John O'Sullivan                     info@johnsmusic7.com 
 


Music

YouTube Eagle 53 playlist (all songs below) click here

3-4-5 major chord progression in Eagle 53 click here

4-5-6-7-8 chord progression in Eagle 53 click here

Failure by The La's in Eagle 53 click here

Here Comes Your Man by The Pixies in Eagle 53 click here

Stone Free by The Jimi Hendrix Experience in Eagle 53 click here

Salsa y Ketchup by Zander Schloss in Eagle 53 click here

Blues in C in Eagle 53 click here

Misirlou in Eagle 53 click here

Sympathy for the Devil by The Rolling Stones
in Eagle 53 click here

Hero by Al Jourgensen and Ministry from the album
Psalm 69 in Eagle 53 click here

Rock and Roll by Lou Reed and
The Velvet Underground in Eagle 53 click here

Guess I'm Falling in Love by Lou Reed and
The Velvet Underground in Eagle 53 click here

12 Bar Blues in Eagle 53 click here

When the **** Hits the Fan by
The Circle Jerks in Eagle 53 click here

Get it on by Marc Bolan and T Rex
in Eagle 53 click here

His Latest Flame by Elvis Presley
in Eagle 53 click here

Girls and Boys by Blur in Eagle 53 click here

White Light / White Heat by Lou Reed and
The Velvet Underground in Eagle 53 click here

E F G A B C D#! E progression in Eagle 53 click here

Hear My Train a Comin' by Jimi Hendrix
in Eagle 53 click here   



 
David's chord...

My favourite chord is a justly tuned 1:2:3:4:5:7 chord.

These numbers show how the frequencies of the notes relate to each other. I derived it
from a 1:2:3:4:5:6:7:8:9 chord. For me any interval narrower than 7:8 has "intolerable"
beating (dissonance). 4's first overtone (8) "beats" against 9's fundamental (9).
5's first overtone (10) beats against 9's fundamental (9). And 8:9 is clearly
dissonant so the 9 has to go. A guitar has only six strings so I could omit
1 and 8 to get 2:3:4:5:6:7. However, intervals narrower than 4:5 have
"tolerable" but very noticeable "beating" (dissonance). 5:6 and 6:7
and 7:8 sound unpleasant (dissonant) to my ears.
If I omit the 6 from the 2:3:4:5:6:7 chord I eliminate the
dissonant intervals. And I could add a 1 at the
bottom to get a "beatless" chord...

1:2:3:4:5:7

Intervals in chords and scales in 12 Tone Equal Temperament (the standard
Western tuning) involving the number 7 are so out of tune as to be
unusable. If the 1:2:3:4:5:7 chord is on E then the notes are...

  E E B E G# D

This chord has four "pitch classes" (note names), E B G# and D.
Major and minor chords have only three "pitch classes".

I'm naming the 1:2:3:4:5:7 "just" chord "David's chord"
after these lyrics in Leonard Cohen's "Hallelujah"...

I've heard there was a secret chord
that David played and it pleased the Lord
but you don't really care for music, do ya?

4 is the root note in the 1:2:3:4:5:7 chord.

Why 4?

Because 4 "keeps the beat"...

1 2 3 4, 1 2 3 4, 1 2 3 4, 1 2 3 4

Minor chords (10:12:15:20:24:30) are "illegal" in my system.
They are "destructive". They sound weaker than a single
note played on its own. They are also rootless
(no 4 to keep time).

Where a minor chord usually occurs I replace it with a 2:3:4 "power chord".
On E that is E, B, E. The high E (4) is the root note, not 2.
On a guitar a minor chord looks like 10:15:20:24:30:40.
If you just use the lowest three notes (10:15:20) you
get a 2:3:4 power chord. 2:3:4 is 10:15:20 divided by 5.
Minor chords are *banned* in my system.

 
Jubal's scale...

My favourite just intonation scale is the 8 to 16 Harmonic Series segment 8:9:10:11:12:13:14:15:16

It can be used for melodic purposes and for chord progressions where the root notes of the
chords (which *must* be rooted, contain a "4") correspond to the the notes, or
a subset of the notes in the scale above.

For me the strongest chord progression is 3:4:5 which is 9:12:15
divided by 3. The 4 in 3:4:5 is the root chord. On a guitar it looks like...

open E major, open A major and barre C# major.

If you play guitar strum...

A A, C# C#, E E, A A, A A, C# C#, E E, A - ,

A A, C# C#, E E, A A, A A, C# C#, E E, A - .

repeat and "stress" the single A chord each time.

The A major chord is *clearly* the "chord of resolution".

The 4:5:6:7:8 chord progression is very strong as well.

4:5:6:7:8 is 8:10:12:14:16 divided by 2.

Both 4 and 8 work as "chords of resolution".

Using all nine notes in the 8:9:10:11:12:13:14:15:16 scale for chord
progressions is "legal", but for me is too complicated.
I recommend two pentatonic scales (five notes, or six if you count the octave, 16).
Omit 11 and 13 as they do not pair well with the other notes, they have no common
factors and cannot be "reduced". E.g. 9, 12 and 15 have a common factor (3) and can be
"reduced" to 3, 4 and 5. The smaller the numbers in an interval the stronger it is.
8, 10, 12, 14 and 16 have a common factor of 2 (4:5:6:7:8).

8:9:10:12:14:16 and 8:9:10:12:15:16 are two nice pentatonic scales/progressions.

Both 8 and 16 work as "chords of resolution".

I'm naming the "8 to 16 Harmonic Series segment scale"
"Jubals's scale" after this verse in Genesis 4:21...

[Jabal's] brother's name was Jubal,
he was the father of all those who play the harp and flute.

Below is a fretboard for guitars, banjo's, mandolins, ukulele's and similar fretted and
stringed instruments where "Jubal's scale" can be played on each string. Note that if
the lowest string is tuned to E 82.5 Hz and the next string is tuned to B 123.75 Hz we
have 8, 9, 10, 11, 12, 13, 14, 15, 16 on E but on the B string we have the numbers
multiplied by 1.5 (3/2)... 12, 13.5, 15, 16.5, 18, 19.5, 21, 22.5, 24 which introduces
a lot of notes that are not octaves above or below 8:9:10:11:12:13:14:15:16.
If a guitar is tuned to E B E G# D F# (David's chord, 2:3:4:5:7:9) on the
B, G#, D and F strings we get *dozens* more "illegal" notes. So if you want to play a
melody you have to use the second E string "only" (4 in the 2:3:4:5:6:7 chord, 165 Hz)
to line up with the chord progression. With 12 Tone Equal Temperament *all* the notes
are octaves above or below or within  a 12 note set. In the diagram below there is only
one fret above the octave (8th) fret which corresponds to 17.0333 (explained below).

A six string guitar should be tuned (frequencies of "open" strings)...

E - 82.5 Hz (2)       
B - 123.75 Hz (3)   
E - 165 Hz (4)        
           G# - 206.25 Hz (5)           
        D - 288.75 Hz (7)           

5 strings, no string at the top, a 2:3:4:5:7 chord.

Strings need to be manufactured to accommodate this tuning because
the "standard" A and D strings have to be tuned two semitones
(approximately) higher and they might "break".  The standard B string
needs to be tuned 269 cents (nearly 3 semitones) higher and that might
break as well. In the meantime I'm going to tune my Jubal's scale guitar
(which is currently being made) so that all the 5 frequencies above are
tuned 203.91 cents (9/8) lower so that the strings don't break. I will
tune the strings to the frequencies above multiplied by 0.88889 (8/9).
I would love to have a 41.25 Hz E bass string at the bottom to make a
1:2:3:4:5:7 chord but this may not be feasible.

The idea is that "David's chord" (2:3:4:5:7) can be played on every note in
"Jubal's scale" by placing your index finger just behind a fret, or playing "open" and strum.
Mandolins, banjo's and ukulele's (4 strings or 4 pairs of strings tuned to unisons)
should be tuned B E G# D (3:4:5:7) and the notes on the E string (4) are the root
notes of the chords. Here is the diagram of the fret arrangement. The nut and
the dotted frets make a 4:5:6:7:8 chord progression.

The first fret (9), fourth fret (12) and seventh fret (15) makes a 3:4:5 progression.



Distances from the nut to each fret are...
0.1111z, 0.2z, 0.2727z, 0.3333z, 0.3846z, 0.4286z, 0.4667z, 0.5z, 0.5303z
where 'z' is the "scale length" of the instrument, the distance
from the nut to the saddle on the lightest string.

Where did 17.0333 in the diagram above come from?
My formula for calculating the "strength" of any just interval n/d is

2/n + 2/d

15:16 is the narrowest "just" interval (apart from 1:1) that has a strength value >= 0.25
which is my "threshold" for good melodic intervals. For intervals used in building
chords (harmony) the threshold is 0.5.  15:16 has a strength value of 0.2583 which
is > 0.25 so the interval is good (in melodies).  16:17 has a strength value of 0.2426
which is < 0.25 so 16:17 is an "illegal" just interval. However I allow a "tempering"
of "good" just intervals by not more than plus or minus 6.77587576937045 cents
which is 256/255 (or 255:256) exactly. If I "temper" 15:16 narrower by
6.77587576937045 cents I get a "just" 16:17 *exactly*. So 17 is "in".
16/15 divided by 17/16 is 256/255.
16 x 16 = 256
15 x 17 = 255
Because 16:17 (really a "tempered" 15:16) is right on the "cusp" (6.77587576937045 cents
away from 15:16) I "tempered" 17 to 17.0333 so that 16:17.03333 is only 3.3879 cents
narrower than a just 15:16. 3.3879 cents is approximately 6.77587576937045 cents
(256/255 or 255:256) divided by 2. So when 17.0333 is paired with 16, 15, 14, 13,
12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, and 1 it makes intervals that are within
3.38794 cents of a "good" just interval (strength >= 0.25).
The 17.0333 is acceptable in melodies but not for chord progressions.

-

Unfortunately you can't play a "just" David's chord on Jubal's scale on a retuned
piano or electronic keyboard. You would need too many keys for practical playing.


Eagle 53 v1 Musical Tuning
 
I worked out the Eagle 53 v1 tuning on the 22nd June, 2016.
It was designed to have major chords on E, A and B
(the I, IV, V progression) on guitar (I play guitar).
By coincidence major chords also occurred on C, F and G
(the I, IV, V progression again),
with the bonus that C7 (2:3:4:5:7) and F7 (2:3:4:5:7) occur as well.

  One of my favourite chord progressions on my Eagle 53 v1 guitar is...

C   C   C   C   C7  C7  C7  C7
F   F   F   F   F7  F7  F7  F7
C   C   A!  A!  F   F   G   G 
C   C   F   F   C   C   G   G 

I use an A! 2:3:4 "power chord" where an A minor chord would
normally be used. As I said above, minor chords
are *banned* in my system.
The "7" chords on C and F look like 2:3:4:5:7.
All the chords (except the open A!) must be "A-shaped" barre chords,
i.e the lowest notes in the chords are on the A string. You can't play a
2:3:4:5:7 chord using an open C and E-shaped barre chords on F and G.

Eagle 53 v1 chords...
Major chords (2:3:4:5:6:8) are all within 1.41 cents of just.
The "7" chords (2:3:4:5:7) are all within 6.2 cents of just.
The 2:3:4 power chords are within 0.07 cents of just.
Again, 4 is the "root note", not 2.

What are cents? A "standard" semitone is 100 cents wide.
An octave (1:2) is 1,200 cents wide.
"Just" means "perfectly in tune".

In 12 Tone Equal Temperament 4:5 and 5:6 are out of tune
by more than 13.6 cents. 2:7, 3:7, 4:7, 6:7 7:8 and 7:9
are out of tune by more than 31 cents.
In Eagle 53 v1 these are all within 6.2 cents
of "just" (where they occur) and 5:7 is also
within 6.2 cents of "just".
I won't use any interval that is
out of tune by more than 6.7758 cents.
6.7758 is 255:256 in cents.
256 is 4 to the power of 4.

Eagle 53 v1 is very close to this "just intonation" tuning...

1/1, 16/15, 9/8, 6/5, 5/4, 4/3, 7/5, 3/2, 8/5, 5/3, 9/5, 15/8, 2/1

It is not new, I found it in a Hypercard stack called
"Just Major with a 7 limit tritone" some time after
I worked it out myself.
It was possibly first created by Robert Rich.

1/1 (0.0 cents) is the key of E
(165 Hz or octaves above and below this)

Eagle 53 v1 is the just tuning above "quantized" to 53 Tone
Equal Temperament. So it is a 12 note "subset" of 53TET.
Why did I choose 53TET?
Because fourths and fifths (3:4 and 2:3) are within 0.07 cents of just.
Major and minor thirds (4:5 and 5:6) are within 1.41 cents of just.
2:7, 3:7, 4:7, 5:7, 6:7, 7:8 and 7:9 are within 6.2 cents of just.

Eagle 53 v1 in cents...

0.0, 113.2075, 203.7736, 316.9811, 384.9057, 498.1132, 588.6792,
701.8868, 815.0943, 883.0189, 1018.8679, 1086.7925, 1200.0

For the last five months I tried different variations of Eagle v1.
I transposed it to another key and changed a few notes.
I had v2, v3, v4, v5 and v1.1 (v1.1 was simply a transposition of v1).
In the end I returned to v1 because it retains the E A B,
C F G major chord I, IV, V progressions and I discovered only
a few days ago that the 4:5:6:7:8 chord progression was
"hidden" in there as well.

The 4:5:6:7:8 chord progression occurs on F in Eagle 53 v1
within 6.2 cents of just...

  F major, A major, C major, D#! (2:3:4), F major.

The D# note in Eagle 53 v1 is 973.585 cents above the F note.
This is within 4.76 cents of a "just" 4:7 (968.826 cents).
In 12 Tone Equal Temperament 4:7 is 1,000 cents,
out of tune by 31.17 cents. So the 4:5:6:7:8 progression
is "impossible" in 12 TET, there's no "7" available.

The 3:4:5 progression occurs on A(4).
open E major, open A major and C#! (2:3:4 power chord)
A major is the root chord.

I "verified" Eagle 53 v1 as my favourite 12 notes
per octave tuning on the 20th June, 2026.

From now on I will revert to its original name... Eagle 53

----

Eagle 53 chords... click here

Eagle 53 chord progressions and scales... click here

Eagle 53 "map"... click here

Here is a diagram of a guitar fretboard fretted for Eagle 53...



The photo below is of my guitar fretted for Eagle 53

Specifications for fret placements for guitars, bass guitars, mandolins,
banjos, ukuleles and similar fretted lutes  click here

Frequencies for piano tuners click here

Fretted violins, violas, cellos and double basses click here

Eagle 53 scl file for retunable electronic midi keyboards click here
On a PC right-click the link to download the file.
On a Mac press ctrl (control) and click the link to download the file.

I'm setting E = 165 Hz as the standard, not A = 440 Hz 

----
 
 Click here to hear Atonement to a Centaur by Chris Vaisvil
in Blue Just Intonation which is similar to Eagle JI v1
 
Click here to launch an app that plays seven strong and
beatless chords in Eagle 53 v1 whose root notes
correspond to 3:4:5:6:7:8:9
 
Click here for a 10 strong chords Eagle 53 v1 app

Click here for a brief article on how I arrived at
my Eagle 53 v1 musical tuning

Click here for Eagle 53 v1 demo tune Uccellini

Click here for Eagle 53 v1 demo tune Bach

Click here to launch an app that demonstrates what can
be done in 19 Equal Divisions of the Octave.

Click here to launch an app that demonstrates the
Hijaz Kar (Arabian) scale in Eagle 53 v1.

Click here to hear a short interlude for clavichord in Eagle 53 v1
by Chris Vaisvil.

Click here to hear John's Tune in Blue JI which is similar
to Eagle JI v1.

My Eagle 53 Tuning group on Facebook...
www.facebook.com/groups/eagletuning


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Here is my "Ultimate Tuning" book. The book describes how
I arrived at my Eagle 53 tuning, the rationale and
the mathematics behind it (for dummies, no prior
knowledge of music theory required). It's also
a gentle introduction to tuning theory.

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Here is my book which describes my "older" rules for musical
composition. The book lists 2196 intervals which I used as
a "map" when verifying Eagle 53.
 
Click here for a free PDF of the book.



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My short take on Just Intonation. Click here to download it.





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