonic scale, do :

1. Play the IV7 chord in root position with one hand.
2. Go up 4 degrees (inclusive), you now play the vii7 chord in 2nd inversion. You have just moved your 2 highest fingers 2 steps down (easy).
3. Go up again, do that 7 times in total and you’ll reach the root degree I. You have played all degrees.
4. Improvise with the other hand, doing bass or melody alternatively, change hand and go to step 1.

Alternative :

- Start on degree ii with a ii7 chord instead of IV7.
- Make arpeggios and pumps with chords, go up/down the scale every two pitches, go chromatic.
- Practice your timing, change chords on different 16th positions.
- Try some alterations, change degrees to go minor / major.
- Take a different keyboard (organ ?) and different rhythms / styles

Recommendations :

- Know what degree you are playing, what chord (color) you are playing.
- Count measures in your head while you play (“1 e & a 2 …”).
- Sing while you play
- Locate the “passing 4″ and the blue notes.
- Record yourself, check what’s wrong from a listener’s point of view

Schedule :

- 10 mn everyday is better than 50 minutes a week.

Modulation Majeur/Mineur sur G
Puis modulation vers Eb via C6/3 
Part A
"Stars shining ..."
G:  G             | Eb7    D7   |
    I             | bvi7   V7   |
"Night breezes ..."
    G             | E7          |
    I             | vi7         |
"Birds singing ..."
    Am7           | Am7b5       |
    ii7           | ii7b5       |
1:"dream a litt..."
    G  Eb7   D7   | D7          |
    I  bvi7  V7   | V7
2:"dream a litt..."
    G  Cm7   D7   | G     ( C6/3 G Bb7/2 )|
    I IVdim7 V7   | I                     |
Pinta sur Eb sur cadence V I
Puis modulation vers G via Eb>D7 (this)
Part B
"Stars falling but  ..."
Eb: Eb  Eb6 | Bb7           | 
    I   I6  | V7            | 
"Still carving ...     "
    Eb  Eb6 | Bb7 Bbdim7    | 
    I   I6  | V7 Vdim7      |
"I'm looking ... "      
    Eb  Eb6 | Bb7           | 
    I   I6  | V7            |
"Just saying this" (fin)
    Eb      | D7     ( Em Am7 D7 ) | 
    I       | viidim7              |

Tuning process

There is a free tuner application for android phones, called “PitchLab”. It can display the root frequency of a sound.  Having a “Hz” number has guided me, it was reassuring on what I was hearing.
Each drum is tuned side by side, with one side on the floor, on a towel.

1. 18″ Bass Drum

- start by the front (batter) side.
- screw with 2 fingers, each screw in good order (1-5-2-6-3-7-4-8).
- take the key and reach the point of no flatness in front of each lug.
- repeat this operation 2 cycles (2 times around the drum using the good order). Go from flatness to first resonance for each lug.
- from this state, go progressively up, by screwing each lug a 1/8 turn.
- ensure that the tuner indicate the same note whatever lug is hit
- I stop at 72 Hz everywhere
- same operation on the back to get 108 Hz (* 2.2)
- come back at the front and set it again to 72 Hz (*1.5)
- fundamental frequency : G1 49 Hz
- calculate the frequency ratios of each side with the fundamental : 72/49 = 1.5 for the front and 108/49= 2.2 for the back

2. 14″ Floor Tom

Same procedure as BD :
- Front side : 98 Hz
- Back side : 98 Hz
- Fundamental frequency : G2 98 Hz
Floor Tom is more sensitive then BD, only a 1/8 turn with the key for a few hertz.
Pitch Lab needs the drum to be tuned by ear first, if they are too much difference between screws, it can be lost, too many harmonics, I guess. It works fine for fine tuning.

2. 12″ Tom

The test stroke has to be very consistent. Seems easier when opposite stroke are tested / tuned pair by pair. One lug can quickly change all the drum.
- Front : 130 Hz
- Back : 130 Hz
- Fundamental frequency : C3 130.8 Hz

2. 14″ Snare

- Front : 196 Hz
- Back : 196 Hz
- Fundamental frequency : G3 196.1 Hz

What’s next … theory.

1. Determine the possible frequency range for each drum.

Write down the lowest frequency that please my ears, tune up until I reach the highest frequency that still sounds good for me (of course, don’t tight to hard, avoid damaging the drum).

Assuming these frequencies are Ok for me on my drumset :
- Bass Drum : from 70 to 110 Hz
- 14″ Tom : from 80 to 100 Hz
- 12″ Tom : from 100 to 130 Hz
- Snare Drum : from 160 to 210 Hz
Next thing could be :

2. Tuning to a specific pitch.

Let’s draw the frequency ranges for each drum on a well tempered scale.
This drawing shows all the intervals that can be done between the drums. Now it’s possible to try having perfect fiths, fourths, major chords, … between them.

Assuming that the style is more important than the pitches, I would lean toward a global “low tuning”, by taste, the sounds seems quieter and the resonance lower.
Assuming the snare drum to be the most heard drum in the set, and that E is a very common pitch on guitars and generally one of the most played (see Measuring the Evolution of Contemporary Western Popular Music).

I would chose E for my snare drum.
Then E too for the bass drum, making a octave unison.
Then make a major chord, with a C in the high range of the 12″ tom and a G on the floor tom.
That also makes the overall set well spread over the available frequency range.
Finally here is the drumset setting :

Tuning to each drum’s natural pitch

I haven’t tried this :
- remove the 2 skins.
- lift the drum with 2 fingers on a lug
- hit the side
- read the frequency on the tuner
- determine the frequencies for each side : use the coefficients determined in the “Tuning process”
- OR try to remember the pitch by ear … good exercise …, OR make a record of it
- tune as described in the “Tuning process”
 

Dampers ideas

- latex caulking
- stuffed animal
- weatherstripe
- fart joke paste
- slim paste (flubber)
- sticky hands
- credit card
- plastic folder
- loosen the lugs next to the snares
- put a cloth on cymbals

Mounting the snares

- Unleash (trigger off), open the clamps.
- Center the snares
- Tighten the fixed clamp
- Unscrew the adjustment screw
- Close the strained (trigger on).
- Tighten the snares
- Tighten the adjustment clamp
Check :
- The adjustment screw allows all settings, from OFF to very tight.
- If the trigger is off and the adjustment fully screwed, then the lightest contact with the snares is obtained.

Playing techniques / training

- one hand 16th flat during 12 bars : 85 bpm
- one hand 16th tip/shank during 12 bars : ? bpm
- paradiddle L R LL R L RR / R L RR L R LL, 16th during 12 bars : ? bpm
- double strokes LL RR : 16th during 12 bars : ? bpm
 

All chords are degrees of the same major diatonic scale, except the last chord which modulates to the relative minor, with a final “major VI” (the I degree of the relative minor scale).
This example sticks to the rule saying that I IV and V degrees have the same functions as vi ii and iii.

Part A
I heard ...               I heard ...
ii7  |  V9  |  I   |  IV9
And so ...                To listen for a while ...
ii7  |  V9  |  vi  |  vi
An there he was ..
ii7  |  V7  |  I   |  III7

Part B
Strumming ...             Singing ...  
vi   |  ii7 |  V   |  I
Killing me ...
vi   |  ii  |  V   |  IV  |  I  | IV  |  bvii7 | VI (fine)

Take a walk on the wild side

D:__:_  Piano
Cadence I IV en passant par ii.
Passer par D (au moins la note) entre C/2 et F.
"intro" :  
 F > C | F^ > C 
"Holly came ..." : 
C/2 | F6 | *2
"Plucked her eyebrows ..." : 
C D | F D |
"Hey babe ..." : 
C/2 | F6 | *2

Bal des Laze

D:02:4 Vox Farf Laze
D:02:4 Reggae 88
    Em | Bm/1 | C7 Am/1 | Bsus4   B   |
G:  vi | iii  | IV ii   | iiisus4 III |

Toto – Rosana

Part A  "All I wanna do ..."
C:   G/2    F: F/2  C    C: G/2
     V         I    V       V

Part B  "Not quite a year since ..."
Bb:  Gm/2  F  Bb/2 ...  Eb/1  Bb/2  F
     vi    V  I         IV    I     V

Chorus : "Meet you all the way ..."
Bb:  F  Bb/2 Eb/1  ...  Eb/1  Bb/2  F
     V  I    IV    ...  IV    I     V

Toto – Hold the line

Part A
A: F#m/2 A/1  Bm7  E/2 
   vi    I    ii7  V

Part B
   Bm7   C#7
   ii7   iii7

A-Ha – Take On Me

Intro
A: Bm  E/2  A  D/2
   ii  V    I  IV 

12 bar blues progression

I  IV I I
IV IV I I
V  IV I V

Here is Jamiroquai’s keyboard chord progression for the tune “Virtual Insanity”.
On the original record, chords are plaid in key of Db.

ii7 in 2nd inversion (the 7th and the 5th are lower pitches).
Rootless 9th chords :
V9 starting with the 9th, the root is only on the left hand.
I9 starting with the 7th, the root is only in the left hand.
IV7 normally (ascending pithces starting from the root).
vii7 starting with the 7th.
bvii7 starting with the 7th.
vi7b5 normally (ascending pithces starting from the root).

Db:  Ebm7 |  Ab9 | Db9 |  GbM7 |  Cm7   |  CbM7   |  Bb7b5
     ii7  |  V7  | I9  |  IV7  |  vii7  |  bvii7  |  vi7b5

The chord progression in the verse is identical to the one found in “Just the two of us” : “I7, vii0, iii”. Both songs can be played on the same chords. “Just the two of us” as two more chords : “ii, V”. We can add them in this song when iterating (see “*5″), we are just adding a quick “ii V I” turnaround progression.

It may be musically more accurate to use the relative minor key on this song because the majority of chords are minors, and the melody is more minor too, but for the purpose of analysis we take the equivalent relative major key. Also we remember that the harmonical functions of vi, ii and iii is equivalent to I, IV and V. “vii” stays appart.

The vii can be “half diminished” (vii0) or “lowered fifth” (viib5) in both songs.

(chorus)
Eb:
Gm  | Gm  | Cm | Cm 
iii | iii | vi  | vi

Gm  | Gm  | Gm | Gm
iii | iii | iii | iii

(verse)
Eb7 | Dm7  | Gm | Gm  |  *5
I7  | vii0 | iii| iii |

Gm  | Gm   | Gm | Gm  |
iii | iii  | iii|  iii|

In key of G (Ray Charles’s recording) :

This tune uses all the degrees of a key.
“Relative Modulation” : Modulations are done within the Major and minor scales of the same key (G here).
Impro on chromatic scale, when in G avoid C (4th) and F (b7th).
GM7 : in second inversion.
G7/D : G7 in second inversion.
A7 : in second inversion.

AABA

intro
GM7 | E7 |  C7 | C#dim7 | D7
I     VI    IV   bIVdim7| V   => I
part A
“Georgia, Georgia ...”
GM7  | B7
I      III 
"the whole day ..."
Em   G7/D | CM7 C#dim7
vi   V      IV  bIVdim7
"Just an old .."
GM7  E9  | A7  D9 
I    VI    ii  V
1:
B7   E9  | A7  D7
III  VI    ii  V       
2:
GM7  C9  | GM7 B7 
I    iv    I   III
part B
“Other arms ...”
Em  Am7 | Em6  C9
vi  iii   vi   IV
Em  Am7 | Em   A7
vi  iii   vi   iii
"still in ..."
Em  Am7   | GM7  F#7
vi  iii     I    VII
Bm7 Bb7b5 | A7   D9
iii         iii  V

This tune is rooted on the second degree of the key it’s played in, written “ii” in functional harmony and refered to as the “dorian mode”.
The first part (A) of the tune is in key of C, with “D” being the root pitch in dorian mode.
The second part (B) is in key of Db, is mainly a transposition of (A) a half step up, with “Eb” being the root in dorian mode.
The bass is starting the tune on a D pitch, then comes the piano voicing and the improvisation.
Piano voicing follows what is called the “fourth voicing” : a chord composed of one pitch, its three following fourth and a third.  For instance, at the beginning of the tune, the piano’s answer to the bass uses these two chords : (E,A,D,G,B) and (D,G,C,F,A) :  a triad chord in second inversion and it’s 6th and 11th played as bass notes.
Improvisation can be performed on different flavors of the D dorian scale, melodic minor, harmonic minor, pure minor, or using the chromatic scale.

C:    Dm7
      ii7
Db:   Ebm7
      ii7

The tune is composed of 3 different keys for modal improvisation.
The part on the Eb key is using the F dorian mode, impro over F dorian, Ab or Ab blues is possible.

(intro)
Eb:  Fm7 | %    | %   | %
     ii7

(A)
Eb:  Fm7  | %    | %   | %
     ii7
Db:  Db7  | %    | %   | %
     I7
C:   Dm+4 | %    | %   | %
     ii+4
Eb:  Fm7  | %    | %   | %      *2
     ii7

(impro)
Ab:  vi7  | %    | %   | %
Gb:  vi7  | %    | %   | %
C:   ii+4 | %    | %   | %
Ab:  vi7  | %    | %   | %     *2

Part A follows the circle of fourths, with a “shortcut” from IV to vii.
Bb + blue notes

(A)
    ♫ Cm  | F ♫ | Bb   | Eb
    ♫ ii  | V  ♫|  I   | IV

    ♫ A0  | D ♫ | Gm   | %     *2
    ♫ vii |iii ♫|  vi  | %

(B)
      D   |  %   |  Gm  | %
      iii | %    | vi   | %

      Cm  |F9/F7 |  Bb  | %
       ii | V    | I    | %

(outro)

      A   |  D   | Gm   | %
      vii | iii  |  vi  | %

     C/Cm |  D   | Gm/A0| Gm
       ii | iii  |vi/vii| vi

The bridge is made of the same chord and melody, lowered 1 step, 3 times, then return to first key.

(intro,A,B)

     DbM7 |  Cm7  | Fm7  | Eb7  Ab7
Db:  IM7  | viib5 | iii  | ii   V


        alternative considering a modulation (the 2 scales can me used for improvisation) : 
Db:  IM7  |Eb: vi | ii   | Db: ii / V

(bridge)
     DbM7 |  BbM7 | AM7  | DbM7
Db:  IM7 | B:IM7 | A:IM7| Db:IM7

“Sequential modulation” : this tune has 4 direct modulations in part B, the same melody played on different keys, going up in pitches, 3 half steps up from Gb to A, then 1 half step up from A to Bb.
Improvisation can be done on common the common pitches between the keys.

(Intro)
F:   F⌂7 |Gb7+4                *4
     I⌂7 |bii7

(A)
     F⌂7 |  %  | G7  |  %
     I⌂7 |  %  | ii7 |  %

     G-7 |Gb7+4| F⌂7 |1:Gb7+4  *2
     ii7 |bii7 | I⌂7 |  %   

(B)
Gb:  Gb⌂7|  %  | B7  |  %
     I⌂7 |  %  | IV7 |  %

A:   F#-7|  %  | D7  |  %
     vi7 |  %  | IV7 |  %

Bb:  G-7 |  %  | Eb7 |  % 
     vi7 |  %  | IV7 |  %

F:   A-7 |  D-7| G-7 |  C7
     iii |  vi | ii  |  V   (turnaround to I)

Here is a quick “making of” a “nsmp” (Nord Sample) sound for Nord keyboards.  It has been created using a Roland RD300sx, Audacity and Nord Sample Editor. Credits & thanks go to them. Please visit the Nord User Forum too.

1. make 3 midi clips playing all the notes at different velocities (80, 105 and 127).

2. Using a DAW Software, play the midi clip and record the audio.
This could be done with “Reaper” (cheaper if not free).
Screenshot :

3. Open “Nord Sample Editor”,
click “Add”, chose the wav file.
click “Assign” in the “Multi Sample Per File Assign” option at the bottom

click “no loop” and “Apply Loop” in the “Apply on All” options at the bottom

do nothing on this screen, Nord has done everything auto !

Just export the instrument to the Nord :

I.  Perception of a vibration

Here is a step by step process of sound creation and perception.

II. Consonance

III. Construction of scales

1. An octave scale :

2. Pythagore’s scale:

It is now possible to use the pure consonance by dividing of multiplying these frequencies by 2 in order to fill all the hearable frequency range from 50Hz to 18000Hz. We will obtain the 9 octaves of the previous scale, and 12 frequencies in each octave.

For instance, between 55 et 110 Hz, we will find 185.625 / 2 = 92.81 Hz.

This operation is repeated 12 times to obtain what is called Pythogore’s scale.

In summary, this scale is obtained by multiplying up to 12 times by 3/2 a given ferquency, and by dividing each frequency by a power of 2 that is high enough to get a result between the given frequency and its double.

Hint: using powers of 2, each frequency in an octave is a pure consonant with the frequency of the same rank in the 8 other octaves.

Here are the frequencies for a Pythagore scale made from a 440 Hz frequency.

This frequencies are used to name music pitches, with these 2 notations : LA= A, SI= B, DO= C, RE= D, MI= E, FA= F, SOL= G.

A    440Hz   * 3/2
=  E    660Hz   * 3/2
=  B    990Hz   * 3/2
=  F#  1485Hz   * 3/2
=  C#  2227Hz   * 3/2
=  G#  3341Hz   * 3/2
=  D#  5011Hz   * 3/2
=  A#  7516Hz   * 3/2
=  F  11274Hz   * 3/2
=  C  16911Hz   * 3/2
=  G  25366Hz   * 3/2
=  D  38049Hz   * 3/2

ATTENTION : The scale above is sorted by order of generation.

Under, we divide these frequencies by the sufficient power of 2 in order to bring back the values in the range of hearable frequencies and between LA 440 Hz and its pure consonant double LA 880 Hz :

A    440Hz
E    660Hz
B    990Hz   / 2 =    495 Hz
F#  1485Hz   / 2 =    742 Hz
C#  2227Hz   / 4 =    556 Hz
G#  3341Hz   / 4 =    435 Hz
D#  5011Hz   / 8 =    626 Hz
A#  7516Hz   / 16 =   469 Hz
F  11274Hz   / 16 =   704 Hz
C  16911Hz   / 32 =   528 Hz
G  25366Hz   / 32 =   792 Hz
D  38049Hz   / 64 =   594 Hz

Under, we sort the pitches in their ascending frequency order :

1.  A   440 Hz
2.  A#  469 Hz
3.  B   495 Hz
4.  C   528 Hz
5.  C#  556 Hz
6.  D   594 Hz
7.  D#  626 Hz
8.  E   660 Hz
9.  F   704 Hz
10. F#  742 Hz
11. G   792 Hz
12. G#  835 Hz

The same scale is found again, defining the same intervals between pitches, but one octave higher :

1.  A  880 Hz
2.  A# 938 Hz
3.  B   990 Hz
4.  C  1056 Hz
5.  C# 1112 Hz
6.  D  1188 Hz
7.  D# 1272 Hz
8.  E  1320 Hz
9.  F  1408 Hz
10. F# 1484 Hz
11. G  1584 Hz
12. G# 1670 Hz

Here is the scale sorted from the lowest to the highest frequency, from pitch C  : C C# D D# E F F# G G# A A# B

Here is the scale with pitches given in the order they were generated above, and from pitch C as well : C G D A E B F# C# G# D# A# F

Hint: On a piano keyboard, the black key and white keys are grouping closely generated pitches.

3. The tempered scale

This is the scale actually used in classical and contemporary western music. The tempered scale can be defined from Pythagore’s scale.

If the generation of Pythogore’s scale is continued, the computation of following frequencies, further than 12, does not give a round multiple of the frequency used to start the scale.

The gap between the 12th frequency and the double of the initial frequency is not equal to the gap between the others 12 frequencies.

This irregularity implies that a given interval of pitches will not produce the same consonance, depending to the pitch from which the interval starts, ie in the same octave or when overlapping two octaves.

For instance : the interval between G in octave 3 and C in octave 4, which makes an interval of 5 pitches), won’t be heard like the interval between C and F both in octave 3, despite the interval is 5 pitches as well.

To overcome this, the 12 frequencies are slightly modified and spread at equal intervals to form a tempered scale. This scane is build by dividing in 12 équals frequency ranges the interval between a frequency and its double (one octave).

To the human sense of hearing, the 9 groups of 12 frequencies are perceived with the same consonance as with Pythogore’s scale. The intervals between pitches are regular, thus the pitches are a cycling suite of octaves with pure consonant pitches across the octaves.

4. Natural scale

The natural scale generated by the suite of the integer multiples of a fundamental frequency (made by the vibration of a body).

For a C pitch at 528 Hz, such natural harmonics are :

1.   528 Hz              => C
2.  1056 Hz  /2 = 528
3.  1584 Hz  /2 = 792    => G
4.  2112 Hz  /4 = 528
5.  2640 Hz  /4 = 660    => E
6.  3168 Hz  /4 = 792
7.  3696 Hz  /4 = 924    => ~A#
8.  4224 Hz  /8 = 528
9.  4752 Hz  /8 = 594    => D
10. 5280 Hz  /8 = 660
11. 5808 Hz  /8 = 726    => entre F et F#
12. 6336 Hz  /8 = 792
13. 6864 Hz  /8 = 858    => entre G# et A
14. 7392 Hz  /8 = 924
15. 7920 Hz  /8 = 990    => B
16. 8448 Hz  /16= 528
17                561    => ~C#
18  594
19  627    => ~D#
20  660
21  693    => ~F
22  726
23                759    => ~F#
24                792
25                825    => ~G#
...

The first column contains the frequencies that are integer multiples of the fundamental C 528 Hz.

In the second column , the frequencies are divided by a power of 2, keeping a pure consonance, greater enough to get a result between the fundamental frequency and its double.

The third column shows the matching pitch in the tempered scale.

We have just build what is called the “perfect chord” y finding the pitches “C,G,E”. That set of pitches is composed of a fundamental and its first 2 natural harmonics.

Second example, for the pitch A (LA) 440 Hz, the harmonics are :
1.  440  Hz             => A
2.  880  Hz /2 = 440
3.  1320 Hz /2 = 660    => E
4.  1760 Hz /2 = 880
5.  2200 Hz /4 = 550    => C#
6.  2640 Hz /4 = 660
7.  3080 Hz /4 = 770    => G
8.  3520 Hz /4 = 880
9.  3960 Hz /8 = 495    => B
10. 4400 Hz /8 = 550
11. 4840 Hz /8  = 605   => D
12. 5280 Hz /8  = 660
13. 5720 Hz /8  = 715   => ~F
14. 6160 Hz /8  = 770
15. 6600 Hz /8  = 825   => ~G#
16. 7040 Hz /16 = 440
17. 7480 Hz /16 = 467   => #A#
...

Again, a perfect chords is identified : “A,C#,E”.

IV. Harmony

In this paragraph we are using the frequencies of a scale to play them successively and simultaneously.

We are making some harmony in the sense that we are paying with frequency rations, as explained above, ie with the degrees of consonance between frequencies.

The game ask us to figure out where we are, where do we come from, and where do we want to go in that “consonance space” made of all the available frequency combinations that we can play.

Having a map may be useful.

This map could be a graphical representation of the scales we built in part II, highlighting the consonance between pitches.

On the tempered scale, the consonance can be highlighted by identifying the included natural scale.

This allows us to sort the frequencies in order of consonance with a given one.

The tempered scale being a cycle, the consonance of two pitches depends on their interval, wherever the two pitches are in that cycle.

ie, because the tempered scale is a cycle, the interval allow us to know the consonance (the frequency ratio) for any couple of pitches.

Let’s apply this on a case in order to see more things about consonance and harmony. The construction of the natural scale for the frequency C 528 Hz (part I.4) defines an order of consonance. Here are the pitches of the tempered scale in that order :

C G E (~A#) D (F/F#) (G#/A) B (~C#) (~D#) (~F) (~F#) (~G#) A

Here are the intervals from C for each pitch, sorted in asending frequency order :

Pitch: C C# D D# E F F# G G# A A# B
Rank : 0 1  2 3  4 5 6  7 8  9 10 11
Intervals between C and
G      : 7 notes
E      : 4 notes
(~A#)  : 10 notes
D      : 2 note
(F/F#) : 5/6 notes
(G#/A) : 8/9 notes
B      : 11 notes
(~C#)  : 0
(~D#)  : 2
(~F)   : 4
(~F#)  : 5
(~G#)  : 7
A ?    : 8

The most consonant pitch with C is G. For any pitch at rank 0, the most consonant pitch in the scale is located at rank 7 (6 pitches up) , then at rank 4 (3 pitches up), and so on to the less consonant one.

Hint: the first two intervals that we just defined, thus the most consonant, are part of the “perfect chord” “C(DO) E(MI) G(SOL)”. They define the intervals of a “third” and a “fifth”.

Here is a proposal of map to represent the observations made on the scales so far in this note.

This map is not an exhaustive view of the possible harmonies, that would have more to do with spectral analysis.

It’s just a way to where we are in that “consonance space” we talked about before, for classical and contemporary western music, that are build on top of the tempered scale.

Music remains a matter of taste, sense of hearing and subjective perception.

V. Bridge to the classical and contemporary music theory

This paragraph is giving the definitions used in classical music theory, based on what we observed and constructed before.

The following “tonal functions” are given to the pitches in tempered scale, due to their consonance with the pitch of rank 1, according to their rank :

1. tonique

2. sus-tonique

3. médiante

4. sub-dominant

5. dominant

6. sus-dominante

7. sensible

The “chromatic scale” is composed of all the 12 pitches we constructed earlier in the tempered scale. It contains all the basic “colors”.

In tha chromatic scale, a “whole tone” is the interval between 3 consecutive pitche and a “half tone” is the interval between 2 consecutive pitches.

The “diatonic scale” is the scale made of the first 7 more consonant pitches of the chromatic scale. It takes the pitches of rank 2,4,5,7,9 and 11 of a chromatic scale. It contains only 2 half tones.

For the pitch C (rank 1), the diatonic scale is containing only the white keys on a piano.

The interval between pitches G and C is a “fifth”, the interval between C et E is a third, and so on …

A “triad” is a chord composed of one pitch and its third and fifth.

A [scale] is a set of pitches separated by intervals.

Each pitch find a [tonal function] in the scale (see above).

The [tonality] or [key] is the first pitch of the scale.

The [mode] indicates which pitch to start from in a scale, thus its closely related to the rank.

Chords are identified on the diatonic scale. A chord name is giving one pitch, the intervals of other pitches to play, and may give as well some voicing indication on how the chord is meant to be plaid.

When an interval number is greater than 7, it has to be simplified or put on the upper octave, ie the « X » – 7 th pitch.

« b » decrease the pitch of a half tone, « # » increase the same amount.

VI. Improvisation and composition

On the map given above, we count 7 pitches * 12 scales = 84 starting locations.

Once we have chose a first pitch among the 84, choosing a second pitch will determine a consonance. The further this second pitch, in the order of consonance we defined above, from the first pitch, the less consonant the plaid result will be. We can walk on the map this way, and put our legs and fingers on successive and simultaneous locations, and, for instance, try to reach the less consonant positions that can still please our perception and taste. There is a huge but finite set of possibilities in our modern music when we chose pitches to play.

Now we have to experiment all the possibilities we talked about, playing some successive pitches (melodies), simultaneous pitches (chords), and maybe identify some progressions and patterns.

VII. Chord Progression

A scale is [harmonized] when a chord has been associated to each pitch.

On a hamonized scale, each chord is a [degree], written I,II,III,IV,V,VI,VII.

When a diatonic scale is harmonized with triads we get the following :

- 3 majord chords corresponding to dregree I, IV and V

- 3 minor chords, corresponding to degrees vi, ii and iii

Degree “vi” is the “relative minor scale” or “eolian mode”.

=> These 2 progressions are similar : they have the same intervals and they cover all the pitches of the scale, thus they can be used to accompany any melody on that scale.

For instance : a I-ii-V progression can replace a I-IV-V (see The Beach Boys’ “Good Vibrations).

+ 1 dimished chord, corresponding to degree VII

To go further we can build others scales and harmonize them, like diatonic minor, melodic minor scales, with 7th chords …

Each chord will find its [fonction] : tension, release, attraction, distraction, … tonique, dominant, sub-dominant,…

Degrees I,IV et V have these functions : “Tonic”, “Dominant” et “Sub-dominant”, written “T,S et D”.

Degrees vi,ii et iii have the same functions, written “t,s et d”.

Using pitches out of the diatonic scale can lead to a [modulation], ie a change of scale.

In that case we talk about a [scale progression] or a [musical form].

VIII. Practical application on some tunes

You’ll find others notes on this website, dedicated to the analysis of tunes, based on the theory explained in the present note.

IX. Annexes

Style

-Reggae :

Basic chord progressions are I, IV, V, and vi occasionally. For instance : A, D, F#m, E pour I, IV, vi, V. The chords can be in any order, so such it start and end on I.

Some exotic scales

Intervals are given in number of half tones.

- orientale
Chaque tetracorde est formé que des groupes de 1/2 tons
2nde augmentés sur chaque tetracorde.
1-3-1 -2- 1-3-1
do réb mi fa sol lab si do

A cause de sa symétrie, le tétracorde supérieur de la gamme
peut constituer le tétracorde d'une même gamme transposée à la dominante :
do réb mi fa sol lab si do
sol lab si do ré mib fab sol
=> Cette gamme évoqué l'Inde.

- hispanique
2-1-2 -2- 2-1-2
do re mib fa sol la sib do

- tzigane
do ré mib fa# sol lab si do
Intervalles : 1 ton - 1/2 ton - 3/2 tons - 1/2 ton - 1/2 ton - 3/2 tons - 1/2 ton

- andalou
do réb mi fa sol lab sib do
1/2 ton - 3/2 tons - 1/2 ton - 1 ton - 1/2 ton - 1 ton - 1 ton

- balkanique
do réb mib fa sol lab si do
1/2ton - 1ton - 1ton - 1ton - 1/2ton - 1ton et demi - 1/2ton

- blues
3, 2, 1, 1, 3, 2

Modes de la gamme diatonique majeure

I ionien C
c-d-e-f-g-a-b
1 - 1 - 1/2 - 1 - 1 - 1 - 1/2
Superman (John Williams)
richesse, optimisme, ouverture

II dorien D
d-e-f-g-a-b-c
1 - 1/2 - 1 - 1 - 1 - 1/2 - 1
médiévale, celtique

III phrygien E
e-f-g-a-b-c-d
1/2 - 1 - 1 - 1 - 1/2 - 1 - 1
Gladiator (Hans Zimmer) :
hispanique

IV lydien F
f-g-a-b-c-d-e
1 - 1 - 1 - 1/2 - 1 - 1 - 1/2
merveilleux, mystère

V mixolydien G
g-a-b-c-d-e-f
1 - 1 - 1/2 - 1 - 1 - 1/2 - 1
The Full Monty (David Lindup)
rock. 

VI éolien A
a-b-c-d-e-f-g
1 - 1/2 - 1 - 1 - 1/2 - 1 - 1
36 quai des orfèvres (Erwann Kermorvant) :
gamme mineure naturelle, sombre, mélancolique

VII locrien B
b-c-d-e-f-g-a
1/2 - 1 - 1 - 1/2 - 1 - 1 - 1
?

Major Modes Harmony

(ionian) I : the 4th is a passing note
(dorian) ii : 
(phrygian) iii : add a lowered 4th. the 2nd and the 6th are passing notes.
(lydian) IV :  raise the 4th a half step
(mixolydian) V :  the 4th is a passing note
(aeolian) vi : none
(locrian) vii : the 2nd is a passing note, and can be flatten.