A Modern Analysis of Practical Music theory
Pentatonic-Style & Hexatonic Style Scales
Pentatonic-Style Scales
Pentatonic Scale: 1 2 3 5 6
Factoral Variants:
1 2 4 5 7
1 3 4 6 7
2 3 5 6 7
1 2 4 5 6
1 3 4 5 7
2 3 4 6 7
Complicated Pentatonic: 2 4 5 6 7
Factoral Variants:
1 3 4 5 6
2 3 4 5 7
1 2 3 4 6
1 2 3 5 7
1 2 4 6 7
1 3 5 6 7
Simple Pentatonic: 1 2 3 4 7
Factoral Variants:
1 2 3 6 7
1 2 5 6 7
1 4 5 6 7
3 4 5 6 7
2 3 4 5 6
1 2 3 4 5
Hexatonic-Style Scales
Variant: 2 3 4 5 6 7
Chaotic Hexatonic: 1 3 4 5 6 7
Variant: 1 2 4 5 6 7
Bright Hexatonic: 1 2 3 4 5 6
Variant: 1 2 3 4 6 7
Variant: 1 2 3 4 5 7
Dark Hexatonic: 1 2 3 5 6 7
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The Chord Spelling, Buidling, and Altering Methodology
Augmented 7 = root followed by a major third followed by a major third followed by a minor third
Major 7 = root followed by major third followed by a minor third followed by a major third
Dominant (7) = root followed by a major third followed by a minor third followed by a minor third
Minor 7 = root followed by a minor third followed by a major third followed by a minor third
Diminished 7 = root followed by a minor thord followed by a
if the chord has ___ interval in it the chord can be played at the ___ scale positions.
1 = any
1.5 = 3, 7
2 = 1, 2, 4, 5, 6
2.5 = 2, 3, 6, 7
3 = 1, 4, 5
4 = any except for 4
4.5 = 4, 7
5 = any except for 7
5.5 = 3, 6, 7
6 = 1, 2, 4, 5
6.5 = 2, 3, 5, 6, 7
7 = 1, 4
if the chord has ___ interval in it the chord can be played at the ___ scale positions.
1 = any
1.5 = 3, 7
2 = 1, 2, 4, 5, 6
2.5 = 2, 3, 6, 7
3 = 1, 4, 5
4 = any except for 4
4.5 = 4, 7
5 = any except for 7
5.5 = 3, 6, 7
6 = 1, 2, 4, 5
6.5 = 2, 3, 5, 6, 7
7 = 1, 4
Augmented (7) = 1 3 #5 7
Major (7) = 1 3 5 7
Dominant = 1 3 5 m7
Minor (7) = 1 m3 5 m7
Diminished (7) = 1 m3 b5 m7
Strictly Diatonic Chords by Scale Position:
(Inversions not listed)
Diatonic Intervals: Interval Factors:
I:
Sus2: 1 2 5 (7) 1 2 5 (7)
Sus4: 1 4 5 (7) 1 4 5 (7)
Add6: 1 3 5 6 (7) 1 3 5 6 (7)
Add9: 1 3 5 (7) 2 1 3 5 (7) 9
Add11: 1 3 5 (7) 4 1 3 5 (7) 11
Drop3: 1 5 (7) 1 5 (7)
Drop5: 1 3 (7) 1 3 (7)
(Drop7): 1 3 5 1 3 5
ii:
Sus2: 2 3 6 (1) 1 2 5 (7)
Sus4: 2 5 6 (1) 1 4 5 (7)
Add6: 2 4 6 7 (1) 1 3 5 6 (7)
Add9: 2 4 6 (1) 3 1 3 5 (7) 9
Add11: 2 4 6 (1) 5 1 3 5 (7) 11
Drop3: 2 6 (1) 1 5 (7)
Drop5: 2 4 (1) 1 3 (7)
(Drop7): 2 4 6 1 3 5
iii: Sus2, Sus4, Sharp5, Add6, Add9, Add11, Drop3, Drop5, (Drop7)
Chord spelling, nashville numbers, and chord charts
a 7 chord starting on interval like 2 is looked at in terms of what steps are teken to get to the intervals which follow it
for example. a 7 chord starting on the 1 of the diatonic scale contains the intervals = 1 3 5 7
a 7 chord starting on the 2 contains the the intervals = 2 4 6 1
in the chord spelling, building, and altering methodology
a 7 chord starting on 2 contains is follwed by m3 5 m7 intervals
it is commonly called a minor 7 chord and the 2 is follwed by a minor 3 interval and eventually a minor 7 interval AKA step
that minor 7 interval ends the chord or arpeggio on the 1 interval of the diatonic scale
the 1 interval, whihc is just before the 2 interval, is the seventh diatonic step acending away from the 2 interval
just like the sevent diatonic step ascending away from the 1 interval is the 7, the diatonic interval just before the 1
in chord spelling however, despite the intervals repeating their pitch class after 7 diatonic steps
after 8 diaotnic steps the starting note is repeating just an octave higher
they are the same note but there's more intervallic space inbetwen those notes. despite sharig note names aka pitch class
they are differentiated in chord spelling bulding, and altering methodology
for example:
the 2 4 6 1 aka the 1 m3 5 m7 7 chord or commonly called the minor 7 chord staring on 2 thus haing a 2 tonal center and also commonly referred to as the mode of dorian
can have a 9 added
that 9 is the interval just above the root interval
in this case of 2 as tonal center that would be 3
if it were 3 then it would be 4
that interval just above the root interval an octave up is that 9
the note name aka pitch class of that interval just above the root note, it's 2nd factor, just one diatonic step up
is the same note as the note an octave up but instead of being called a 2 it is called a 9
often, despite the 2 4 6 triad containing the 2 as root followed by a minor third step to reach the 4th diatonic interval that minor third step is just called a third
especially with use of chord charts and nashville numbers, which is how many musicians that actually work do music
it is common terminology to say a 7 chord on ii ( meaning a minor 7th chord staring on 2 as discussed above.)
instead of specifying such it is implied that you already know that the 7 will be a minor 7 when playing on the 2
this is where the chord spelling building and altering methodology thrives
is in tandum with nashville numbers and chord charts
for example:
IV: 7 drop5 add9 (major third and major 7 are implied
vii: 7 drop5 (minor third and minor 7 are implied, diminished fifth would be implied too but removed)
iii: 6 (sometimes instead of being called add 6 they are just called 6 chords, goes for 9, 11, & #11 as well
this methodology can be used systematicallly to figure a list of we honed on chords for each interval of the scale
not only is the system practical, it is possible to use computationally with other methodologies in music
parallel movement is easy to track using this methodology
and it is so synchronistic because of some special rootedness of consecutive thirds
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Special inversions that don't fit into the first, second, & third inversion methodology
the first second third and even fourth inversions apply to the to traditional chord spelling, building, and altering
methodolgy, which allows musicians and composers to computationally evaulate chord structure for each interval of the scale
by naming and categorizing components of chords and how those components might possible be added,
moved, and deducted in a rational way.
7 5 2 Transparent
the intervals two diatonic steps down and two diatonic steps up
1 6 3 Intense
the intervals one pentatonic step down and two pentatonic steps up
or the intervals two diatonic steps down and two diatonic steps up
6 3 1 Pipey
the intervals two pentatonic steps down and one pentatonic step up
the intervals three diatonic steps down and two diatonic steps up
1 5 3 Mean
the intervals two pentatonic steps down and two pentatonic steps up
the intervals three diatonic steps down and two diatonic steps up
5 7 3 Sour
1 4 7 Sectioned ***
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placing a tonal center in between consecutive diatonic thirds to form another pentatonic-style scale
thirds alloe instrumentalist great maneuverability but can also be used for highly efficient
and pleasant tonal center shifting
3 5 7 2 4 & 4 2 7 5 3
6 1 3 5 7 & 7 5 3 1 6
2 4 6 1 3 & 3 1 6 4 2
5 7 2 4 6 & 6 4 2 7 5
1 3 5 7 2 & 2 7 5 3 1
4 6 1 3 5 & 5 3 1 6 4
7 2 4 6 1 & 1 6 4 2 7
1 4 6 5 3 & 3 5 6 4 1 (specially sequenced)
Highly efficient Smooth Tonal Center Shifting using diatonic thirds adjacent to a tonal center
select a tonal center of your scale
this coul be the diatonic scale or a reduced version of such
and move backwards four diatonic steps to figure out
a good alternate tonal centers to move to thorughout the composition the music moves
for example:
2 -> 5 -> 1 -> 4
in the low register of an instrument use the starting tonal center
maybe in higher tuned instruments start with a resulting tonal center though
and instead of conituning up consecutive double third steps
star over from the beginning after three or five values for tonal center
turn back and go around, or just flat out strt over.
try plaing the root of a tonal center than moving a good alternate
then move to a note that's the second factor of a good chord for the original tonal center
often you don't need to move too many alternate tonal centers but it does happen where you get to run through a bunch
on and around these tonal centers find arpeggios & chords which match
2 as essential primary center corresponds to
these consecutive diatonic thirds pentatonic-style scales (one forwards & one backwards
5 7 2 4 6 & 6 4 2 7 5
and other variants of pentatonic-style scale with 2 as tonal center
arrange the diatonic scale in consecutive fourths and fifths, 4 1 5 2 6 3 7
deductions possible = instead of traditional 4 & 7 are 1 & 3 as well as 5 & 6
resulting in pentatonic style scales:
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Scale Deductions:
Deduct Intervasl from the chromatic, diatonic, hexatonic, and pentatonic scales using non pentatonic and hexatonic-style decutions
through chord building
consecutive intervals
consecutive factors
deductions can be utilized at different areas of the frequency spectrum.
select a starting note in the bass to base this off of and at rational intervals change which notes are being deducted from the diatnic scale.
for the best results use three deduction variations that deduct two notes for 3 pentatonic variant scale sets
make sure the two alternate pentatonic style deductions are of the intervals a fifth above and a fourth below those intervals
make the bass note this is based off of the 4th, 1st, or 5th of the ionion diatonic scale of the key being used
switch between deductions used at consecutive 4ths, ocatves, or 5ths.
for example:
deduct 3 & 7 from B1 until E2
deduct 4 & 7 from E2 until D2
deduct 1 & 5 from D2 until G2
switching can be improvised and whenver crossing a consecutive fourth a different but rational deduction se can be used.
it is slightly sadistic to cut notes out of your chords rather than just not using deducted notes as roots, but that can be construed as sadistic too.
Keeping to one diatonic scale is great.
When working with deductions and trying to use chords, if an interval is being deducted and you wish not to use it, you can take that interval from the interval of the root, or the closest to the root as possible
that has a sound and almost brings out animal sounding, dynamic sentiments.
another strategy would just be to take that interval from the deducted note
say i want to do a 1 3 7 chord on 1
but I want to deduct 3
i can use the 3rd factor from 7 in the diatonic scale, the 2nd, making a 1 7 9 chord (the 9 is the 2nd factor relative to a root but an octave up)
or I could use the 3rd factor from the the 3 to get a 1 5 7 chord
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Consecutive Intervals of the Chromatic Scale:
1 m2 2 m3 3 4 t 5 m6 6 m7 7 = Chromatic Scale
^---^---^---^---^---^---^---^---^---^---^---^----Consecutive Seventh <-&-> Minor Seconds
^-------^-------^-------^-------^-------^--------
^-----------^-----------^-----------^------------
^---------------^---------------^----------------
^-------------------^-------------------^--------
------------^-------------------^----------------
----^-------------------^-------------------^----
----------------^-------------------^------------
--------^-------------------^--------------------
^-----------------------^------------------------
^---------------------------^--------------------
--------^---------------------------^------------
----------------^---------------------------^----
------------------------^------------------------
----^---------------------------^----------------
------------^---------------------------^--------
--------------------^----------------------------
^-------------------------------^----------------
----------------^--------------------------------
^-----------------------------------^------------
------------^------------------------------------
^---------------------------------------^--------
--------------------------------^----------------
------------------------^------------------------
----------------^--------------------------------
--------^----------------------------------------
^-------------------------------------------^----
----------------------------------------^--------
------------------------------------^------------
--------------------------------^----------------
----------------------------^--------------------
------------------------^------------------------
--------------------^----------------------------
----------------^--------------------------------
------------^------------------------------------
--------^----------------------------------------
----^--------------------------------------------
D A E B F# C# G# D# A# F C G = Linearized Circle of Fourths<-&->Fifths
| | | | | | | | | | | |
4 1 5 2 6 3 7 t m2 m6 m3 m7 = Circle of Fourths <-&-> Fifths-Style Chromatic Scale (Consecutive Fourths <-&-> Fifths)
^---^---^---^---^---^---^---^---^---^---^---^----Consecutive Fourths <-&-> Fifths (4 1 5 2 6 3 7 t m2 m6 m3 m7 repeat)
^-------^-------^-------^-------^-------^--------Consecutive Minor Sevenths <-&-> Seconds (4 5 6 7 m2 m3 repeat)
^-----------^-----------^-----------^------------Consecutive Minor Thirds <-&-> Sixths (4 2 7 m6 repeat)
^---------------^---------------^----------------Consecutive Minor Sixths <-&-> Thirds (4 6 m2 repeat)
^-------------------^-------------------^--------Consecutive Minor Seconds <-&-> Sevenths (4 3 m3 2 m2 1 7 m7 6 m6 5 t repeat)
------------^-------------------^----------------
----^-------------------^-------------------^----
----------------^-------------------^------------
--------^-------------------^--------------------
^-----------------------^------------------------Consecutive Tritones (4 7 repeat)
^---------------------------^--------------------Consecutive Sevenths <-&-> Minor Seconds (4 t 5 m6 6 m7 7 1 m2 2 m3 3 repeat)
--------^---------------------------^------------
----------------^---------------------------^----
------------------------^------------------------
----^---------------------------^----------------
------------^---------------------------^--------
--------------------^----------------------------
^-------------------------------^----------------Consecutive Thirds <-&-> Minor Sixths (4 m2 6 repeat)
----------------^--------------------------------
^-----------------------------------^------------Consecutive Sixths <-&-> Minor Thirds (4 m6 2 repeat)
------------^------------------------------------
^---------------------------------------^--------Consecutive Seconds <-&-> Minor Sevenths
--------------------------------^----------------
------------------------^------------------------
----------------^--------------------------------
--------^----------------------------------------
^-------------------------------------------^----Consecutive Fifths <-&-> Fourths
----------------------------------------^--------
------------------------------------^------------
--------------------------------^----------------
----------------------------^--------------------
------------------------^------------------------
--------------------^----------------------------
----------------^--------------------------------
------------^------------------------------------
--------^----------------------------------------
----^--------------------------------------------
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Numbering Non-Diatonic Intervals Using the Chromatic Scale Arranged in Consecutive Fourths and Fifths:
4 1 5 2 6 3 7 t m2 m6 m3 m7
| | | | | | | | | | | |
4 1 5 2 6 3 7 10 8 11 9 12
Assigning Pentatonic & Diatonic Scale Position Identities to Non-Diatonic Intervals using Scale Position Sequencing Methodology:
4 1 5 2 6 3 7 10 8 11 9 12 = Reference Chromatic Scale Arranged in Consecutive Fourths and Fifths
4 1 5 2 6 3 7 4 1 5 2 6 = Transient-Style Diatonic (TSD)
3 1 5 2 6 3 1 5 2 6 3 1 = Transient-Style Traditional Pentatonic Deduction (TSTPD)
5 1 5 2 6 3 6 3 6 2 5 1 = Balanced (B)
4 1 5 2 6 3 7 3 6 2 5 1 = Oscillating-Style Diatonic (OSD)
5 1 5 2 6 3 6 2 5 1 5 1 = Oscillating-Style Traditional Pentatonic Deduction (OSTPD)
Example of what the term "transient-style" refers to: 1 2 3 1 2 3 1 2 3... These behave like stairs
Example of what the term "oscillating-style" refers to: = 1 2 3 2 1 2 3 2... These behave like hills
Use this experimental method to assign Diatonic, and Pentatonic scale position identities to non-diatonic intervals, and the diatonic intervals dedcuted of pentatonic and hexatonic scales.
It is also advised to experiment with assigning intervals to non-diatonic intervals and the deducted intervals of the pentatonic and hexatonic scales by assigning them scale position identities based on the methodology of chord Spelling...
(Major minor, and 7 triads; augmented, major, minor, dominant, and diminished 7 chords (quiads); b5, #5, & #11 chords; alterations like add 6, 9, & 11; drop (7), 5, 3, & (1).)
For optimally pleasent results, select three consecutive intervals of any scale postion sequence, and simply move around to each interval playing the assigned or default scale postion identity.
For example: Use 2 6, and 3 of the TSTPD Sequence above & the Key of A#
Play around with 6(G) as tonal center/root and then move to 2(C) as root, then move back to 6(G), and then onto 3(D)
2, 6, & 3 = subdominant, tonic, & dominant factors, even if the corresponding intervals are not of 4, 1, 5 relationship, but using this optimal method of Scale Position Sequencing, they will most likely be, even if not in the key of the 1.
There are some unique subdominant, tonic, and dominant factor sets though.
For example: the 3, 1, & 5 as well as the 1, 5 & 6 of the TSTPD.
Optimal Tonal Center Shifting Patterns:
4 = subdominant factor
1 = tonic factor
5 = dominant factor
4->1->5
5->1->4
1->4->5
1->5->4
4->5->1
5->4->1
[This Scale Position Sequence Methodology began from a simple improvisation version of scale position sequencing.
Here's an example:
In the Key of A#:
The 1 is played pentatonically as if a 1
the 2 is played pentatonically as if a 3
skip the 3 and play the 4 as if a 2 pentatonically.
There are many, many ways of using this as well as the Scale Position Sequencing Methodology.
Rather than simply 4,1, 5 factor tonal center shifting, but such is the least experimental way.
Sequencing Scale Postions Identities, and Tonal centers like in both of the A# examples works because the traditional pentatonic deductions beautifully reduce the scale resulting in deductive semi-atonalism, while allowing for easy constructive atonalism
where the music allows for various tonal center shifts, and the appealing to of multiple tonal centers atonally through use of strange chord progressions, chords, and arpeggios.
The Complicated and Simple Pentatonics have alternate names corresponding to the Scale Postion Sequencing Methodology...
(5 & 6 are the new identies assigned to the 4&7 in the OSTPD sequence.)
4 1 5 2 6 3 7 = Traditional Pentatonic
^-----------^---Deduct These
4 1 5 2 6 3 7 = Complicated or Transient Pentatonic (Corresponding with TSTPD below)
^-------^-----Deduct These
4 1 5 2 6 3 7 = Simple or Oscillating Pentatonic (Corresponding with OSTPD below)
^---^-------Deduct These
4 1 5 2 6 3 7 = Chaotic or Unorganized Hexatonic
^---------Deduct This
3 & 1 are the new identites assigned to the deducted 4 & 7 in the TSTPD sequence.
5 & 6 are the new identities assigned to the 4 & 7 in OSTPD.
This incidence although conflicting in some ways because the 1 & 3 as well as the 5 & 6 are deducted in the context of the Simple & the Complicated pentatonic-style scales, and assigned in the context of the Scale Position Sequencing Methodology.
Those more dissonant deductions are actually even more justified because of this incidence, and scale postion sequencing is more rational as well.
These contingencies do highlight the circular, reccurring, and balanced nature of the spectrum of pitch, as well as the highly mathmatical, equilateral 12-note Western Music system.
And that confliction might represent a rationalization of some sort as to why deducting the 4 & the 7 is more optimally pleasureful, and practical than deducting the 1 & 3 or the 5 & 6, despite them relating to eachother in their symmetry according arranging the diatonic scale by consecutive fourths and fifths, (4 1 5 2 6 3 7).
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Sequencing Scale Positions and deductions along the pitch spectrum
Use or deduct different notes at different postions along the pitchspectrum
Envelope & Waveform Methodology:
1 3 5 7 = attack, hold, sustain, release.
This goes for all of the triad and quiad sequences: minor, diminished, dominant, and augmented.
This is more about the format.
Now, if 1 3 5 7 = a sine wave, then changing the intervals based on this concept thus alters the wave shape which the triad or quiad appeals to, if it does.
This premise implies that the 1 3 5 7 quiad which chord spelling is centered around represents an important thread of the diatonic scale
Chord spelling & alteration uses this principle of 1 3 5 7, even when working with a chords that are not the 1. For example a 7 chord on the ii, (also known as a minor 7 chord or quiad) = 1 m3 5 m7 in chord spelling methodology.
This ii q would, according to Envelope & Waveform Methodology represent a different waveform or wave shape.
Each chord then resembles what various instruments in having different timbres, consisting of different harmonic textures, of course being bound to the timbre, and possible harmonic occurences of the instrument.
Trumpets for example, use arious frequency ratios use to produce the fundamental of each note, like a human voice.
Then there's all the harmonics.
What this methodology is saying, that just like a person's physiology changes how their voice makes a note, as well as all of the frequency ratios and harmonics involved in that, chord, comprised of multiple frequency ratios as well pertains to various waveforms and thus back to what can produce those wave forms
All of the chracteristics of a waveform are at play here.
This includes rise, decay, dampen, fall, and evolve. (Noteworthy phenomena in audio engineering, and FM synthesis.)
Feasibly this would include phemoena of articulation like sticcato, legato, portamenta. Silence and loudness however seem to be the phenomena that would not be feasibly related to via intervasl but, using a non- diatonic note like a m2 as a drone or noise somewhat represents relative silence, so who knows where the conceptual limits are.
These and many more could be implied by which intervals are used in a chord sequence, possibily even directly correlating to the chords possible via traditional chord spelling and alteration methodology.
If 1 3 5 1 corresponds to a sine wave than 1 6 3 1 corresponds to a ramp wave.
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Inversions
1st inversion of 1 3 5 7 is 3 1 5 7
2nd inversion of 1 3 5 7 is 5 1 3 7
3rd inversion of 1 3 5 7 is 7 1 3 5
1st = 1 6 3 5
1st->2nd = 1 6 4 3
1st->3rd = 1 4 2 6
1st->2nd->3rd = 1 6 4 2
1st->3rd->2nd = 1 7 3 5 *
1st->3rd->3rd = 1 3 6 4
1st->2nd->3rd->3rd = 1 7 5 3
1st->3rd->2nd->3rd = 1 4 3 6
1st->3rd->3rd->3rd = 1 5 7 3
1st->2nd->3rd->3rd->3rd = 1 6 5 3
1st->3rd->2nd->3rd->3rd = 1 3 6 5
1st->3rd->2nd->3rd->3rd->3rd = 1 4 6 2
2nd = 1 4 6 3
2nd->1st = 1 5 3 7 **
2nd->3rd = 1 6 2 4
2nd->1st->3rd = 1 2 6 4
2nd->3rd->3rd = 1 5 3 6
2nd->1st->3rd->3rd = 1 5 6 3
2nd->3rd->3rd->3rd = 1 3 7 5 ***
2nd->1st->3rd->3rd->3rd = 1 6 3 4
3rd = 1 2 4 6
3rd->3rd = 1 3 4 6
3rd->3rd->3rd = 1 3 5 6
3rd->3rd->3rd->3rd = 1 3 5 7
[* ** *** = Foremost Inversion of a cycle of inversions that coincides with another, unlisted cycle of inversions.
Cycles refer to consecutively implemented third inversions which make the last factor of the inversion the first.
For example: 1st->3rd->2nd, 1st->3rd->2nd->3rd, 1st->3rd->2nd->3rd->3rd, 1st->3rd->2nd->3rd->3rd->3rd are a set that cycle by one factor of the inversion
When improvising with an instrument in hand it is common to simply move the root note of a chord to the top and the next lowest note becomes the root
typically triads and 7 chords are used.
Such improvisational inversions are accounted for in the list of inversions above because of what 3rd inversions provide
to make best use of the above list of inversions it's best to take one inversion style from the list and learn the shape/sequence of intervals of that inversion style for each scale position since the above are chord spelling chords that operate on a facotoral basis.
For example: in 1 6 5 3 the 6 means the 6th factor of the diatonic scale above the root. If the root is the 2nd factor of the ionian diatonic scale then that 6 means the 7th of the ionian diatonic scale.
Learning these are like learning vocabulary and the purpose is to have access and comprehensive understanding of these while improvising.
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Assigning Non-Diatonic Intervals Scale Position Identities Using Adjacent Intervals:
1 m2 2 m3 3 4 #4b5 5 m6 6 m7 7
1 1or2 2 2or3 3 3or4or5 4or5 5 5or6 6 6or7 6or7or1
Flipping, Replacing, & Reassigning Non-Diatonic and Diatonic Intervals Using Enantiomer Methodology: :
1 2 3 4 5 6 7 :Assign the non-diatonic intervals pentatonic, or diatonic scale position identites using Adjacent Intervals, or Scale Position Sequencing Methodology
m2 m3 m6 m7 then flip, replace, or reassign the identiy of diatonic interval to the new identity of the non-diatonic interval.
*--*
1 2 3 4 5 6 7
m2 m3 m6 m7
Sets become more and more experimental from here on, but if it's rational it's right.]
1 2 3 4 5 6 7
m2 m3 m6 m7
1 2 3 4 5 6 7
m2 m3 m6 m7
*-----* *--*
1 2 3 4 5 6 7
m2 m3 m6 m7
1 2 3 4 5 6 7
m2 m3 m6 m7
*-----*
1 2 3 4 5 6 7
m2 m3 m6 m7
*-----*
1 2 3 4 5 6 7
m2 m3 m6 m7
1 2 3 4 5 6 7
m7 m6 m3 m2
1 2 3 4 5 6 7
m7 m6 m3 m2
*-----*
1 2 3 4 5 6 7
m7 m6 m3 m2
1 2 3 4 5 6 7
m7 m6 m3 m2
*-----*
1 2 3 4 5 6 7
m7 m6 m3 m2
1 2 3 4 5 6 7
m7 m6 m3 m2
*-----* *--*
1 2 3 4 5 6 7
m7 m6 m3 m2
*--*
1 2 3 4 5 6 7
m7 m6 m3 m2
(*---* = The Diatonic intervals and their corresponding non-diatonic intervals that can be flipped to create a new interval set.)
[Explanation: In traditional chord/arpeggio spelling, add 9 chords refer to the relative 2 factor of the diatonic scale an octave above the root of the chord being addressed.
In simpler terms, a 9 is just a 2 an octave up.
For roots of the diatonic scale with a relative 2 factor that is a whole-tone above the root (1, 2, 4, 5 , & 6) the relative 9 factor is a 2 interval.
For roots of the diatonic scale with a relative 2 factor that is a half-step above the root (3, & 7) the relative 9 factor is a m2 interval.
When Numbering Non-Diatonic Intervals Using the Chromatic Scale Arranged in Consecutive Fourths and Fifths, (4 1 5 2 6 3 7 10 8 11 9 12), 9 = m3.
A composition can suddenly flip and any diatonic intervals being used can become the non-diatonic intervals and their newly assigned scale position identities.
One of the
For example:
Before Flip: After Flip:
1 2 3 4 5 6 7 m3 2 m2 4 m7 m6 7
m3 m2 m7 m6 1 3 5 6
Different Enantiomer Sequences can be used octave by octave; that was the base idea; using the first two enantiomer sequences as they are octave by octave.
In one octave the non-diatonic notes have identities that correspond to the diatonic interval above it...
For example: m3 gets treated as if a 3. Then in the next octave it corresponds to the interval below it, (the m3 gets treated like a 2)
This strategy for patternizing intervals and their scale position identities became usable for not just flipping, and skewing intervals, but reassigning and replacing them as well.
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Interval matching
1 8 2 9 3 4 10 5 11 6 12 7
1 5 2 6 3 7 11 8 12 9 13 10
4 1 5 2 6 3 7 10 8 11 9 12
1 2 3 4 5 6 7 8 9 10 11 12
0 1 2 3 4 5 6 7 8 9 10 (11)
For example: the 7th column of numbers implies some sort of facotral, and intervallic relationship between the numbers 10, 11, 7, & 6.
The chromatic scale ( 1 8 2 9...), the chromatic scale arranged in consecutive fourths & fifths (4 1 5 2...), and the list of natural numbers (1, 2, 3, 4...) are some of the most profound sequences of numbers in all of nature.
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Music Math can help you to commit to an idea, so have at it!
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BPM:
Using averages of intervals, and frequencies to find BPMS appels to our sense of time and place.
BPMs based on the character attributes of numbers appeals use through math and physics. These BPMs tend grip us from deep within, being more essential and less obscurely relevant.
The same phenomena would be observed if the SI unit for time were based on frequecnies of light, sound, or simple number concepts. The alligning of our SI unit for timekeeping to how Cesium atoms absorb light, tranfrom it into energy and are displaced from atoms that were not kicked, is a strangely obscure phenomena to tie timekeeping to and as a result the SI value for the econd is smoother than it is gripping and captivating.
I use a tap tempo app to find the tempo I'm feeling
then multiply that bpm by 60 to find its frequency (1Hz = 60bpm)
i divide the value by 2 until I am in a reasonable frequency range for pitch, becuase doing so does not affect the pitch class of the frequency, like with the double time example
multiplying and dividng by 2 yields the same note when working with frequency and the concept is the same for
I look at a chart with all the frequencies in A4 = 400 Western standard just tuning
and I find what frequency that BPM correlates to
if it is in between frequencies that and in the song those frequencies are strange intervals of the key of the song you can calculate BPM in cents like with frequency.
Another measure for pitch is cents.
in between A4 and A5 there are 1200 cents
in between A4 and A#4 there are 100 cents
this is a very helpful tool to use in audio and music.
We can correlate BP the same exact way.
the 440Hz multiplied by 60 = 26,400 BPM
divide 26,400 by 2 until you get 103.125BPM
do the same for A#4 = 466.16Hz and get 109.25625BPM
109.2562 - 103.125 = 6.13125BPM
6.13125BPM = 100 BPM cents
If your value was in between A4=440Hz and A#4=466.16Hz, say 444.4 Hz, then subtract 440 from 444.4 and get 4.4
since 6.13125 = 100 BPM cents
4.4 = ?
cross multiply and divide like we did before when we working out of 60
6.13125? = 440 (this happens to be the same number we subtracted from 466.16Hz, that is just a coinidence)
440 / 6.13125 = 71.76350662589195 BPM cents
I then round this BPM cents value to a number sequencing number. For BPM I would use a clarity numbers since tempo is an essential feature not a complex one
maybe use the clip style mixture number sequence here 77.771 BPM Cents and reverse calculate
sice 6.13125Hz = 100 BPM cents
divide it by 100 to get .0613125 and multiply that by 77.771 to get 4.76870844375Hz
add 4.76870844375 to the BPM of A4 = 440HZ to get 444.76870844375 Hz.
Multiply that by 60 to get 26,686.122506625 BPM
divide that by 2 to get into the appropriate BPM range for music and around what we figured befor for the BPM of 440Hz
We get 104.2426660415039 BPM
My DAW let's me have three decimal place values
I like to round down because it's a rule of thumb that we like to rise up to notes in music
I would use 104.242 BPM not 104.243 BPM because of that.
Rounding down like this as a rule of thumb plays on the concept of headroom. It's additive headroom and it is noticable, so round down if you have to round for any parameter!
You can truncate any value if desired
instead of using all of those decimal place values I have for tempo I could truncate it 104.230 BPM or repeat the first number for added number sequencing for 104.231 BPM, or the previous number which gives us 104.242 BPM again. you can also skew a middle ground if the numbers allow. There's a lot of maleability but the point is synchronize your number choices to something profound.
This math of translating a number out of 100, 10, or 1 to it's equivlalent out of 60, 120, 1200, or 12 is commonly denoted by ratios.
42 is to 100 as ? is to 60
can be represented as 42/100 =?/60
but these equations can also be represented as 42:100 = ?:60
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Complimentary Number Skews:
The following complimentary number sets use ratios to establish relationships between two numbers, particularly regarding the base-12 number system.
Although not actually frequency ratios, complimentary number ratios can be used for countless applications in audio engineering and music parameters, including functioning as frequency ratios.
two digit nmbers can be repeated and mixed for example: 11161616 is or 5777777.
complimentary numbers can be stacked into multi digit numbers
say we pair 2 and 7 together we can stack 2 into the 40's 424242 or 42474747
the numbers input to the DAW for various parameters directly affect the electrical signal embedded with music that goes to the speaker
these numbers affect that signal and having numbers that flow intervallically unlock very important textures and attributes to modern music
My favorite complimentary number skews
applying complimentary number skews to the intervals of the Western 12-note pitch-class system
Numerically
12:7 12 (octave +4 diatonic steps)
11:6 11 (octave +3 diatonic steps)
10:5 10 (octave +2 diatonic steps)
9:4 9 (octave +1 diatonic step)
8:3 8 (octave)
7:2 7
6:1 6
5:12 5
4:11 4
3:10 3
2:9 2
1:8 1
Chromatically
12:7 m7
11:6 m6
10:5 tritone
9:4 m3
8:3 m2
7:2 7
6:1 6
5:12 5
4:11 4
3:10 3
2:9 2
1:8 1
and this works because you are dividing a whole into intervals
which represent prominent points of the spectrum of possible interval (frequency ratios)
and 12 is especially good at this.
If you were to use a 9 interval system you might be able to figure out how to apply it to other phenomena like we have with political affiliation
12 just happens to do so hyper-practically, meaning it allows for the most variety, equilbrium, and maleability
just like a 12-interval set of colors hyper practically represents highly pertinent houses of color which is quite evident. This is because 12 correlates to the dodecahedron perfectly equilateral polyhedron with 12 pentagonal faces and 12 vertices(points) which is rare in geometry.
if math is zoomed in logic and perfection there means precision on a small scale
and intervals are not zoomed in logic but also not zoomed out logic, and their perfection means precision to the naked eye, and in layman's world
then geometry is zoomed out logic and perfection within means precision on an obscure magnituide. The rarity of perfect shapes like the dodecahedron means that something is going on with the number 12 that makes it a pertinent value in ordering a broad spetrum of possibilites, and variables.
For example:
1:8
2:9
3:10
4:11
5:12
6:1
7:2
8:3
9:4
10:5
11:6
12:7
To use this for chords I can select four of these based on how I like the number as the interval for a chord.
Right now I really like.
1:8
4:11
9:4
Using the values on the left of each ratio can be used produce chords and arpeggios that consist of the 1, 4, & 9 scale degrees of any scale position.
The values on the right of can also be used in the exact same way, as a 4, 8, & 11 arpeggio or chord based on any scale position and moving in steps of the scale.
It's also possible to use all of the values on the left&right for one arpeggio or chord that in this case would consist of 1,4,8,9,11.
I like to avoid the 4th and 7th factors of the diatonic scale to keep my chords and arpeggio consistent of the pentatonic.
For example if were playing that 1, 4, 9 chord or arpeggio on the 1st scale postion of the diatonic scale, because, first of all, the 4th and 7th factors and also the 4th and 7th scale postions relative to the 1st scale position.
I would shift away from the 4th to the 5th. If we were using this chord or arpeggio on the second scale position the 4th is the 3rd factor of the ditonic scale away from the root and the 7th is the 6th factor, so we wouldn't need to shift any of the factors of our arpeggio or chord sequence.
I prefer to shift the step up to result in larger interval limits from the scale position which the arpeggio or chord is based from.
You can also shift up when travelling up and down when travelling down, or do so when you want to shift in those directions.
These are my favorite sets.
1:4
2:5
3:6
4:7
5:8
6:9
7:10
8:11
9:12
10:1
11:2
12:3
1:3
2:4
3:5
4:6
5:7
6:8
7:9
8:10
9:11
10:12
11:1
12:2
1:6
2:5
3:4
4:3
5:2
6:1
7:12
8:11
9:10
10:9
11:8
12:7
1:9
2:10
3:11
4:12
5:1
6:2
7:3
8:4
9:5
10:6
11:7
12:8
1:7
2:8
3:9
4:10
5:11
6:12
7:1
8:2
9:3
10:4
11:5
12:6
It seems that this last complimentary number skew works especally well because of it's balanced contigency.
Notice how on the number skew above that starts off with 1:6 has values that move in opposite directions.
When doing this, usually inversive results occur.
In this list there 2:5 as well as 5:2, and this inversiveness occurs for each value
The last complimentary number list, which starts with 1:7, has this inversiveness although it ascends. It's balanced.
In no way are these lists superior though, I enjoy so many of these lists musically and for audio engineering. This concept amazes me.
Using this 1:7 list as a standard, however, it seems that
5:11 & 11:5 = Harmonic Content
7:1 & 1: 7 = Clarity; Natural-Sounding
4:10 & 10:4 = Smoothness
use these numbers for the different utilities in audio engineering specifically, but be sure to explore the many possible number skews as well.
Typically it seems that the 1, 4, & 5 bo
we can take the most essential number patterns of the base-12 number system and make them into complimentary number sets.
4 1 5 2 6 3 7 10 11 9 12 (the circle of fourths and fifths)
1 2 3 4 5 6 7 8 9 10 11 12 (the counting scale)
1 8 2 9 3 4 10 5 11 6 12 7 (the chromatic scale)
=
1:4:1
2:1:8
3:5:2
4:2:9
5:6:3
6:3:4
7:7:10
8:10:5
9:8:11
10:11:6
11:9:12
12:12:7
any of these can be ranched off into a counting scale based complimentary number skew, those seem to be very effective.
based on 12:7 we can create a complimentary number skew list.
1:8
2:9
3:10
4:11
5:12
6:1
7:2
8:3
9:4
10:5
11:6
12:7
8:4
7:1
6:5
5:2
4:6
3:3
2:7
1:10
12:8
11:11
10:9
9:12
the circle of fourths and fifth sequence (on the right set of values in each ratio)
and the counting scale (on the left) have been synchronized in a particular manner.
number sets can be done in compund ratios as well
8:7:4:3:2:1
7:8:1:6:8:8
6:9:5:2:1:2
5:10:2:5:7:9
4:11:6:1:12:3
3:12:3:4:6:4
2:1:7:7:11:10
1:2:10:12:5:5
12:3:8:9:10:11
11:4:11:11:4:6
10:5:9:8:3:12
9:6:12:10:9:7
matrixes are amazing as well
1 2 3 4 1 5
4 5 6 : 2 6 3
7 8 9 7 10 8
10 11 12 11 9 12
the attributes of the numbers correlate quite profoundly according columns of these matrixes
1:4 as 3:6 similarly to how 4:2 and how 5:3
coiling the number sequences seems to have an affect as well.
for example:
4 1 5
3 6 2
7 10 8
12 9 11
Number Sequencing
444.444 = repeating numbers style number sequencing = smoothness = I use these on highpass, lowpass, shelves EQ parameters as well as mix, volume, gain, time based parameters. They make good values for everything from frequency to values of miliseconds.
494.949 = complimentary style number sequencing = clarity = i use these for notches, bells, and for the q intensity. Sometimes it's a case by case basis when you want a smoothness number or a clarity number for a parameter.
for a mixture of the two use:
4.449 clip style sequences
4.999 lip style sequences
even repeating the first value a few times before switching values is logical and occurs when doing calculations with these numbers
111.8888
101010.5555
105105.101010555
etcetera.
however using the number styles
188.888
181.818
111.111
888.888
and possibly 111.118
are in my opinion for time being most effective
they interesect and correlate within the closed system of a DAW that is tuned into well-tuned systems of time, frequency, and ratio keeping
These systems are well balanced and positive AKA pleasent
set a low pass filter to -6dB per octave & 12712Hz
then set a shelf to
Inverting these complimentary numbers yields pleasent results as well that I used primarily until discovering the set I listed first.
technically the set I used previous
Use number sequences
for time based parameters:
multiply a number, (the lower the better), by 600 for the value of a number sequence in miliseconds
for example:
105 is to 100 as ? is to 60
105 multiplied by 60 divided by 100 = 63
63 = what 105 is to 100
if you wanted a tempo (BPM) from the number sequence 105 then stop here. 63 is your BPM, you can muliply it by 2 for its tempo in double time which is is equivalent to multplying A4 = 440Hz by 2 to get A5 = 880Hz.
Tempo is parallel to frequency in this way
to get your sequence number in milliseconds
multiply 63 by 1000 to get 63,000.
for a smaller value instead of dividing by 2 you can move place values
instead of 105 you would be using the number sequence 10.5 and you would get 6,300 miliseconds
Use 1.05 and you would get 630 miliseconds.
instead of 105 to 100 you can also do 105 to 1 for amazing results as well.
if you feel out a value for a parameter by turning the knob or fader manually, (instead of typing in numbers) and want to find something nearby you can reverse the process
say i have my release on an expander plugin set to 453.6 miliseconds and I want to find a number that comes from a specific number sequence like the list above I could divide 453.6 by 1000 then multiply by 100 and divide it by 60 to get .756
i can round .756 to .777 and then multiply by 60 divide by 100 and multiply by 1000 (or just multiply by 600 to be speedy) and get 466.2 miliseconds.
however, transforming the number sequences into miliseconds is not really necessary the original number sequences like 222.2 or 277.7 or 272.7 or 222.7 work really well too.
Using number sequences to produce values for parameters validifies the theory of intervallic metaphysics.
The numbers themselves have attriutes and are parts of patterns that come out in finetuned closed systems like timekeeping, frequency, ratios, and percentages.
many other number sequence strategies can be used, for example:
9 1
8 2
7 3
6 4
5 5
4 6
3 7
2 8
1 9
these are clarity numbers too they have a sparkle or a polish to them when used
there's a lot of rational number sequences.
i like using number sequences when finding BPMs too
they make good BPMs themselves but I like to use them when