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 ********** 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 ******** 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 *** ********* 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. 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: ********** 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 ********** 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 ----------------------------------------^-------- ------------------------------------^------------ --------------------------------^---------------- ----------------------------^-------------------- ------------------------^------------------------ --------------------^---------------------------- ----------------^-------------------------------- ------------^------------------------------------ --------^---------------------------------------- ----^-------------------------------------------- ********** 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). ********** 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. 1 4 5 6 3 a beautiful sequence ********** 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. ********** 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. **************************************************** Music Math can help you to commit to an idea, so have at it! Using Frequency Ratios Correlating BPM to the concept of Cents