Just to say that in some microtonal systems of music, then C# differs from Db, and B# can differ from Cb, and the order of the pitches varies depending on the size of the fifth. So in some systems it goes, in increasing pitch, as
B B# Cb C C# Db D
and others it goes
B Cb B# C Db C# D
And you get double flats and double sharps as well, and in some systems, they can be different in pitch again from any of the notes generated so far.
That's because you get to the sharp or the flat by going around the circle of fifths, like this:
... Fbb Cbb Gbb Dbb Abb Ebb Bbb
Fb Cb Gb Db Ab Eb Bb
F C G D A E B
F# C# G# D# A# E# B#
F## C## G## D## A## E## B##
...
As you see, that gives you 35 distinct pitches. So - in a tuning system where you can get to all the pitches by going around a circle of fifths, if it has more than 35 notes, you would need to go even further to triple flats or triple sharps to notate them all.
In some tuning systems like 72 equal, then it's simple, it's got the same 12 equal size of fifth, at 700 cents. So - you can't get to all the notes by a circle of fifths. Instead it is built up of six distinct "circles of fifths" each with twelve notes, shifted by a sixth of a semitone each.
Similarly for 24 equal (quarter tones) and any other multiple of 12 equal pitches. For those systems, then Cb is the same as B and B# the same as C just as for twelve equal.
For other ones - let's stick to equal temperaments for now, or mild temperings of equal temperaments - it depends on whether the interval you choose as the fifth of your tuning is flatter or sharper than the 12 tone fifth.
If the interval you choose as a fifth is flatter than the twelve equal 700 cents, then that works like meantone. 31 equal is a prime example.
31 equal temperament
This is the circle of fifths in 31 equal
Where Bx is short for B##
So, B## is identical to Dbb,
Starting from D, the notes are
D, Ebb, D#, Eb, D## = Fbb,
E, Fb, E#, F, Gbb, F#
Which you can make a bit easier to understand and notate using extra symbols for half flat, and half sharp, so you don't need to bother about reading double flats and double sharps - just know that Ebb is the same as D half #, and Dx (or D##) is the same as E half flat, etc.
That's from Paul Rapoport's article on it here: About 31-tone equal temperament
Then the other type of tuning you get is one with the fifth sharper than the 12 tone fifth.
A simple example here is nineteen equal. For that, just write out the notes like this
C C# Db D D# Eb E E# Fb F F# Gb G G# Ab A A# Bb B B# Cb
Now tune them all equally spaced - and that's the 19 equal system. Because it uses fifths sharper than 12 equal, then the sharps and flats work more in the order you expect.
It's a reasonably interesting tuning, and guitar players particularly like tunings like this because it's reasonably intuitive - and 19 frets are not that many - if you want to try your first microtonal guitar fretting, this might be a starting point.
With 19 pitches, of course, it also has 19 distinct key signatures you can modulate to.
Seventeen equal is similar, except, you equate F# to Gb and B# to Cb - then as before, spread the notes out equally but as you have only 17 of them then you end up with a 17 equal system. It's another interesting tuning.
When you look at equal divisions of the octave generally - some numbers of equal divisions give you fifths that are flatter than twelve equal so are "meantone like".
In meantone you make the fifths flat so that in the cycle C G D A E then the E is a pure harmonic 5/4. That's "quarter comma" meantone - and other systems with flat fifths are classified according to how much of the comma you temper out.
In Schismic temperament you get to the 5/4 from the "other side" as it were, approximating 10/1 using six fifths (instead of four) and raising all the fifths in pitch rather than lowering them.
So - if the fifth is flatter than twelve equal, you tend to classify it by reference to the syntonic comma, e.g. quarter comma meantone, sixth comma meantone (needs six fifths to get a syntonic comma flat) and so on.
If it is sharper, you tend to classify it by reference to the Schisma in the same way. So those are called "Shismic temperaments".
The meantone type ones are the ones with the sharps and flats in a rather unintuitive order, like 31 equal. While the schismic ones have them in a more intuitive order like 19 equal.
Some equal divisions of the octave give you a choice of two possible fifths, one flatter than 700 cents for a meantone type tuning, and one sharper for a Schismatic type notation system, so you have a choice of two ways of notating the same pitches.
(here EDO stands for Equal Divisions of the Octave. If you only ever discuss divisions of the octave, you can also write it as 53-et, or "equal temperament).
If you use pure fifths, then C# is just 23.46 cents sharp of C, a bit under a quarter of a semitone.
The circle eventually closes up almost (never can exactly) after 53 notes.
But - then - the system of sharps and flats gets a bit clumsy - because when you get beyond the first 12 notes, the new notes are only a fifth of a tone away from the ones you already have, and so if you used this notation for D## you would get D to D# as a semitone (approximately) and D# to D## as a fifth tone (approximately).
But there have been attempts to make a reasonably unified system of notation for all tunings - using various ways to notate pitches by the closest pitch in a notation which combines together the seven note names with adjustments to them by a syntonic comma.
The syntonic comma is 81/80, or 21.51 cents, is useful because if you take for instance the Pythagorean E, and go down by a syntonic comma, you get a pure 5/4 E.
So by adjusting various Pythagorean note names by the syntonic comma you can notate any of the five limit harmonic series intervals such as 10/9, 8/5, 15/8 etc.
By using various other commas, you can also notate pitches based on the seventh, and eleventh harmonic.
Eventually you get notation like this:
Where those various symbols are related to the comma adjustments, together with some conventional sharps or flats.
They also developed ways of using Sagittal with linear temperaments (made up from circles of fifths) and developed a consistent set of notation systems for all the small numbered ETs up to, I think over 100 pitches per octave. They sometimes have more than one way to notate the same pitches, of course, to take account of the possibility of both schismatic and meantone type ways of notating the same pitches.
It was a multi year project mainly undertaken by George Secor and Dave Keenan, with contribution also by Gene Ward Smith and was one of the main topics of discussion on the Tuning Forum for some years.
Sagittal microtonal notation
In this answer I have just scratched the surface of a vast subject, which is a matter of much discussion by microtonal experts and with a lot of material available to read on it as well nowadays.
I am not at all expert on it, myself, but perhaps with these starting points, if you are interested, you can find out more.
I thought someone should have a go at providing a microtonal answer to this question :).
If you are expert in any of this and spot any mistakes in what I just said, don't hesitate to correct - suggest an edit or comment!
B B# Cb C C# Db D
and others it goes
B Cb B# C Db C# D
And you get double flats and double sharps as well, and in some systems, they can be different in pitch again from any of the notes generated so far.
That's because you get to the sharp or the flat by going around the circle of fifths, like this:
... Fbb Cbb Gbb Dbb Abb Ebb Bbb
Fb Cb Gb Db Ab Eb Bb
F C G D A E B
F# C# G# D# A# E# B#
F## C## G## D## A## E## B##
...
As you see, that gives you 35 distinct pitches. So - in a tuning system where you can get to all the pitches by going around a circle of fifths, if it has more than 35 notes, you would need to go even further to triple flats or triple sharps to notate them all.
TWELVE EQUAL TYPE SYSTEMS
In some tuning systems like 72 equal, then it's simple, it's got the same 12 equal size of fifth, at 700 cents. So - you can't get to all the notes by a circle of fifths. Instead it is built up of six distinct "circles of fifths" each with twelve notes, shifted by a sixth of a semitone each.
Similarly for 24 equal (quarter tones) and any other multiple of 12 equal pitches. For those systems, then Cb is the same as B and B# the same as C just as for twelve equal.
For other ones - let's stick to equal temperaments for now, or mild temperings of equal temperaments - it depends on whether the interval you choose as the fifth of your tuning is flatter or sharper than the 12 tone fifth.
MEANTONE TYPE SYSTEMS
If the interval you choose as a fifth is flatter than the twelve equal 700 cents, then that works like meantone. 31 equal is a prime example.
31 equal temperament
This is the circle of fifths in 31 equal
Where Bx is short for B##
So, B## is identical to Dbb,
Starting from D, the notes are
D, Ebb, D#, Eb, D## = Fbb,
E, Fb, E#, F, Gbb, F#
Which you can make a bit easier to understand and notate using extra symbols for half flat, and half sharp, so you don't need to bother about reading double flats and double sharps - just know that Ebb is the same as D half #, and Dx (or D##) is the same as E half flat, etc.
That's from Paul Rapoport's article on it here: About 31-tone equal temperament
SHISMIC TYPE TUNINGS
Then the other type of tuning you get is one with the fifth sharper than the 12 tone fifth.
A simple example here is nineteen equal. For that, just write out the notes like this
C C# Db D D# Eb E E# Fb F F# Gb G G# Ab A A# Bb B B# Cb
Now tune them all equally spaced - and that's the 19 equal system. Because it uses fifths sharper than 12 equal, then the sharps and flats work more in the order you expect.
It's a reasonably interesting tuning, and guitar players particularly like tunings like this because it's reasonably intuitive - and 19 frets are not that many - if you want to try your first microtonal guitar fretting, this might be a starting point.
With 19 pitches, of course, it also has 19 distinct key signatures you can modulate to.
Seventeen equal is similar, except, you equate F# to Gb and B# to Cb - then as before, spread the notes out equally but as you have only 17 of them then you end up with a 17 equal system. It's another interesting tuning.
MEANTONE AND SCHISMIC TYPE TUNINGS
When you look at equal divisions of the octave generally - some numbers of equal divisions give you fifths that are flatter than twelve equal so are "meantone like".
In meantone you make the fifths flat so that in the cycle C G D A E then the E is a pure harmonic 5/4. That's "quarter comma" meantone - and other systems with flat fifths are classified according to how much of the comma you temper out.
In Schismic temperament you get to the 5/4 from the "other side" as it were, approximating 10/1 using six fifths (instead of four) and raising all the fifths in pitch rather than lowering them.
So - if the fifth is flatter than twelve equal, you tend to classify it by reference to the syntonic comma, e.g. quarter comma meantone, sixth comma meantone (needs six fifths to get a syntonic comma flat) and so on.
If it is sharper, you tend to classify it by reference to the Schisma in the same way. So those are called "Shismic temperaments".
The meantone type ones are the ones with the sharps and flats in a rather unintuitive order, like 31 equal. While the schismic ones have them in a more intuitive order like 19 equal.
SOMETIMES YOU CAN USE BOTH, DEPENDING ON WHAT YOU CHOOSE TO CALL YOUR "FIFTH"
Some equal divisions of the octave give you a choice of two possible fifths, one flatter than 700 cents for a meantone type tuning, and one sharper for a Schismatic type notation system, so you have a choice of two ways of notating the same pitches.
PURE FIFTHS - 53-ED0
(here EDO stands for Equal Divisions of the Octave. If you only ever discuss divisions of the octave, you can also write it as 53-et, or "equal temperament).
If you use pure fifths, then C# is just 23.46 cents sharp of C, a bit under a quarter of a semitone.
The circle eventually closes up almost (never can exactly) after 53 notes.
But - then - the system of sharps and flats gets a bit clumsy - because when you get beyond the first 12 notes, the new notes are only a fifth of a tone away from the ones you already have, and so if you used this notation for D## you would get D to D# as a semitone (approximately) and D# to D## as a fifth tone (approximately).
SAGITTAL - PROJECT TO CREATE A UNIFIED MICROTONAL SYSTEM OF NOTATIONS
But there have been attempts to make a reasonably unified system of notation for all tunings - using various ways to notate pitches by the closest pitch in a notation which combines together the seven note names with adjustments to them by a syntonic comma.
The syntonic comma is 81/80, or 21.51 cents, is useful because if you take for instance the Pythagorean E, and go down by a syntonic comma, you get a pure 5/4 E.
So by adjusting various Pythagorean note names by the syntonic comma you can notate any of the five limit harmonic series intervals such as 10/9, 8/5, 15/8 etc.
By using various other commas, you can also notate pitches based on the seventh, and eleventh harmonic.
Eventually you get notation like this:
Where those various symbols are related to the comma adjustments, together with some conventional sharps or flats.
They also developed ways of using Sagittal with linear temperaments (made up from circles of fifths) and developed a consistent set of notation systems for all the small numbered ETs up to, I think over 100 pitches per octave. They sometimes have more than one way to notate the same pitches, of course, to take account of the possibility of both schismatic and meantone type ways of notating the same pitches.
It was a multi year project mainly undertaken by George Secor and Dave Keenan, with contribution also by Gene Ward Smith and was one of the main topics of discussion on the Tuning Forum for some years.
Sagittal microtonal notation
THIS ONLY SCRATCHES THE SURFACE
In this answer I have just scratched the surface of a vast subject, which is a matter of much discussion by microtonal experts and with a lot of material available to read on it as well nowadays.
I am not at all expert on it, myself, but perhaps with these starting points, if you are interested, you can find out more.
I thought someone should have a go at providing a microtonal answer to this question :).
If you are expert in any of this and spot any mistakes in what I just said, don't hesitate to correct - suggest an edit or comment!
About the Author
Robert Walker
Writer of articles on Mars and Space issues - Software Developer of Tune Smithy, Bounce Metronome etc.Studied at Wolfson College, Oxford
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