Notes on Scales and Tones and Modes and Keys by Fred Bisshopp

One might say C-Major is the mother of all (western) scales and keys, so there we begin.

Table 1: Names of things and names of names and more names and some adjectives
↓	for all major scales | names | degrees | modes snd scales | keys, triads and arpeggios
	Solfeggio			Degree	a.k.a.	Mode	a.k.a. (scale)	Tonic	Others
C	do -- re -- mi -- fa -- sol -- la -- ti -- do	-- di -- ri -- -- fi -- si -- li -- --	-- ra -- me -- -- se -- le -- te -- --	I	tonic	Ionian	Major	C-Maj	C6 Cmaj7
D				II	supertonic	Dorian	Minor	D-min	Dm6 Dm7
E				III	mediant	Phrygian	--	E-min	Em7
F				IV	subdominant	Lydian	--	F-Maj	F6 Fmaj7
G				V	dominant	Mixolydian	Dominant	G-Maj	G6 G7
A				VI	submediant	Aeolian	N-minor	A-min	Am7
B				VII	subtonic	Locrian	H-diminished	B-dim	Bm7-5
C				Notes on the columns
↓		Among diagrams that contain 'o' for tones that are 'played' and '|' for tones that are not, every one of the twelve major scales is represented by the pattern, o|o|oo|o|o|oo . For the C-major scale, the '|'s are the black keys, and these are the names of the white keys.
Sol-fa		The first column has singable names of the tones in any major scale -- the 'o's in the diagram above. The second and third columns have some of the names of the '|'s, first as sharps, then as flats.
Degree		This is another way to name the 'o's in order of increasing tones. For example, IV-chords, V-chords and V7-chords are important to major- and minor-key harmony. Arabic numerals will also be used.
a.k.a		These are other references to the 'o's, sometimes nouns and sometimes adjectives; 'tonic', 'subdominant' and 'dominant' are the most common ones.
Mode		These are names of scales, taken from the names of some of the Church modes (Dark ages --- Renaissance). The mode starts with the note in the first column and runs through notes in C-major; e.g. Aeolian: a|oo|o|oo|o|a. Sometimes there is a change of key, as in A-Dorian, which uses the notes of G-Major, beginning with A, in the order, a|oo|o|o|↑o|a . (Note the use of '↑' for the f-sharp.)
(scale)		N-minor is shorthand for Natural minor (a.k.a. Ethiopian) and H-diminished, for Half-diminished. The major and minor keys are usually associated with the Ionian and Aeolian modes, but Mixolydian (major), Dorian (minor) and Phrygian (minor) are worthy of note.
Tonic		The tonic key and triad are the first, third, and fifth notes of the mode. Some examples are c|-|o-|o (Major), a|-o|-|o (minor), b-|o|-o (diminished) and o|-|o-||↑ (augmented -- oops, not in any mode).
Others		These are playable chords and arpeggios. Augmented chords, two of the minor sixths, and diminished sevenths cannot be played with the tones in a major scale. (Dm6-5 is Ddim7, but '-5' is A-flat in the Dorian mode.) Much more about the names of chords will come later.

In the realm of Browser Science certain symbols, like ♣, ♦, ♥ and ♠, are relatively easy to use in html-documents. Most irritatingly, however, they don't always work, and the lists of convenient ones do not contain symbols for 'sharp', 'flat' or 'natural'. So it will be convenient to use '↑', '↓' and '-' or 'o' for those symbols, bearing in mind that the first two may not always work either. When they work it's easy to live with since ↑ and ↓ also suggest 'augmented' and 'diminished', i.e. 'sharp' and 'flat'. As has already been introduced above, '↑', '↓', 'o' or '-', and '|' will also be used to indicate 'sharp', 'flat', 'natural', and 'not played' in diagrams of scales, but sometimes, when it is only the intervals that matter, '↑', '↓' and 'o' or '-' will all be represented as 'o' or '-'. In diagrams of intervals, triads and some chords, the 'o's are played, the '-'s are there, but not necessarily played, and the '|'s are there too, but not played.

Let it be noted that the word, tone, is used in several different ways; the first, already introduced, is to refer to a specific pitch that can, or should be sung or played by an instrument. At a given pitch, different voices, different instruments or different registers of instruments (including voices) will produce sounds that vary in tone or timbre, according to the presence or absence of various overtones or harmonics. Finally, in any twelve-tone scale, two tones (pitch) that are separated by exactly one tone are said to be separated by an interval of one tone. With diagrams, o|o is a (playable) tone (interval), oo is a semitone, o|-|o is a bitone, and either of o|-|-|o or o-|-|-o is the famous tritone.

The other common way to describe intervals is by reference to the position of a tone in a scale, and the conventional choices are major scale (Ionian mode) or dominant scale (Mixolydian mode). The only difference is the seventh tone, which is diminished by a semitone in the dominant scale. The minor scales (Dorian, Phrygian, Aeolian modes) also have a seventh tone that is two semitones below the octave, as indicated in the following table.

Table 2: Common intervals (major scale)
interval pattern sol-fa second o|o do-re, fa-sol third o|-|o do-mi, fa-la minor third o|-↓ re-fa, mi-sol, la-do fourth o|-|-o do-fa, sol-do fifth o|-|--|o do-sol, fa-do	interval pattern sol-fa sixth o|-|--|-|o do-la, re-ti (minor) seventh o|-|--|-|-↓ re-do major seventh o|-|--|-|-|o do-ti octave o|-|--|-|-|-o do-do tenth o|-|--|-|-|--|-|o do-(do)-mi
Note: Only some truly useful examples of sol-fa intervals have been included.

Note that dominant and minor sevenths are often referred to simply as sevenths, especially in the naming of chords (apeggios too), where C7 and Cm7 mean 'play the b↓' and Cmaj7 means 'play the b'.

Some other intervals and different names of intervals are formed by adding the qualifiers, diminished or augmented, to signify decreased or increased by a semitone. Thus the tritone is a diminished fifth and an augmented fourth, the minor third is a diminished third and an augmented second; the major seventh is an augmented seventh, a diminished seventh is a sixth, the semitone is a diminished second, and augmented chords have an augmented fifth. And, just for fun, the minor third is also a sesquitone.

It is pretty well known (see refs) that sol-fa, once learned, has little to do with specific tones -- 'do' can be sung as any of the twelve tones, and with no conscious effort, one can then sing the 'o's from 'do' to 'do' and back to produce a major scale in any of the twelve keys. What is perhaps less well known is that with minimal effort, one can also learn to sing 're' to 're' and the other five modes as well. So it isn't twelve scales we can sing without names for sharps or flats, it's eighty-four.

In any classification of scales that may or may not be modes of major scales it is far simpler to do them eighty-four at a time than to consider a myriad of special cases. To make identification of 'new' scales easier, patterns will be shown as pairs of elements, with the pair, |o|o| and oo|o|oo, for any mode of any major scale. To make it easier yet, any '↑' or '↓' that appears in two octave patterns of modified C-major scales will be replaced by an 'o' after its first occurrence.

Table 3: Elements of once-modified C-major scale patterns. (Unmodified: |o|o| and oo|o|oo.)
tone pattern elements f↑ o|o|o|↑o|o|oo|o|o|oo|o|oo |o|o| and oo|o|oo c↑ ↑o|oo|o|o|o|oo|oo|o|o|o|o |o|o|o| and oo|oo d↑ o||↑oo|o|o|oo||ooo|o|o|oo |o|o| and oo||ooo g↑ o|o|oo||↑o|oo|o|oo||oo|oo |oo|o| and oo||oo a↑ o|o|oo|o||↑oo|o|oo|o||ooo |o|oo| and o||ooo	tone pattern elements b↓ o|o|oo|o|o↓|o|o|oo|o|oo|o |o|o| and oo|o|oo e↓ o|o↓|o|o|o|oo|oo|o|o|o|oo |o|o|o| and oo|oo d↓ o↓||oo|o|o|ooo||oo|o|o|oo |o|o| and ooo||oo a↓ o|o|oo|o↓||oo|o|oo|oo||oo |o|oo| and oo||oo g↓ o|o|oo↓||o|oo|o|ooo||o|oo |oo|o| and ooo||o
By inspection, the f↑- and b↓-elements are not 'new', and the c↑- and e↓-elements are the same. Thus there are exactly seven once-removed (from major) scales, with one coincidence (c↑ or e↓).

There are four cases where the choice of one sharp or flat makes it necessary to modify a second tone. They are: c↓ → b↓, f↓ → e↓, b↑ → c↑ and e↑ → f↑. If the second modification is not made, the result is a six-tone scale.

Table 4: The first modification demands a second one.
tones	pattern	elements	mode-name	a.k.a.
c↓ b↓	↓||o|oo|o|o↓o||o|oo|o|ooo	|oo|o| and ooo||o	Ionian -1-7	Mixolydian -1 (g↓)
f↓ e↓	o|o↓↓||o|o|oo|ooo||o|o|oo	|o|oo| and ooo||o	Ionian -3-4	-
b↑ c↑	↑o|oo|o|o||↑oo|oo|o|o||oo	|oo|o| and o||ooo	Ionian +1+7	-
e↑ f↑	o|o||↑↑o|o|oo|o||ooo|o|oo	|o|oo| and o||ooo	Ionian +3+4	Lydian +3 (a↑)
The two scales that are only once removed are identified in the last column.

The convention for assignment of mode-names is that they always refer to modes of the C-major scale, as it was in the 15th century. The modified tones in the patterns in Tables 3 and 4 take place at the indicated position(s) in the Ionian mode, and sharps and flats are indicated by '+' and '-'. That may not be the best way to learn a new scale, so, just in case, the tables also tell which tones are modified and how.

Before looking at other twice-modified scales, note that a pair of modifications that contains f↑ or b↓ is one modification of a G- or F-major scale, and the elements for those scales are unchanged by the change of key, so there will be no new pairs of elements. Again, each distinct pair of elements is common to eighty-four members of a family of patterns. So here are elements for some of the twice-modified scales:

Table 5: Elements of twice-removed scales.
tones pattern elements c↑ d↑ ↑|↑oo|o|o|o|o|ooo|o|o|o|o |o|o|o|o| and o-o d↑ g↑ o||↑oo||↑o|oo||ooo||oo|oo ||ooo|| and oo|oo d↑ a↑ o||↑oo|o||↑oo||ooo|o||ooo ||ooo|| and ooo|o c↑ a↑ ↑o|oo|o||↑o|oo|oo|o||oo|o |oo|oo| and o||oo g↑ a↑ o|o|oo||↑|↑oo|o|oo||o|ooo |ooo|o| and oo||o	note pattern elements d↓ e↓ o↓|↓|o|o|o|ooo|o|o|o|o|oo |o|o|o|o| and o-o d↓ a↓ o↓||oo|o↓||ooo||oo|oo||oo ||ooo|| and oo|oo d↓ g↓ o↓||oo↓||o|ooo||ooo||o|oo ||ooo|| and o|ooo e↓ g↓ o|o↓|o↓||o|oo|oo|oo||o|oo |oo|oo| and oo||o g↓ a↓ o|o|oo↓|↓||oo|o|ooo|o||oo |o|ooo| and o||oo
Not included are c↑g↑ (= a↓), e↓a↓ (= g↑), and sharp-flat pairs --- all discussed in an appendix.

And now we get to go looking for their names:

Table 6: Some known names of once-removed scales (incomplete list)
tones	pattern	mode-name	other names
e↓ or c↑	c | o ↓ | o | o | o | o c d | o o | o | o | o | ↑ d	Ionian -3 Dorian +7	Melodic Minor (ascending), Jazz Minor, Hawaiian
	g | o | o o | o ↓ | o | g a | o | ↑ o | o o | o | a	Mixolydian -6 Aeolian +3	Melodic Major, Hindu, Spanish
	f | o | o | o o | o ↓ | f g | o | o | ↑ o | o o | g	Lydian -7 Mixolydian +4	Lydian Dominant, Overtone, Bartok
	↓ | f | o | o | o o | o ↓ f | o | o | o | ↑ o | o f	Phrygian -1 Lydian +5	Lydian Augmented
	b o | o ↓ | o | o | o | b ↑ d | o o | o | o | o | ↑	Locrian -4 Ionian +1	Super Locrian
	d ↓ | o | o | o | o o | d e o | o | o | o | ↑ o | e	Dorian -2 Phrygian +6	Javanese
a↓	c | o | o o | o ↓ | | o c	Ionian -6	Harmonic Major
g↑	a | o o | o | o o | | ↑ a	Aeolian +7	Harmonic Minor, Mohammedan
	d | o o | | ↑ o | o o | d	Dorian +4	Ukranian Minor, Mesheberakh, Romanian Minor, Lydian Minor
	e o | | ↑ o | o o | o | e	Phrygian +3	Spanish Gypsy, Phrygian Dominant, Freygish, Hijaz, Israeli, Jewish, Major Phrygian, (Spanish)
d↑	a | o o | | ↑ o o | o | a	Aeolian +4	Minor Gypsy
	e o | o | o | o o | | ↑ e	Phrygian +7	Neapolitan Minor
d↓	c ↓ | | o o | o | o | o c	Ionian -2	Semi-Persian
So far, no other names for modes of Ionian -5 (g↓) or +6 (a↑).

The klezmer scale, Adonoi Molokh, is just the dominant scale (Mixolydian or Ionian -7); it is unusual because of the way it is used, harmonically speaking.

Table 7: Some known names of twice-removed scales (incomplete list)
tones	pattern	mode-name	other names
d↓e↓ or c↑d↑	c ↓ | ↓ | o | o | o | o c e o | o | o | o | ↑ | ↑ e	Ionian -2-3 Phrygian +6+7	Neapolitan
	g | o | o o ↓ | ↓ | o | g b | ↑ | ↑ o o | o | o | b	Mixolydian -5-6 Loccrian +2+3	Arabian, Major Locrian
	f | o | o | o o ↓ | ↓ | f a | o | ↑ | ↑ o o | o | a	Lydian -6-7 Aeolian +3+4	Lydian Minor (major)
d↓a↓ or d↑g↑	c ↓ | | o o | o ↓ | | o c e o | | ↑ o | o o | | ↑ e	Ionian -2-6 Phrygian +3+7	Double Harmonic Major, Arabic, Bysantine, Gypsy (Major)
	g ↓ | | o o ↓ | | o o | g b o | | ↑ o o | | ↑ o | b	Mixolydian -2-5 Locrian +3+6	Oriental
	f | o ↓ | | o o ↓ | | o f a | o o | | ↑ o o | | ↑ a	Lydian -3-6 Aeolian +4+7	Hungarian Gypsy, Hungarian Minor, Double Harmonic Minor, Gypsy Minor, Algerian (1 to 8 of 11 tones)
g↓e↓	f ↓ | | o | o o | o ↓ | f	Lydian -2-7	Romanian Major
c↑a↑	g | | ↑ o | ↑ o | o o | g	Mixolydian +2+4	Hungarian (Major)
d↑a↑	b o | | ↑ o o | o | | ↑ b	Locrian +3+7	Persian
There are bound to be lots of others.

There is an unusual scale that isn't here yet. The Enigmatic Scale appears in print every now and then (see reference below), and it was Verdi who took on the challenge to harmonize it and did so in his Ave Maria (1898). As a C-scale it is c↓||o|↑|↑|↑oc (ascending), and it looks like it could be as many as four-times removed from C-major. In fact it is three-times removed, and the pattern that shows it is f↓||o|o|↑|↑of, for Lydian -2+5+6 (g↓ c↑ d↑).

Some of the internet sites that were consulted to find these names are:
http://www.dolmetsch.com/pianochords.htm (huge source!)
http://docs.solfege.org:81/3.15/C/scales/modes.html
www.medianmusic.com/ScaleForeign.html
en.wikipedia.org/wiki/Category:Musical_scales
everything2.com/user/chromaticblue/writeups/Musical+scales
incolor.inebraska.com/tgannon/txts/scales.txt
http://www.standingstones.com/theorcnr.html
en.wikipedia.org/wiki/Enigmatic_scale Not all sources are in perfect agreement about the names, but the differences are remarkably slight, so arriving at some sort of consensus was relatively easy. Three of the offending names are: 'Gypsy', 'Hungarian' and 'Spanish'. For the first two the fix is to follow the name with an optional 'Major', for those that have major thirds, and a required 'Minor', for those with minor thirds. But take care: 'Minor Gypsy' and 'Gypsy Minor' are different scales. 'Spanish' appears twice, both times with major thirds, so the suggestion is that it should not be used for Phrygian +3 (g↑) which, after all, has a lot of other names, including 'Spanish Gypsy'. 'Neapolitan Minor' and 'Neapolitan' are different minor scales that are like harmonic and melodic minor scales, except for a '-2' that shifts the mode fron Aeolian to Phrygian. And finally there is 'Lydian Minor', a source of confusion that has been around for several years now. The problem is that Google: Lydian Minor finds links to pages that define the scale as Lydian -6-7 in one key or another and other links to questions from other parties who find that confusing. What Google doesn't find is http://www.dolmetsch.com/pianochords.htm where the scale is defined as Lydian -3-7 . The triad of Lydian -6-7 is a major chord, and that of Lydian -3-7 is a minor chord, so I prefer Lydian Minor for the name of the second scale. That almost clears it all up, but for this: Lydian -3-7 looks like it is twice removed from the modes, but Dorian +4 is the same scale. It is described as the fourth mode of harmonic minor in the Dolmetsch source, and my favorite name for it is 'Mesheberakh'.

In the course of putting this together it became clear that guitarists have an abiding interest in unusual scales. They might prefer diagrams in which '|' looks like a fret, and the 'o's in the major scale, o| |o| |o|o| |o| |o| |o|o, are fingerprints. Here are a few practical examples:

Table 8: Four familar scales
E B G D A E	- 1 - - - - - 1 - - 1 - - 1 - - - -	| | | | |	- 2 - - 2 - - - - - - - - 2 - - 2 -	| | | | |	- - - - - - - 3 - - 3 - - - - - - -	| | | | |	- - - - 4 - - 4 - - 4 - - 4 - - 4 -	| | | | |	- - - - - - - - - - - - - - - - - -	Ionian (major)	ti - - - re la mi - - -	| | | | |	do sol - - - - - - fa do	| | | | |	- - - - - - mi ti - - - - - -	| | | | |	- - - la fa do sol re	| | | | |	- - - - - - - - - - - - - - - - - -
E B G D A E	- - - - - - - 1 - - - - - - - - - -	| | | | |	- 1 - - 1 - - 2 - - 1 - - 1 - - 1 -	| | | | |	- - - - 2 - - - - - - - - - - - - -	| | | | |	- - - - - - - 4 - - 4 - - 3 - - 3 -	| | | | |	- - - - 4 - - - - - - - - 4 - - 4 -	Aeolian (minor)	- - - - - - ti - - - - - - - - -	| | | | |	la mi do sol re la	| | | | |	- - - fa - - - - - - - - - - - -	| | | | |	- - - - - - re la mi ti	| | | | |	- - - sol - - - - - - fa do
E B G D A E	- 1 - - - - - 1 - - 1 - - - - - - -	| | | | |	- 2 - - 2 - - 2 - - - - - 1 - - 1 -	| | | | |	- - - - - - - - - - 3 - - - - - - -	| | | | |	- - - - 4 - - 4 - - 4 - - 4 - - 3 -	| | | | |	- - - - - - - - - - - - - - - - 4 -	Ionian -3 or Aeolian +6+7 (minor)	si - - - ti fi - - - - - -	| | | | |	la mi do - - - re la	| | | | |	- - - - - - - - - si - - - - - -	| | | | |	- - - fi re la mi ti	| | | | |	- - - - - - - - - - - - - - - do
E B G D A E	- 1 - - - - - 1 - - - - - - - - - -	| | | | |	- 2 - - 2 - - 2 - - - - - 1 - - 1 -	| | | | |	- - - - 3 - - - - - 2 - - - - - - -	| | | | |	- - - - - - - 4 - - 4 - - 3 - - 3 -	| | | | |	- - - - - - - - - - - - - 4 - - 4 -	Aeolian +7 (minor)	si - - - ti - - - - - - - - -	| | | | |	la mi do - - - re la	| | | | |	- - - fa - - - si - - - - - -	| | | | |	- - - - - - re la mi ti	| | | | |	- - - - - - - - - - - - fa do

• Ionian is the only scale I have learned from a professional guitarist; the rest are just whatever feels most comfortable to me. In this (also Phrygian, Lydian and Locrian) the lateral position of the hand is fixed, and the only muscles in play are the fast and accurate ones at the inner side of the forearm.
  
  
• Aeolian and most other scales require lateral shifts of (hand) position, and such shifts can be anticipated by squeezing the fingers (same fast muscles) to move -4- or -1- to a place where it anchors the shift, thus to give the slower muscles a little help with getting the new position right. In Aeolian, for example, the one-tone interval that would normally be 1| |3 on the D-string is squeezed to 1| |4, and the one-tone 1| | |4 shift to/from the G-string is much easier than the 1| | |3 stretch would have been.

Table 9: Six less familar scales
E B G D A E	- 1 - - - - - 1 - - - - - 1 - - - -	| | | | |	- 2 - - 2 - - - - - - - - 1 - - 2 -	| | | | |	- - - - 3 - - 3 - - 2 - - - - - - -	| | | | |	- - - - - - - 4 - - 4 - - 3 - - 4 -	| | | | |	- - - - - - - - - - - - - 4 - - - -	Ionian -6 (major)	ti - - - re - - - mi - - -	| | | | |	do sol - - - - - - fa do	| | | | |	- - - le mi ti - - - - - -	| | | | |	- - - - - - fa do sol re	| | | | |	- - - - - - - - - - - - le - - -
E B G D A E	- - - - - - - 1 - - - - - 1 - - - -	| | | | |	- 1 - - 1 - - - - - 1 - - 1 - - 2 -	| | | | |	- - - - 2 - - 3 - - - - - - - - - -	| | | | |	- - - - - - - 4 - - 4 - - 3 - - 4 -	| | | | |	- - - - 4 - - - - - - - - 4 - - - -	Aeolain +3 or Ionian -6-7 (major)	- - - - - - re - - - mi - - -	| | | | |	do sol - - - te fa do	| | | | |	- - - le mi - - - - - - - - -	| | | | |	- - - - - - fa do sol re	| | | | |	- - - te - - - - - - le - - -
E B G D A E	- 1 - - - - - - - - - - - 1 - - - -	| | | | |	- 2 - - 2 - - - - - - - - 1 - - 2 -	| | | | |	- - - - 3 - - 2 - - 2 - - - - - 3 -	| | | | |	- - - - - - - 4 - - 4 - - 3 - - - -	| | | | |	- - - - - - - - - - 4 - - 4 - - - -	Ionian -2-6 (major)	ti - - - - - - - - - mi - - -	| | | | |	do sol - - - - - - fa do	| | | | |	- - - le mi ti - - - ra	| | | | |	- - - - - - fa do sol - - -	| | | | |	- - - - - - - - - ra le - - -
E B G D A E	- 1 - - 1 - - 1 - - - - - - - - - -	| | | | |	- 2 - - 2 - - 2 - - - - - - - - 1 -	| | | | |	- - - - 3 - - - - - 2 - - 2 - - - -	| | | | |	- - - - - - - - - - 4 - - 3 - - 3 -	| | | | |	- - - - - - - - - - - - - 4 - - 4 -	Aeolian +4+7 (minor)	si ri ti - - - - - - - - -	| | | | |	la mi do - - - - - - la	| | | | |	- - - fa - - - si ri - - -	| | | | |	- - - - - - - - - la mi ti	| | | | |	- - - - - - - - - - - - fa do
E B G D A E	- 1 - - - - - - - - - - - - - - - -	| | | | |	- 2 - - 2 - - 1 - - - - - 1 - - 1 -	| | | | |	- - - - 3 - - - - - 2 - - - - - 2 -	| | | | |	- - - - - - - 4 - - 3 - - 3 - - - -	| | | | |	- - - - - - - - - - 4 - - 4 - - 4 -	Phrygian +7 or Aeolian -2+7 (minor)	si - - - - - - - - - - - - - - -	| | | | |	la mi do - - - re la	| | | | |	- - - fa - - - si - - - te	| | | | |	- - - - - - re la mi - - -	| | | | |	- - - - - - - - - te fa do
E B G D A E	- 1 - - - - - - - - 1 - - - - - - -	| | | | |	- 2 - - 2 - - 1 - - - - - 1 - - 1 -	| | | | |	- - - - - - - - - - 3 - - - - - 2 -	| | | | |	- - - - 4 - - 4 - - 4 - - 4 - - - -	| | | | |	- - - - - - - - - - 4 - - - - - 4 -	Phrygian +6+7 or Aeolian -2+6+7 (minor)	si - - - - - - fi - - - - - -	| | | | |	la mi do - - - re la	| | | | |	- - - - - - - - - si - - - te	| | | | |	- - - fi re la mi - - -	| | | | |	- - - - - - - - - te - - - do

• Ionian -6 introduces two fingering problems that sometimes occur in removed scales. They are the combinations, o| | |o and o|o| |o|o. The sesquitone, 1| | |4 on one string, is no problem, but the semitone that usually precedes or follows it is a stretch in any string change but G to/from B. The fix is usually to bury the sesquitone in a string change. No such fix works for the other pattern: the Mr Spock fingering, 1|2| |3|4, is too painful for me, and attempts to bury the central tone in 1|2|3|4 lead to other, worse problems. My best offer is the slide, 1|1| |3|4, that appears in this scale and several others.
  
  
• Aeolian +7 and Ionian -6 are harmonic scales, minor and major. Harmonic Minor is very popular; Harmonic Major, in my opinion, is greatly underrated.
  
  
• Ionian -3 and Aeolian +3: are melodic scales, minor and major, and they are pretty.
  
  
• Aeolian +4+7 and Ionian -2-6 are known as Hungarian Gypsy and Bysantine , among many names They are two of the double harmonic scales that have two spectacular sesquitones. It's the end of the line for seven-tone scales --- there aren't enough black keys to do another minor third. (There is a six-tone triple harmonic scale with six black keys in the elements, ||oo||oo|| and oo. A fearfully symmetric thing it is, and ugly too --- if it were mine to name it might be Grendel, or maybe Grendel's Mother.)
  
  
• Phrygian +7 and Phrygian +6+7 are minor scales, harmonic and melodic, and their names are Neapolitan Minor and Neapolitan (also minor). Personally, I am not particularly fond of the 1| |3|4|4 in Neapolitan or the 1|1| |3|4 that appears elsewhere, but I prefer them to the uncomfortable stretches they avoid.
  
  
• The guitar fingerings were altogether unplanned --- just couldn't resist them; couldn't resist the sol-fa diagrams either.

(Next digression: tuning strings with harmonics, might appear here, but for now it is in Appendix 2.)

The next planned contribution to these notes is about harmony, and the beginning of that is the business of naming musical chords and describing their contents. Some helpful references are:

http://www.dolmetsch.com (again, huge!)
http://www.standingstones.com/theorcnr.html

The several meanings of tone figure in describing chords, as follows: A chord is both a set of several tones (pitch) and an agreed upon abbreviation for that, and the abbreviations are more closely based upon the tones (interval) in Table 2. That leaves the choice of intervals (Table 2) or degrees (Table 1), but nevermore tones, to describe the contents of a chord. The only place it makes a difference is in seventh and ninth chords where, for example, C7 means either 'play the seventh interval (b↓)' or 'play the diminished seventh degree (-b)'. (Of course, none of us thinks that way when we practice our sixth, seventh and major seventh arpeggios every day --- that lives in another part of the mind, where this kind of chatter matters little.) Here are more examples:

Table 10: Sixth, seventh and ninth chords and their contents, both ways.
type	example	intervals	degrees	example	intervals	degrees
major	C6	1 3 5 6	1 3 5 6	C9	1 3 5 7 9	1 3 5 -7 9
	C7	1 3 5 7	1 3 5 -7	C7-9	1 3 5 7 -9	1 3 5 -7 -9
	Cmaj7	1 3 5 +7	1 3 5 7	Cmaj9	1 3 5 +7 9	1 3 5 7 9
minor	Cm6	1 -3 5 6	1 -3 5 6	Cm7	1 -3 5 7	1 -3 5 -7
	Cm9	1 -3 5 7 9	1 -3 5 -7 9	CmM7	1 -3 5 +7	1 -3 5 7
diminished	Cdim7	1 -3 -5 -7	1 -3 -5 --7	Cdim9	1 -3 -5 -7 -9	1 -3 -5 --7 -9
also	Cm7-5	1 -3 -5 7	1 -3 -5 -7	CmM9	1 -3 5 +7 9	1 -3 5 7 9

Notes:

• '+' and '-' are often used to signify 'augmented' and 'diminished' for intervals and degrees alike. Sometimes, but not in HTML documents, the symbols for 'sharp' and 'flat' are used instead.
• In the third line, it appears that 'maj' refers to the augmented seventh interval or the seventh degree of a major scale; the same goes for 'M' in the fifth and seventh rows. 'maj' will usually be replaced by 'M' from here on.
• On the Google-hit scale the importance of a diminished-ninth chord is about 1/200th of that of a diminished-seventh chord. Its inclusion here indicates that 'dim' conveys diminish the seventh and ninth intervals alike.
• A diminished seventh interval is a twice-diminished seventh degree. To say '--7' for a twice-diminished seventh degree is clumsy, and '6' (either way) is almost always used instead. The problem with that is that the use of '6' for a diminished-seventh interval causes confusion, so '--7' (degrees) will be used here, but not very often.
• The even rarer (1/300th) diminished-sixth chord (e.g. Cdim6) would probably have a diminished sixth (-6, either way), but there isn't much point to introducing that abbreviation -- explanation to follow, pretty soon.

The choice of the style of describing tones in a chord is open, and my personal choice (closer to the fingers) is degrees, so degrees it will be from here on. That the abbreviations appear to be based on intervals just doesn't matter. What the abbreviations reflect is that in the music we usually play, dominant and minor sevenths (e.g. C7 and Cm7) appear far more often than major sevenths (e.g. Cmaj7 and CmM7) (nobody uses Cmmaj7). We simply prefer to use the longer abbreviations for the less common chords. The following table has all the chords on the last pages of "The Worlds Greatest Legal Fake Book" and just a few others.

Table 11: Common chords and notation for them
chord	degrees	e.g.	chord	degrees	e.g.
major	1 3 5	C	sixth	1 3 5 6	C6
seventh	1 3 5 -7	C7	major seventh	1 3 5 7	Cmaj7, CM7
minor	1 -3 5	Cm	minor sixth	1 -3 5 6	Cm6
minor seventh	1 -3 5 -7	Cm7	minor major seventh	1 -3 5 7	CmM7
diminished	1 -3 -5	Cdim, C0	diminished seventh	1 -3 -5 --7	Cdim7, C07
augmented	1 3 +5	Caug, C+	augmented seventh	1 3 +5 -7	Caug7, C+7
ninth	1 3 5 -7 9	C9	major ninth	1 3 5 7 9	Cmaj9, CM9
minor ninth	1 -3 5 -7 9	Cm9	fifth chord	1 5 8	C5, C(no 3)
suspended chord	1 4 5	Csus, Csus4	seventh suspended	1 4 5 -7	C7sus, C7sus4
suspended chord	1 2 5	Csus2	seventh suspended	1 2 5 -7	C7sus2
example 1	1 -3 +5 -7	Cm7+5	example 2	1 -3 -5 -7	Cm7-5
example 3	1 3 5 -7 -9	C7-9	six nine chord	1 3 6 9	C69, C96
The fifth chord is also called a power chord. The seventh tone of Cm7-5 is a semitone higher than that of C07.

The three examples shown above are called extended chords --- they show a common device for adding a few more degrees or intervals (no '7's here) to a standard abbreviation.

In practice, the notes in a chord are not always played all at once, and players that use the abbreviations are expected (allowed is better) to improvise an accompaniment to whatever else is going on. For that purpose, another convention is that a single note (usually an octave low) appears within parentheses; e.g. (C) or (G) and sometimes (E) may appear above or below a melody line along with C's that indicate C-major chords. Also used is the convention that 'chord/note' means an inversion of the standard chord in which the lowest note is specified; e.g. C7/G contains the notes of C7 in the order, G B↓ C E .

And now, back to Cdim6, probably the notes, C E↓ G↓ A↓. It's quite pretty, and it sounds familiar; in fact it is far better known as A↓7/C. Another kind of coincidence among chords is Cdim = Cm-5 , and still another is A↓6x5 = Fm/A↓. In that the 'x5' signifies 'don't play the fifth degree', and the result is somewhat more harmonious than the A↓+ that appears in harmonizations of the F Harmonic-minor scale;

Next topic: chords that can be played with the notes in various scales.

Table 12: Chords in some modes and once-removed scales.
X	Ionian (major)	X	Aeolian (minor)	X	Aeolian +7 (harmonic)	X	Dorian +4 (mesheberakh)	X	Phrygian +3 (freygish)
C	X X6 XM7	A	Xm Xm7	A	Xm XmM7	D	Xm Xm6 Xm7	E	X X7
D	Xm Xm6 Xm7	B	X0 Xm7-5	B	X0 X07	E	X X7	F	X X6 XM7
E	Xm Xm7	C	X X6 XM7	C	X+ XM7+5	F	X X6 XM7	G↑	X0 X07
F	X X6 XM7	D	Xm Xm6 Xm7	D	Xm Xm6 Xm7	G↑	X0 X07	A	Xm XmM7
G	X X6 X7	E	Xm Xm7	E	X X7	A	Xm XmM7	B	X0 X07
A	Xm Xm7	F	X X6 XM7	F	X X6 XM7	B	X0 X07	C	X+ XM7+5
B	X0 Xm7-5	G	X X6 X7	G↑	X0 X07	C	X+ XM7+5	D	Xm Xm6 Xm7
same chords - different modes				same chords - different modes

Table 13: More chords in once- and twice-removed scales.
X	Ionian -6 (harmonic)	X	Ionian -3 (melodic)	X	Aeolian +3 (melodic)	X	Lydian -3-6 (double)	X	Ionian -2-6 (double)
C	X XM7	C	Xm Xm6 XmM7	A	X X7	F	Xm XmM7	C	X XM7
D	X0 X07	D	Xm Xm6 Xm7	B	X0 Xm7-5	G	X-5 X6-5 X7-5	D↓	X X7 XM7
E	Xm Xm7	E↓	X+ XM7+5	C↑	X0 Xm7-5	A↓	X+ XM7+5	E	Xm Xm6
F	Xm Xm6 XmM7	F	X X6 X7	D	Xm XmM7	B	X0x3 X07x3	F	Xm XmM7
G	X X6 X7	G	X X7 X+ X+7	E	Xm Xm6 Xm7	C	X XM7	G	X-5 X6-5 X7-5
A↓	X+ X+7	A	X0 Xm7-5	F	X+ XM7+5	D↓	X X7 XM7	A↓	X+ XM7+5
B	X0 X07	B	X0 Xm7-5	G	X X6 X7	E	Xm Xm6	B	X0x3 X07x3
different chords						same chords - different modes

Next digression: scales that can be paired with the various chords (almost complete now). The purpose of the extended chords is to provide accompanists with information about extra notes that can or should be be used in chords, and there is no reason the idea can't be extended to indicate what scales should be used in the improvisation of descants and other secondary voices. So here is a modest proposal:

• Let an unmodified X always refer to the Ionian scale in the key of X (we do that anyway), and call it the X-major scale.
• Let Xm always refer to the Aeolian scale in the key of X (we usually do that too), and call it the X-minor scale.
• Let Xo always refer to the Locrian scale in the key of X (think 'o' from Locrian or from the abbreviation, X0, for a diminished chord), and call it the X-diminished scale.

Along with the use of intervals and degrees of X (Table 10) there are other ways assign tones to degrees in chords and in the accompanying scales. Some of the other possibilities are indicated below.

Table 14: Sixth, seventh and ninth chords and their contents, other ways.
type	name	degrees of X	degrees of Xm	degrees of Xo	name	degrees of X	degrees of Xm	degrees of Xo
sixths	X6	1 3 5 6	---	---	Xm6	1 -3 5 6	1 3 5 +6	---
major	X7	1 3 5 -7	---	---	X9	1 3 5 -7 9	---	---
	XM7	1 3 5 7	---	---	XM9	1 3 5 7 9	---	---
minor	Xm7	1 -3 5 -7	1 3 5 7	---	Xm9	1 -3 5 -7 9	1 3 5 7 9	---
	XmM7	1 -3 5 7	1 3 5 +7	---	XmM9	1 -3 5 7 9	1 3 5 +7 9	---
diminished	Xo7	1 -3 -5 --7	1 3 -5 -7	1 3 5 -7	Xo9	1 -3 -5 --7 -9	1 3 -5 -7 -9	1 3 5 -7 9

Again a choice is needed, and what will be adopted is

• Let the adjustments in extended chords (scales) be specified by degrees in whatever scale is implied by what appears to the left of the adjustments. The scale will often be X (Ionian), Xm (Aeolian), Xo (Locrian),or Xa ( = X+ = X+5), but other adjustments can occur when the basic chord has '6', '7' or '9' in its abbreviation.

When we are not being pedantic, we associate X and Xm with major and minor scales, but according to Table 1 the Minor scale is the Dorian mode. In the proposed notation the Minor scale in the key of X is Xm6 or Xm+6 or X-3-7. ( B↓ is the sixth degree of Dm (D-Aeolian), it is augmented to the sixth degree of D(-Ionian) in (D-)Dorian, and that works for any key.) The situation with the Xdim chord is more confusing because of the several ways a scale can be associated with it along with the name 'half-diminished' that is sometimes used to refer to the Locrian made. In short, I prefer to adopt 'X-diminished scale' as defined above. For the seven modes we now have

Table 15: Church modes played as scales in the key of X, Xm or Xo
key							first note							sharp or flat
Ionian			Dorian			Phrygian			Lydian			Mixolydian			Aeolian			Locrian
X	do	---	Xm+6	la	fi	Xm-2	la	te	X+4	do	fi	X-7	do	te	Xm	la	---	Xo	ti	---

To help with the discussion of scales that go with other chords, note that for scales,

X7 = X9 = X-7,   X6 = Xa-5 = X,   X5 = Xx3 or Xmx3,
Xm = Xm7 = Xm9 = X-3-6-7,   XmM7 = Xm+7 = X-3-6,   Xm6 = Xm+6 = X-3-7,
Xo = X-2-3-5-6-7 = Xm-2-5 and   Xo7 = Xo-7 = X-2-3-5-6--7 = Xm-2-5-7.

It will be suggested now that mode-names should sometimes be abandoned in favor of what might be called chord-names, which are names based upon modifications of the scales associated with X, Xm, Xo and sometimes Xa ( = X+5 ), using standard chord abbreviations. The reason for doing that is that X-major and X-minor scales are so much more familiar to those of us who practice them every day than are the modes, and Xo and Xa are relatively familiar. The trade-off is between more modifications of more familiar scales and fewer modifications of less familiar ones; singers just might prefer the mode-names. Some sixths and sevenths are also included among the basic chords.

Table 16: basic scales
mode-name	type	chord	alternates	scale-name
Ionian	major	X	X5, X6	major
Mixolydian	major	X7	X-7	dominant
Aeolian	minor	Xm	Xm7, X-3-6-7	minor
Aeolian +6	minor	Xm6	Xm+6	minor sixth
Locrian	diminished	Xo	Xm-2-5	diminished
Locrian -7	diminished	Xo7	Xo-7	diminished seventh
Ionian +5	augmented	Xa	X+5	augmented
Again there is no theory involved in X7 = X-7, Xm6 = Xm+6 or Xo7 = Xo-7, just conventions.

Table 17: formerly once-removed scales
mode-name	type	chord	alternates	scale-name
Ionian -3	minor	Xm6+7	Xm+6+7, X-3	melodic minor
Aeolian +3	major	X7-6	X-6-7, Xm+3	melodic major
Mixolydian +4	major	X7+4	X+4-7	Bartok
Lydian +5	major	Xa+4	X+4+5	Lydian augmented
Locrian -4	diminished	Xo-4	- - -	Super Locrian
Dorian -2	minor	Xm6-2	- - -	Javanese
Ionian -6	major	X-6	- - -	harmonic major
Aeolian +7	minor	Xm+7	- - -	harmonic minor
Dorian +4	minor	Xm6+4	- - -	Mesheberakh
Phrygian +3	major	X7-2-6	Xm-2+3	Freygish
Aeolian +4	minor	Xm+4	- - -	Minor Gypsy
Phrygian +7	minor	Xm-2+7	- - -	Neapolitan Minor
Ionian -2	major	X-2	- - -	Semi-Persian

Table 18: formerly twice-removed scales
mode-name	type	chord-name	alternates	a.k.a.
Ionian -2-3	minor	Xm6-2+7	X-2-3	Neapolitan
Mixolydian -5-6	- - -	X7-5-6	Xo+2+3	Arabian
Lydian -6-7	major	X7+4-6	Xm+3+4	Lydian Minor
Ionian -2-6	major	X-2-6	- - -	double harmonic major
Mixolydian -2-5	- - -	X7-2-5	Xo+3+6	Oriental
Aeolian +4+7	minor	Xm+4+7	X-3+4-6	double harmonic minor
Lydian -2-7	major	X7-2+4	- - -	Romanian Major
Mixoydian +2+4	major	X7+2+4	- - -	Hungarian (Major)
Locrian +3+7	- - -	Xo+3+7	X-2-5-6	Persian

---

From here on things are still pretty disorganized, and there will be quite a few changes to follow.
First item: http://howmusicreallyworks.com/Index.html is a helpful and entertaining reference that just might talk us into buying the whole book (890 pgs) by Wayne Chase. (I haven't yet, but I might.)

We will need the old circle of fifths, of course, and here it is, stretched out with overlaps out to seven sharps or flats.

Table 19: related major and minor keys
minor	-Am	-Em	-Bm	Fm	Cm	Gm	Dm	Am	Em	Bm	+Fm	+Cm	+Gm	+Dm	+Am
major	-C	-G	-D	-A	-E	-B	F	C	G	D	A	E	B	+F	+C
order	-7	-6	-5	-4	-3	-2	-1	0	+1	+2	+3	+4	+5	+6	+7
notes	edc bagf	edc bag	ed bag	ed ba	e ba	e b	b		f	f   c	fg  c	fg   cd	fga  cd	fga   cde	fgab  cde
Key-signatures with more than seven sharps or flats are sometimes used, but not very often.

In the first two rows the tone to the right of Xm or X is the fifth degree in the X-minor or X-major scale, and the tone to the left is the fourth degree. The interval to the left is also a fifth (interval) if the fourth degree is diminished by an octave; it is usually called the fifth below while the interval to the right is called the fifth (above is not said). If '\' and '/' are used to indicate increase and decrease in pitch, C\G and G/C are fifths while F\C and C/F are fifths below ---- all relative to the key of C. The row marked 'index' shows the numbers of sharps (+) or flats (-) in the scales, and the last row indicates the way in which sharps or flats are added as one moves from one key to another. Where the index is ±6 the scales are the same, so +Dm (-Em) and +F (-G) can be expressed either way, and scales that would have more than seven sharps or flats are almost always written the other way. For example, in Chopin's Prelude (op 28, No 15) in D-flat major there is an interlude in D-flat minor that is scored as +Cm (four sharps), rather than -Dm (eight flats).

In Table 19 the 'magnfied' keys are the ones most closely related to C (more about Am to follow), and to transpose all that, one need only slide the 'magnifier' to the left or right. For example, in the key of E-flat the closely related keys are   Fm-A   Cm-E   Gm-B  .

In comparison with previous discussions of removed scales, it might be said that the scales to the left and right of X in Table 19 are once-removed harmonically from X, and in the vertical direction the neighboring scales are modes of one another, with no change of the index. Note also that X and Xm, with indices that differ by ±3, are not so close to one another, but (X,X-3) and (Xm,Xm+3) are close major-minor pairs. (Does that mean anything, Max? - Maybe.)

The information in the last row of Table 19 shows the orders in which sharps or flats are added as key-changes are made, and it may be worth expanding the description of it.

Table 20: Placement of sharps and flats in key-signatures.
sharps c 2 +c d 4 +d e f 1 +f g 3 +g a 5 +a b c D Bm E +Cm G Em A +Fm B +Gm Sixth and seventh sharps : +e (f) and +b (c)		flats c 4 -d d 2 -e e f 5 -g g 3 -a a 1 -b b c -A Fm -B Gm -D -Bm -E Cm F Dm Sixth and seventh flats : -c (b) and -f (e)

So the next topic is sequences of chords (or keys or scales), and we start with a collection of examples (pay little attention to this - many changes to follow).

C-major   C G	G\G-7 , F\G7 , E\C , D\G7 , C\C , G\C , C\C

The box at the left specifies a tonic key and a part (not much here) of the circle of fifths. To the right are two voices, bass\chord, where 'bass' includes low-pitch notes (keyboard l.h., cello, etc). The extended chord, G-7, and G7 both indicate the Mixolydian mode. The G7 strongly suggests playing the seventh degree of the dominant scale.

C-major   C G	G/G-7/b , F\G7/b , E\C/c , D\G7/d , C\C/c , G\C/c , CBAGC\C/c

This time inversions are specified. and a simple bass run is indicated.

C-major Am F C G	C\C/ceg , A\Am/cea , F\F+4/dfa , G\G7/bfg , C\C/ceg

Would that be the heart and soul of major-key harmony? It's pretty old, and it seems appropriate that it be modal. Writing out the inversions serves to indicate the notes that are shared by neighboring chords. If those notes are sustained, the accompaniment seems to contain a second voice.

Ionian Dm F Am C Em G Bo	C\C/ceg , A\Am/cea , F\F/cfa , D\Dm/dfa , G\G/dgb , E\Em/beg , B\Bo/bdf , G\G7/bfg , C\C/ceg

The use of 'Ionian' (or X-Ionian) is meant to imply that all the scales are modes of the tonic key.

Some more examples are
Aeolian (A-minor)
A\Am/a - D\Dm/a - F\F/a - E\Em7/b - A\Am/a - D\Dm/f - C\C/e - B\Bo/d -
A\Am/c - G\G/b - A\Am/c - D\Dm/a - F\F/a - E\Em7/b - A\Am/e =

F-Harmonic Minor (Lydian -3-4-6)
F|Fm/f - B↓|B↓m/f - D↓|D↓/f - C|C7/e - F|Fm/f - B↓|B↓m/c↑ - A↓|A↓+/c - G|G07/B↓ -
F|Fm/A↓ - E|E07/G - F|Fm/A↓ - B↓|B↓m/F - D↓|D↓/F - C|C7/E - F|Fm/A↓ =


Related topics -- more cadences in major keys, minor keys and harmonic minor keys -- key to key (and back) modulations -- harmonization of melodies (fakebook style).

Table 20: Resources
title	author	publisher	topic(s)
The Source	Steve Barta	Hal Leonard	more scales, chords
Harvard Concise Dictionary of Music	Don Michael Randel	Harvard U Press	many things
The Fiddler's Fake Book	David Brody	Oak Publications	modes
The Compleat Klezmer	Henry Sapoznik	Tara publications	modes, harmony
The World's Greatest Legal Fake Book	Dedicated to Herman Steiger	Warner Brothers Publications	common chords
Musicophilia	Oliver Sacks	Vintage Books	our minds, read these!
This is Your Brain on Music	Daniel J. Levetin	Plume (Penguin)
How Music REALLY Works!	Wayne Chase	Roedy Black Publishing, Vancouver, Canada	many more things

Appendix 1: All of the once- and twice-removed scales:

To facilitate the elimination of unproductive cases, recall that any pair of modifications that includes b↓ or f↑ cannot lead to a twice-removed scale. After f↓→e↓ and b↑→c↑ (table 4) the remaing sets of flats and sharps are (d↓,e↓,g↓,a↓) and (c↑,d↑,g↑,a↑). Immediately we have it that the numbers of flat-pairs and sharp-pairs are each 4x3/2=6, and the number of mixed-pairs is 16 or less. Among pair-wise tables, (c↑d↑ by d↓e↓) and (g↑a↑ by g↓a↓) are special because they both have but one element that isn't impossible. The first has only c↑e↓, and the second has only g↓a↑. That brings the number of scales that do not appear in tables 3, 4 and 5 to ten. The collected results are here:

Table 21: The thirty once- and twice-removed scales, with coincidences:
d↓ : o↓||oo|o|o|ooo||oo|o|o|oo : |o|o| and ooo||oo e↓ : o|o↓|o|o|o|oo|oo|o|o|o|oo : |o|o|o| and oo|oo a↓ : o|o|oo|o↓||oo|o|oo|oo||oo : |o|oo| and oo||oo g↓ : o|o|oo↓||o|oo|o|ooo||o|oo : |oo|o| and ooo||o	d↑ : o||↑oo|o|o|oo||ooo|o|o|oo : |o|o| and oo||ooo c↑ : ↑o|oo|o|o|o|oo|oo|o|o|o|o : |o|o|o| and oo|oo g↑ : o|o|oo||↑o|oo|o|oo||oo|oo : |oo|o| and oo||oo a↑ : o|o|oo|o||↑oo|o|oo|o||ooo : |o|oo| and o||ooo
Once-modified scales that are not once-removed are not included.
e↓ a↑ : o|o↓|o|o||↑oo|oo|o|o||ooo : |oo|o| and o||ooo b↑ c↑ : ↑o|oo|o|o||↑oo|oo|o|o||oo : |oo|o| and o||ooo d↓ g↑ : o↓||oo||↑o|ooo||oo||oo|oo : ||oo|| and oo|ooo d↑ g↑ : o||↑oo||↑o|oo||ooo||oo|oo : ||ooo|| and oo|oo d↓ g↓ : o↓||oo↓||o|ooo||ooo||o|oo : ||ooo|| and o|ooo c↑ a↑ : ↑o|oo|o||↑o|oo|oo|o||oo|o : |oo|oo| and o||oo e↓ g↑ : o|o↓|o||↑o|oo|oo|o||oo|oo : |oo|oo| and o||oo g↓ a↓ : o|o|oo↓|↓||oo|o|ooo|o||oo : |o|ooo| and o||oo d↑ g↓ : o||↑oo↓||o|oo||oooo||o|oo : ||oooo|| and o|oo	c↑ g↓ : ↑o|oo↓||o|o|oo|ooo||o|o|o : |o|oo| and ooo||o f↓ e↓ : o|o↓↓||o|o|oo|ooo||o|o|oo : |o|oo| and ooo||o d↑ a↓ : o||↑oo|o↓||oo||ooo|oo||oo : ||oo|| and ooo|oo d↓ a↓ : o↓||oo|o↓||ooo||oo|oo||oo : ||ooo|| and oo|oo d↑ a↑ : o||↑oo|o||↑oo||ooo|o||ooo : ||ooo|| and ooo|o e↓ g↓ : o|o↓|o↓||o|oo|oo|oo||o|oo : |oo|oo| and oo||o c↑ a↓ : ↑o|oo|o↓||o|oo|oo|oo||o|o : |oo|oo| and oo||o g↑ a↑ : o|o|oo||↑|↑oo|o|oo||o|ooo : |ooo|o| and oo||o d↓ a↑ : o↓||oo|o||↑ooo||oo|o||ooo : ||oooo|| and oo|o
g↓ a↑ : o|o|oo↓|||↑oo|o|ooo|||ooo : |o| and ooo|||ooo d↓ e↓ : o↓|↓|o|o|o|ooo|o|o|o|o|oo : |o|o|o|o| and o-o	c↑ e↓ : ↑o↓|o|o|o|o|ooo|o|o|o|o|o : |o|o|o|o| and o-o c↑ d↑ : ↑|↑oo|o|o|o|o|ooo|o|o|o|o : |o|o|o|o| and o-o
Twice-modified scales that are not twice-removed are not included. All that remains is to count the coincidences; there are eight, so the table (plus |o|o| and oo|o|oo) describes 23x84=1932 scales for us to practice. Please don't tell my piano teacher about this.

For anyone who really wants to carry this any further: in B-major, two diminished '↑'s leaves three '↑'s, and in D↓-major, two augmented '↓'s leaves three '↓'s, but that isn't quite enough to take care of all cases. There are more mixed sharp-flat combinations, and Ionian +1+6+7, for example, is three-times removed. (The 'worst case', d↓↓↓↓|||||↑↑↑↑d, is eight-times removed.)
If we add 'x' to indicate a note never played and allow both ↑ and ↓ to operate on the same note, we also have

Table 22: A few six- and eight-tone scales ---- three-times removed
tones	pattern	mode-name	other names
d↓e↓cx	↓ | ↓ | f | o | o | o | ↓	Dorian -1-2x7	Whole tone
c↑d↑ex	f | o | o | o | ↑ | ↑ | f	Lydian +5+6x7	Whole tone
e↓g↓g↑	b o | o ↓ | o ↓ | ↑ o | b	Locrian -4-6+6	Diminished (halfstep-wholestep)
a↓a↑c↑	f | o ↓ | ↑ o | ↑ o | o f	Lydian -3+3+5	Diminished (wholestep-halfstep)

Appendix 2: Frets and harmonics and guitars
We begin with a few things about the twelfth-root of 2, also written as 21/12, and approximately equal to 1.06 (quite close). A better estimate is 1.059463 whose twelfth power is approximately 1.999998, with a corresponding reciprocal, 0.943874, whose twelfth power is approximately 0.499998 . Those are quite good enough to generate the following:

frets
0	1	2	3	4	5	6	7	8	9	10	11	12
powers of some numbers
1	1.06	1.1236	1.1910			1.4185						2.0122
	1.0595	1.1225	1.1892	1.2599	1.3348	1.4142	1.4983	1.5874	1.6818	1.7818	1.8877	2.0000
	.94387	.89090	.84090	.79370	.74915	.70711	.66742	.62996	.59460	.56123	.52973	.50000
1 minus the powers of (1/2)1/12
0	.05613	.10910	.15910	.20630	.25085	.29289	.33258	.37004	.40540	.43877	.47027	.50000
the fractions, 1/5, 1/4, 1/3 & 1/2
				.20000	.25000		.33333					.50000

• Sorry about all those numbers --- only the first few digits matter much, except to make it clear that the fractions other than 1/2 can be near, but not the same as numbers in the previous row.
• The first row of numbers shows the powers needed to replace an approximation, r (e.g. 1.06), of 21/12 by an improved approximation, r(1+δ), where (1+δ)=(11+(2/r12))/12. (Newton's method, yes, but nobody really needs to know that.)
• The entries in the second and third rows of powers are reciprocals; they have been calculated from improved approximations and rounded to five digit accuracy.

The results are collected in the diagram,

	One octave of some guitar strings (approximate spaces):
nut	1	2	3	4	5	6	7	8	9	10	11	12
e	f	-	g	-	a	-	b	c'	-	d'	-	e'
B	c	-	d	-	e	f	-	g	-	a	-	b
G	-	A	-	B	c	-	d	-	e	f	-	g
D	-	E	F	-	G	-	A	-	B	c	-	d
A'	-	B'	C	-	D	-	E	F	-	G	-	A
E'	F'	-	G'	-	A'	-	B'	C	-	D	-	E
D'	-	E'	F'	-	G'	-	A'	-	B'	C	-	D
0				1/5	1/4		1/3					1/2

On a perfect guitar (and most guitars are nearly perfect):

• the twelfth fret is located at one half the length of the open string,
• the frequencies of tones played by stopping the string at the frets are multiples of the open-string frequency --- namely, the powers of 21/12, as specified above,
• distances from the nut to the frets are multiples of the open string length --- namely, 1 minus the powers of (1/2)1/12, as specified above,
• the loudest of the open-string harmonics are made by restraining, but not stopping the string at the fractions of the open string length that are shown above.

Note 1: Stopping the string at the kth fret means depressing it firmly between that fret and the fret to the left (or the nut), so the string is free to vibrate to the right, and nothing happens to the left. Doing that on a string whose open string frequency is X will be indicated by the notation, X(k) (with X(0) for the open string).

Note 2: Sounding the harmonic at the location 1/k, relative to the open string length, means touching the string lightly there, and that allows the string to vibrate both to the left and right, but not at 1/k. Doing that on a string whose open string frequency is X will be indicated by the notation, X(1/k) (also with X(0) for the open string).

Note 3: The harmonic at 1/k can also be sounded at 1-(1/k) and sometimes at other locations.

Note 4: The octaves we need to know about run from C' (two octaves low) to C (one octave low) to c (middle c) to c' (one octave high). (Notes in the octave of c" are not often attempted on guitars, but with harmonics they can be played.)

Now we can discuss several ways to tune guitars, including ways that also work for some stringed instruments that don't have frets (the nearly perfect ones).

First, there is the matter of whether or not this guitar is worth tuning. On any string, the tones, X(12) and X(1/2), should be the same (to be written as X(12) = X(1/2)), so the first thing to do is to check,

is E'(12) = E'(1/2) ?? and is e(12) = e(1/2) ??
If not, and the instrument has a moveable bridge, the position of the bridge can usually be adjusted to produce the X(12) = X(1/2) unisons on all the strings, and this guitar may be worth tuning. If not, and the instrument has a fixed bridge, get rid of it.

The tuning of most stringed instruments is relative, and the pitch of one note (e.g. a=440 cps) comes from elsewhere (meter, pitchpipe, tuning fork, singer with perfect pitch, other instrument); it will be assumed that A'(0) has been set. Then the familiar tuning scheme that uses fretted notes is:

tune A'(0) , A'(0) /= E'(5) , A'(5) \= D(0) , D(5) \= G(0) , G(4) \= B(0) , B(5) \= e(0)
In that, '/=' or '\=' means adjust the tension of the string to the right to give a unison.

In tuning with harmonics it helps to know what notes are being played. The situation for A'(1/2) is that the octave is being played both to the left and to the right, and the finger can be removed, leaving the string in motion, still sounding the tone, A'(1/2) = A'(12) = A. When the finger is removed (carefully) from the fretted A'(12) the sound persists at a considerably lower volume, but that only works at the twelfth fret. Next in the sequence defined by powers of 1/2 is A'(1/4), in which the octaves of A'(1/2) are bisected to produce A'(1/4) = A(1/2) = a. Likewise, A'(1/8) = A(1/4) = a(1/2) = a', but that one is pretty hard to play.

The harmonic, A'(1/3), divides the A'-string into three octaves of a note that is not any octave of A'. To see what the note is, consider throwing away the left-most octave by stopping the string and playing the (1/2)-harmonic of what is left. To do that, stop the string at the seventh fret to produce, in effect, an E-string, and then restrain the fretted string at its midpoint with the right index finger while plucking it somehow. The result is E(1/2) = e. Caution: the pitch of E(1/2) is ever so slightly lower than that of A'(1/3) --- more about that to follow. For now the unison for tuning the E'-string by harmonics is A'(1/3) /~ E`(1/4), and the '~' serves to indicate the small deviation from an exact tuning.

Also available, but not necessarily used, is the harmonic, G(1/5), which divides the G-string in five octaves, and it occurs when the string is restrained at a point that is very near the fourth fret. The unison that occurs there is G(1/5) \~ B(1/4).

From that we construct the two schemes for harmonic tuning:

tune A'(0) , A'(1/3) /~ E'(1/4) , A'(1/4) \~ D(1/3) , D(1/4) \~ G(1/3) , G(1/5) \~ B(1/4) \~ e(1/3)
tune A'(0) , A'(1/3) /~ E'(1/4) \= e(0) , A'(1/4) \~ D(1/3) , D(1/4) \~ G(1/3) , e(1/3) /~ B(1/4)

• If we have an oboe to give us an a = 440 cps, a /= A'(1/4) is a handy way to tune A'(0).
• For the D-tuning, in which the lowest string is tuned to D', and the interval is a fifth rather than a fourth, A'(1/3) /~ E'(1/4) is replaced by A'(1/2) /~ D'(1/3), and the rest follows the first of the harmonic tunings. Otherwise, one can use any standard tuning scheme and then lower the tone of the E'-string with D(0) /= D'(1/2).

There are other tuning schemes, including ones that use both stopped strings and harmonics. The ones that use only harmonics have the advantage that the unison can be tuned (with a free left hand) while both strings are ringing. It takes less time, and I prefer it even though it is not an exact twelve-tone (tempered) tuning. (Do the arithmetic; introduce cents; musette registers on accordions; beats; maybe theremins; maybe harmoniums and vox humana.)

But watch out, and don't try any of this on your piano ! (Issues: 88 keys vs 6 strings; circle of fifths/octaves; tension and stiffness; a = 440, a(1/2) ~ 881 and a(1/4) ~ 1764 (refs: Johnson & Young and maybe http://piano-lessons-riverside-ca.com/Lessons/pianotuning.html))