METHODS TO LEARN MUSICAL FREE IMPROVISATION-COMPOSITION WITH NEW PERCEPTIONS

This book is for learning music composition and improvisation , based on more abstract mathematics of the music, the rhythm and the musical instrument and with new musical practice. It is a new awareness and method to link mental perception-images, the creation of feelings and finger actions. The true goal of composition and improvisation is the existential process of creating and listening it. It is both individually healing and socially celebrating.

Sunday, January 17, 2016

1. Guitar tuning. In what is the standard tuning optimal if at all ? Melodic optimal and chord optimal tunings. Alternative tunings and comparison with violin, mandolin, and other instruments

(SEE ALSO POST 177 ABOUT WHICH TUNING IS APPROPRIATE FOR WHICH STYLE OF PLAYING STYLE OF PLAYING )

The Latin names of the 7 notes come from an ancient latin hymn to saint John.

The first stanza is:

Ut queant laxis
resonare fibris
Mira gestorum
famuli tuorum,
Solve polluti
labii reatum,
Sancte Iohannes.

It may be translated: So that your servants may, with loosened voices, resound the wonders of your deeds, clean the guilt from our stained lips, O Saint John.

A paraphrase by Cecile Gertken, OSB (1902–2001) preserves the key syllables and loosely evokes the original meter:

Do let our voices
resonate most purely,
miracles telling,
far greater than many;
so let our tongues be
lavish in your praises,
Saint John the Baptist.

Another poem with English words for these syllables is the next

Do
Resonate
Miracles
Fabulous
Solving
Latent
Silence

Then there is also the 12-notes syllables for C, C#, D D#, E F F# G G# A A# B C

Do, Di, Re, Ri, Mi, Fa, Fi, Sol, Sil, La, Li, Si, Do

It seems that the tuning of the guitar in intervals of pure 4ths with one string in major 3rd, comes from the family of string instruments known to us today as the family of Viola da Gamba which was originated in in Spain in the mid to late 15th century. But the guitar as instrument is of ancient origin.

(see e.g. https://en.wikipedia.org/wiki/Viol )

The tuning of the guitar is not at all optimal for playing, that is why there are many alternative tunings.

1) The optimal tuning for solos, for the size of the guitar (66 cm scale length) would be to be tuned by 4ths (5 semitones) between all strings (E2,A2,D3,G3,C4,F4), as is with other instruments like bass, Puertorican Cuatro etc. This is so because of the size of our hand that comfortable it can cover at most 5 frets of the fretboard and the uniformity of the tuning helps the fingers move vertically in the fretboard easily.

As a contrast when we are dealing with a mandolin or violin due to its small size of the fretboard, the size of the human hand can cover comfortably 7 frets, and that is why, mandolin it is tuned on 5ths, (7 semitones). It is known though that the mandolin is not optimal for chord shapes, that are rather uncomfortable compared to the guitars.

This all 4ths tuning is used for centuries in other instruments (like Bass and 5-double string puertorican cuatro) . I believe that both types of tuning (standard and all 4ths) must be used by a guitarist . Knowing the guitar in all 4ths tuning makes you know better the guitar in the standard tuning. So it should be used in a supplementary way not exclusive way, as an advanced (post graduate) understanding of the standard tuned guitar. I want to say that I tuned my 12-string guitar to all 4th tuning just recently , and I have positive views about the all 4ths tuning . I still keep also a 6-string guitar with the standard tuning to follow friends in groups when playing the guitar , but when I want to compose music ,( because composition does not require instrument skills but only selection of chord and melodies and then writing them in the computer) I prefer the all 4th tuning guitar . Because it represents on its fretboard by the triads, in a perfect way the 24-cycle of chords with their relatives and intervals of 4ths distance cycle (see post 34 in my current blog ) . I use the 12-strings guitar not mainly for chords (as it is tiring by the required finger pressure on double strings chords ) but for melodies, and all 4th tuning made them very much easier! I use as chords only partial 3-notes (triads) or 4-notes chords , and all 4th tuning made them easier too. Chords with more notes than 3 or 4 not chosen by the composer but forced voicing due to the standard tuning of the guitar maybe a disadvantage too! Using the 12-string guitar with all 4th tuning is like using a 6-string bass or the popular Puertorican instrument Cuatro for solos which is also tuned in all strings by perfect 4ths. The only optimal thing which is lost is the 4-string chord shapes on the highest 4 strings of guitar in a standard tuning (ukulele chords), but for this I keep a 4-string tenor guitar with the tuning DGBE (see post 16 of my current blog).

See the next video of a guitarist discussing the all 4th tuning on the guitar

https://www.youtube.com/watch?v=W73StQ7kP0A

OPTIMAL GUITAR TUNING FOR CHORDS PLAYING MAINLY

2) The optimal known tuning for 3-note chords is the tuning in major 3rds (4 semitones) . It gives 3-note chords mostly in only 1, or 2 frets of the fretboard. In addition is it is uniform tuning allowing to transfer shapes vertically among the strings (see e.g. https://en.wikipedia.org/wiki/Major_thirds_tuning and http://ralphpatt.com/ ). And it must be said that 6-strings is not at all the optimal for chords. 3 or 4 notes chords played on 6 strings will require repetitions and inversions in the intervals of the chord which significantly alters the original intended sound of the chord. The optimal number of chords is of course 4 strings.

2.1) An even more optimal but unkown tuning for the 6-string guitar when chord-playing is the main target and not so much solo playing is and even better by alternating minor and major 3rds. In semitones for the 6 strings 4-3-4-3-4 or 3-4-3-4-3
E.g. Bb2- D3-F3-A3-C4-E4 or F2-A2-C3-E3-G3-B3 or A2-C3-E4-G4-B4-D4
THIS MAY BE CALLED THE HARMONIC TUNING OF THE GUITAR AS IT IS BASED ON THE HARMONIC 2-CTAVES 7-NOTES SCALE (see also post 79)
The latter is the most natural open tuning. There the same shape for major and minor chords and only 3 of them and in only one or frets compared to the 6 in the standard tuning guitar. If we want also dominant and major 7nth chords we use again only 2 frets. The same with the aug chords Only the dim7 chords require 3 frets. Because of the symmetry of the tuning among the strings, the relations of relative chords and also chords in the wheel of 4ths is immediate to grasp also geometrically. Of course when we say shape of chords as it is standard in jazz, we do not play all 6-strings but only 3 or 4 strings.

The easiness with which one can improvise melodies within a diatonic scale (all notes within 3 frets and in a very symmetric zig-zag pattern) together with 3-notes chords of the scale (gain all chord patterns within 3-frets) is unsuprassed.
At the same time , the easiness with which one can me diatonic scale modulations, chromatic (1 semitone apart) or by changing a minor to a major chord and vice versa and continuing in a relevant diatonic scale is unsurpassed again!

This harmonic tuning by alternating minor-major 3rds, allows, for all 4-notes chords of e.g. the D major scale in the 3rd octave (c3,d3,e3,f3,g3,a3,b3), Cmaj7->Em7->G7->Bm6->Dm7->Fmaj7->Am7 in 1st normal position across the fretboard, something not possible with the standard tuning of the guitar. In the standard guitar it is possible only by 2nd or 3rd inversion, or by shifting to the 4th octave or 2nd octave. Therefore there are important very natural voicing of the 4-notes chords of the 3rd octave that we miss with the standard tuning and it is in a single octave!

(One similar example is the keyboard of some accordions or concertinas or bandoneons, where 4 notes increase in pitch by one semitone at a skew line (neither vertical neither horizontal) an then we shift to the next line. It is entirely equivalent with tuning strings by major 3rds (4 semitones ) , where the skew line of buttons plays the role of the string. It is not only saving mores space in thiw way, that having more range , but also the chords are played easier forteh same reasons as that above. )

The standard guitar tuning is mixed and somewhere in between these two optimals as it is required to paly both solos, and chords. But it is not equally optimal between the two. In other words one might say that a 6-string guitar with the standard tuning is about 70% optimal for chords and 30% optimal for solos.

THE STANDARD GUITAR TUNING IS ALMOST SUB-OPTIMAL IN REPRESENTING IN A QUITE SYMMETRIC WAY THE WHEEL OF 4THS IN THE CHORDS. PROBABLY THE BEST IN IN REPRESENTING IN A QUITE SYMMETRIC WAY THE WHEEL OF 4THS BY CHORDS IN NORMAL POSITION IS THE REGULAR TUNING BY 4THS. BUT IT HAS DIFFICULT SHAPES FOR THE INVERSIONS, WHILE THE STANDARD NOT SO MUCH.

3) Nevertheless if we restrict to 4 strings only so as to play 3 or 4 notes chords, then the tuning of the last 4 strings of the guitar is indeed optimal as it can play more compactly the dominat seventh, and major 7th chords, than the tuning by the major thirds (4 semitones), although the 3-notes chords are less good than the 3rd major tuning. The optimal aspect of the higher 4-strings in the standard guitar tuning it is that it can play within 3 frets all the three inversions of major and minor triads (normal , 1st inversion, 2nd inversion) with roots on any of the 3 higher strings , and also the 7th of these chords, the diminished 7nth and the augmented too. This cannot be achieved with the all major 3rds tuning or all perfect 4th tuning. Therefore I would consider 4-strings guitars with the standard tuning (D3,G3,B3,E4) as indeed a very optimal tuning for 4-notes jazz chords on 4 strings. This suggests also an alternative tuning to the guitar so as to have the lower 4-strings chord-optimal and the higher 4-strings solo-optimal. Such a tuning would have the 5th and 4 the strings tuned on a major 3rd interval (4 semitones) E.g.
(D3:D2, G3:G2, B3:B2, E4:E3, A3:A3, D4:D4 or C3:C2, F3:F2, A3:A2, D4:D3, G3:G3, C4:C4 which is closer to the range of mandocello or G3:G2, C4:C3, E4:E3, A4:A3, D4:D4, G4:G4 which chords in the upper 4 strings identical with those ofthe ukulele) thus a bit lower or higher range than the standard tuning. If the standard tuning can be marked as 70% optimal for chords and 30% for solo, then the above tuning is 60% optimal for cords and 40% optimal for solos! The instruments like Ukulele, tenor guitar (chicago tuning), Greek 4-string Buzuki etc are quite optimal in tuning for 4-notes chords. Actually the term gui-tar comes from the word 4-strings in sanskrit language. We may compare them with the tuning of the mandocello C2-G2-D3-A3.

4) A modern 12-string guitar it is as if two different instruments: It is like a Puerto Rican Cuatro on the 4 lower strings (good for solos), and like 4-double strings ukulele or 4-double strings (good for chords)! I often use a plastic clip to tie and mute either the lower 2 strings of the guitar, when I want to use the higher 4 strings mainly for chords, or the higher 2 strings when I want to use its lower 4 strings for solos! Similar things can be said for the 6-string guitar! The lower 4 strings of the guitar are poor for chords as they support only the shapes A, and E as movable (not open chords) in the fretboard (and possibly Cmajor-shape Cminor-shape that both are requiring 4 frets though!). While the 4 higher strings support all the 3 shapes D,A,E major or minor as movable (not open chords) on the fretboard in a more convenient way for the fingers requiring only 3 frets! This is a bit strange of course, as we are used to play solos on higher strings and chords lower pitch than the solo! This suggests also an alternative tuning to the guitar so as to have the lower 4-strings chord-optimal and the higher 4-strings solo-optimal. Such a tuning would have the 5th and 4 the strings tuned on a major 3rd interval (4 semitones) E.g.

(D3:D2, G3:G2, B3:B2, E4:E3, A3:A3, D4:D4 ) which is without sharps but a bit lower range than the standard tuning. To play this guitar which is more optimal , we prefer to mute either the 2 higher strings (when playing chords) or the 2 lower strings when playing solos, and we do that either with a plastic clip or with the left hand.
The idea that a 12-string guitar is good for solos (double strings produce great solos, while 12-strings are hard to press for chords) , while a 6-string guitar good for melodies, is apparent in the next double guitar back-to-back a 6-string and a 12-string. The player in the next video turns it around to play chords with the 6-string and melodies with the 12-string!
https://www.youtube.com/watch?v=ED2cFK9OEwA

5) 6-STRING GUITAR RE-ENTRANCE TUNING (BARITONE GUITAR) . I have discovered also a re-entrance tuning (similar to what they use in ukuleles) for the 6-string guitar so as to know haw to play such a guitar if you know how to play the 6-string guitar in the standard tuning. The shape of the chords are isomorphic. but it gives a lower sound to the guitar , with simpler and harmonically more perfect chords voicing, without really going lower than the E2 of the guitar of the standard tuning. This re-entrance 6-string guitar tuning is the (A2, D3, G2, C3, E3, A3). The higher 4 strings of this tuning is an octave lower than the tenor ukulele tuning. Of course one has to change all the strings of the guitar to do it properly. I used a guitar A2-string, and an E3-string for the lower two strings, and a E2-string, A2-string, D3-string, and G3-string for the higher 4 strings. This tuning of the guitar is almost exact for the baritone human voice, which is considered to be the G2-G4 . That is why we call this guitar tuning the BARITONE GUITAR.
Obviously if one know how to paly a 6-string guitar with the standard tuning knows also how to play this re-entrance 6-string guitar and vice-versa.

6) 12-STRING GUITAR RE-ENTRANCE TUNING (BARITONE and BRIGHT 12-string GUITAR)
There are two similar e-entrance tuning forthe 12-string guitar too. The most direct is instead of the standard (E3:E2, A3:A2, D4:D3, G4:G3, B3:B3, E4:E4 ), the (E4:E3, A4:A3, D2:D3, G2:G3, B2:B3, E3:E4 ), and the more BRIGHT and higher pitch in the average tuning , the (A3:A2, E4:E3, G3:G2, C3:C4, E3:E4, A3:A4). The higher 4-pairs tuning (G3:G2, C3:C4, E3:E4, A3:A4) I use for a tenor mandola (see post 67).
Obviously if one know how to play a 12-string guitar with the standard tuning knows also how to play this re-entrance 12-string guitars (baritone and bright) and vice-versa.

We may compare the previous tuning with the Greek lute tuning.
The Greek Islands lute tuning which has re-entrance t the lower pair and it is

(C3:C4, G2:G3, D3:D4, A3:A3)

and the Cretan Lute which again has re-entrance at the lower pair and it is

(G3:G3, D3:D3, A3:A3, E3:E3)

Modern technology has created electronic fretboards as e.g. in the Eigenharp, where the right hand ti strike the string is not used anymore, only the left hand that touched the string on the fret.

https://www.youtube.com/watch?v=f9qPf31xYnY

And the Udar

https://www.youtube.com/watch?v=gpUKgRI3knw

Electric guitar without strings
https://www.youtube.com/watch?v=xs6jzjlrBVc

Ancient Greek Kithara?
https://www.youtube.com/watch?v=6adj7Xoo9Us

The next is a chart of the tuning of the various celebrated instruments

http://www.hora.ro/assets/images/tuning.png

See also The Online encyclopedia of instruments tuning

http://tunings.pbworks.com/w/page/22530600/Guitar%20tunings

For the Common Octaves symbolism see
https://en.wikipedia.org/wiki/Octave#/media/File:Common_Octave_Naming_Systems.png

The relation of the frequency f1 , tension T and and string length l is given by the next formula

https://www.youtube.com/watch?v=gbP6h0dtcQw

and μ=m/L is the one-dimensional density of the string (m is the mass of the string), while the numerator of the previous quotient is the standing wave velocity on the string. When we press our fingers of the frets, the numerator is considered constant as both the mass and length change or in other words the density and tension of the string remain constant. So finally the FREQUENCY IS INVERSELY PROPOSITIONAL TO THE LENGTH OF THE STRING. Or F=A/L, where F is the frequency (fundamental) ,L the length of the string and A is a constant depending on the tension and density of the string.

http://hyperphysics.phy-astr.gsu.edu/hbase/waves/string.html

The previous formula gives a formula for the fret width corresponding to a semitone of the 12-notes Bach-chromatic scale. This should be an increase in the frequency by one semitone, which is the 12th root of 2 (as 12 semitones make an octave which is an increase of the frequency by 2) . So the corresponding decrease of the length of the string should be F*(2^(-1/12))=A/(L*x), we solve for x, then x=2^(-1/12). That is at each new fret, the length decreases, from L to L*2^(-1/12). For example if L=66 cm which is standard scale length for a guitar, then 66cm*(2^(-1/12)=62.2957 cm, so the 1st fret would be a width of 66cm-62,2957cm=3.7042 cm .
On the other hand for say a Tambur with scale length L=79cm, the first fret would be 79cm-79cm*(2^(-1/12))=79cm*(1-2^(-1/12)=79cm*0.05612569=4.4339 cm etc In General if the scale length is C, then the 1st fret will have length C*0.05612569. So is calculated any new fret. In this calculation is assumed that the string is parallel to the fretboard, and is not taken in to account the angle which the string has with the fretboard. But for more accurate calculations the latter must be taken in to account too.

We give an example of calculations of the right frets fora tambur which had variable frets,of scale length 79 cm.

FRETBOARD DIMENSIONS	79	SCALE LENGTH IN CM
Number of fret	Lemgth from lower string end till fret	Multipler	Length from upper string end till fret
1	74.5661,	0.056125687	4.4339
2	70.3810	0.109101282	8.6190
3	66.4308	0.159103585	12.5692
4	62.7023	0.206299474	16.2977
5	59.1831	0.250846462	19.8169
6	55.8614	0.292893219	23.1386
7	52.7262	0.332580073	26.2738
8	49.7669	0.370039475	29.2331
9	46.9737	0.405396442	32.0263
10	44.3373	0.438768976	34.6627
11	41.8488	0.470268453	37.1512
12	39.5000	0.5	39.5000
13	37.2830	0.528062844	41.7170
14	35.1905	0.554550641	43.8095
15	33.2154	0.579551792	45.7846
16	31.3512	0.603149737	47.6488
17	29.5916	0.625423231	49.4084
18	27.9307	0.646446609	51.0693
19	26.3631	0.666290036	52.6369
20	24.8834	0.685019738	54.1166
21	23.4868	0.702698221	55.5132
22	22.1686	0.719384488	56.8314
23	20.9244	0.735134226	58.0756
24	19.7500	0.75	59.2500

We give also an example of 1st approximation calculations of the frets for a guitar which has variable frets, of scale length 66 cm.

FRETBOARD DIMENSIONS	66	SCALE LENGTH IN CM
Number of fret	Length from lower string end till fret	Multipler	Length from upper string end till fret
1	62.2957	,0.056125687	3.7043
2	58.7993	0.109101282	7.2007
3	55.4992	0.159103585	10.5008
4	52.3842	0.206299474	13.6158
5	49.4441	0.250846462	16.5559
6	46.6690	0.292893219	19.3310
7	44.0497	0.332580073	21.9503
8	41.5774	0.370039475	24.4226
9	39.2438	0.405396442	26.7562
10	37.0412	0.438768976	28.9588
11	34.9623	0.470268453	31.0377
12	33.0000	0.5	33.0000
13	31.1479	0.528062844	34.8521
14	29.3997	0.554550641	36.6003
15	27.7496	0.579551792	38.2504
16	26.1921	0.603149737	39.8079
17	24.7221	0.625423231	41.2779
18	23.3345	0.646446609	42.6655
19	22.0249	0.666290036	43.9751
20	20.7887	0.685019738	45.2113
21	19.6219	0.702698221	46.3781
22	18.5206	0.719384488	47.4794
23	17.4811	0.735134226	48.5189
24	16.5000	0.75	49.5000

4-FACTORS IN GUITAR OR SIMILAR INSTRUMENTS THAT WILL MAKE THE STRINGS SOUND OUT OF TUNE ON OTHER FRETS EVEN IF TUNED CORRECTLY AS FREE OPEN STRINGS AND EVEN IF THE FRETS ARE IN CORRECT DISTANCES

1) THE ANGLE OF THE STRING TO THE SURFACE OF THE FRETBOARD IS TOO BIG (when we press on a fret). This is usually solved by good design of the instrument by the instrument manufacturer.

2) THE SCALE LENGTH AS MEASURED FROM THE UPPER TO TO THE LOWER END OF THE STRINGS IS NOT THE CORRECT AND DOUBLE THE LENGTH FROM THE UPPER END OF THE STRING TILL THE 12TH FRET. (this applies to instruments with variable lower bridge or end of the string). For variable lower-right bridge this is solved by re-adjusting the position of the lower-right bridge on the surface of the body.

3) THE STRING CHANGES SIGNIFICANTLY ITS TENSION AS WE PRESS ON IT FROM THE UPPER FRETS TILL THE LOWER FRETS (this is because fret distances are calculated with the assumption of constant tension of the string. It appears especially on instruments with pairs of strings, where as open strings the two strings are correctly tuned, but when we press them on a fret they sound out of tune reflexively between them. This is because of one of the strings may have lower tension than the other and when we press it with the finger the lower tension string gets pressed more and increases its tension relative to when it was open.). This is usually solved by the manufacturer of sets of double strings e.g. of 12-string guitars that should chose for the double strings that differ by an octave such width and density so that both the thick and lower and thin and higher strings require the same pressure as the fingers presses them on a fret.

4) THE AMPLITUDE OF VIBRATION OF THE STRING IS SIGNIFICANT DIFFERENT AMONG FRETS, WHICH MAKES THE "EFFECTIVE" LENGTH OF THE VIBRATING STRING, NOT THE GEOMETRIC LENGTH FROM THE FRET TILL THE LOWER BRIDGE. This is one of the reasons that in expensive guitars the lower bridge is at a greater distance from the upper bridge, in the bass strings. Strictly speaking though all frets should be shifted slightly at the lower strings to the right, as the lower bridge is also. This can be solved by careful design of the fretboard by he instrument manufacturer

See also about the Pythagorean tuning

https://en.wikipedia.org/wiki/Pythagorean_tuning

A video about the geometry of consonance
https://www.youtube.com/watch?v=UcIxwrZV10A

A video with amazing resonance patterns of various frequencies

https://www.youtube.com/watch?v=wvJAgrUBF4w

https://www.youtube.com/watch?v=1yaqUI4b974

https://www.youtube.com/watch?v=CGiiSlMFFlI

Differences between flamenco and a classical guitar

https://www.youtube.com/watch?v=_dtPZ4sarG8

HOW TO KNOT NYLON STRINGS TO INSTRUMENTS DESIGNED FOR METAL STRINGS WITH A LOOP AT THE END.

We only need of course to know how to knot with a nylon string a knot at one of its end so as to create a loop and hang the string. One of the many appropriate knots is the loop-knot as seen in the next video

https://www.youtube.com/watch?v=FUMgzaOMylk
also
https://www.youtube.com/watch?v=Cierg3aoB5Y

NOTES FREQUENCIES

("Middle C" is C₄ )

Note	Frequency (Hz)	Wavelength (cm)
C₀	16.35	2109.89
C^#₀/D^b₀	17.32	1991.47
D₀	18.35	1879.69
D^#₀/E^b₀	19.45	1774.20
E₀	20.60	1674.62
F₀	21.83	1580.63
F^#₀/G^b₀	23.12	1491.91
G₀	24.50	1408.18
G^#₀/A^b₀	25.96	1329.14
A₀	27.50	1254.55
A^#₀/B^b₀	29.14	1184.13
B₀	30.87	1117.67
C₁	32.70	1054.94
C^#₁/D^b₁	34.65	995.73
D₁	36.71	939.85
D^#₁/E^b₁	38.89	887.10
E₁	41.20	837.31
F₁	43.65	790.31
F^#₁/G^b₁	46.25	745.96
G₁	49.00	704.09
G^#₁/A^b₁	51.91	664.57
A₁	55.00	627.27
A^#₁/B^b₁	58.27	592.07
B₁	61.74	558.84
C₂	65.41	527.47
C^#₂/D^b₂	69.30	497.87
D₂	73.42	469.92
D^#₂/E^b₂	77.78	443.55
E₂	82.41	418.65
F₂	87.31	395.16
F^#₂/G^b₂	92.50	372.98
G₂	98.00	352.04
G^#₂/A^b₂	103.83	332.29
A₂	110.00	313.64
A^#₂/B^b₂	116.54	296.03
B₂	123.47	279.42
C₃	130.81	263.74
C^#₃/D^b₃	138.59	248.93
D₃	146.83	234.96
D^#₃/E^b₃	155.56	221.77
E₃	164.81	209.33
F₃	174.61	197.58
F^#₃/G^b₃	185.00	186.49
G₃	196.00	176.02
G^#₃/A^b₃	207.65	166.14
A₃	220.00	156.82
A^#₃/B^b₃	233.08	148.02
B₃	246.94	139.71
C₄	261.63	131.87
C^#₄/D^b₄	277.18	124.47
D₄	293.66	117.48
D^#₄/E^b₄	311.13	110.89
E₄	329.63	104.66
F₄	349.23	98.79
F^#₄/G^b₄	369.99	93.24
G₄	392.00	88.01
G^#₄/A^b₄	415.30	83.07
A₄	440.00	78.41
A^#₄/B^b₄	466.16	74.01
B₄	493.88	69.85
C₅	523.25	65.93
C^#₅/D^b₅	554.37	62.23
D₅	587.33	58.74
D^#₅/E^b₅	622.25	55.44
E₅	659.25	52.33
F₅	698.46	49.39
F^#₅/G^b₅	739.99	46.62
G₅	783.99	44.01
G^#₅/A^b₅	830.61	41.54
A₅	880.00	39.20
A^#₅/B^b₅	932.33	37.00
B₅	987.77	34.93
C₆	1046.50	32.97
C^#₆/D^b₆	1108.73	31.12
D₆	1174.66	29.37
D^#₆/E^b₆	1244.51	27.72
E₆	1318.51	26.17
F₆	1396.91	24.70
F^#₆/G^b₆	1479.98	23.31
G₆	1567.98	22.00
G^#₆/A^b₆	1661.22	20.77
A₆	1760.00	19.60
A^#₆/B^b₆	1864.66	18.50
B₆	1975.53	17.46
C₇	2093.00	16.48
C^#₇/D^b₇	2217.46	15.56
D₇	2349.32	14.69
D^#₇/E^b₇	2489.02	13.86
E₇	2637.02	13.08
F₇	2793.83	12.35
F^#₇/G^b₇	2959.96	11.66
G₇	3135.96	11.00
G^#₇/A^b₇	3322.44	10.38
A₇	3520.00	9.80
A^#₇/B^b₇	3729.31	9.25
B₇	3951.07	8.73
C₈	4186.01	8.24
C^#₈/D^b₈	4434.92	7.78
D₈	4698.63	7.34
D^#₈/E^b₈	4978.03	6.93
E₈	5274.04	6.54
F₈	5587.65	6.17
F^#₈/G^b₈	5919.91	5.83
G₈	6271.93	5.50
G^#₈/A^b₈	6644.88	5.19
A₈	7040.00	4.90
A^#₈/B^b₈	7458.62	4.63
B₈	7902.13	4.37

METHODS TO LEARN MUSICAL FREE IMPROVISATION-COMPOSITION WITH NEW PERCEPTIONS

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Sunday, January 17, 2016

1. Guitar tuning. In what is the standard tuning optimal if at all ? Melodic optimal and chord optimal tunings. Alternative tunings and comparison with violin, mandolin, and other instruments

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