Extended meantone notation: Difference between revisions

:''This page is currently reworked at [[User:PiotrGrochowski/Extended meantone notation]], see also [[{{TALKPAGENAME}} #under construction|Talk:Extended meantone notation #under construction]]''.

Most musicians are familiar with the [[Circle-of-fifths notation|circle of fifths]]. In general, notes use 7 base note letters, along with sharps, double-sharps, flats, and double-flats (and beyond):

... F𝄫 – C𝄫 – G𝄫 – D𝄫 – A𝄫 – E𝄫 – B𝄫 – F♭ – C♭ – G♭ – D♭ – A♭ – E♭ – B♭ – F – C – G – D – A – E – B – F♯ – C♯ – G♯ – D♯ – A♯ – E♯ – B♯ – F𝄪 – C𝄪 – G𝄪 – D𝄪 – A𝄪 – E𝄪 – B𝄪 ...

 

In some tunings (such as 24edo and 31edo), sharps can be split in half. Thus, some notes can be notated using semisharps and semiflats or with [[ups and Downs Notation|ups and downs]].

 

For example, in 31edo, the chromatic scale becomes:

 

 

Note that the base note letters alternate.

 

The meantone circle of fifths, however, has no single semisharp or semiflat. In extended meantone notation, a sharp is split into 2 different parts, the diesis and the kleisma.

 

Sharps and flats denote raising and lowering by a given number of chromatic semitones. The diesis is the difference between adjacent accidentals (e.g. C♯–D♭ and D♯–E♭), while the kleisma is the amount by which B♯ exceeds C♭ and E♯ exceeds F♭.

! rowspan="2" | Interval

! rowspan="2" | Number of fifths<br>(move up to raise,<br>move down to lower)

| ♯

| ♭

| Chromatic semitone

| 7

| ↑

| ↓

| 12

| 19

A meantone chromatic semitone consists of one diesis and one kleisma. Note that in most meantone tunings, the diesis and kleisma are roughly a quarter tone. The diesis represents the [[just intonation|just]] intervals [[128/125]] and [[648/625]], while the meantone kleisma represents [[15625/15552]] or [[3125/3072]]. In [[septimal meantone]] (meantone temperaments where 7/4 is an augmented sixth), the diesis also represents [[36/35]], [[50/49]], and [[64/63]], while the kleisma also represents [[49/48]] and [[245/243]].

An octave is made of 19 dieses and 12 kleismas. Unlike semisharps and semiflats, the diesis and kleisma can be generalized to other tunings:

! rowspan="2" | EDO

! rowspan="2" | Comma fraction

! rowspan="2" | Explanation

! Chromatic<br>semitone

! Diatonic<br>semitone

| &minus;1

| [[12edo|12]]

| {{frac|11}} comma

| Chromatic semitone is equal to kleisma,<br>diesis is tempered out

| [[19edo|19]]

| {{frac|3}} comma

| Chromatic semitone is equal to diesis,<br>kleisma is tempered out

| [[26edo|26]]

|  

| rowspan="2" | Diesis is larger than chromatic semitone,<br>kleisma is negative

| [[31edo|31]]

| {{frac|4}} comma

| [[43edo|43]]

| {{frac|5}} comma

| [[55edo|55]]

| {{frac|6}} comma

| [[50edo|50]]

| {{frac|2|7}} comma

There are of course notational equivalences:

 

[[9–odd–limit]] intervals and their notation relative to C:

| E♭

| A♭

| style="border-left: 5px solid black;" | A♯<br>B♭↓

| D♯<br>E♭↓

| F♯<br>G♭↓

| E𝄫<br>D↑

| B𝄫<br>A↑

| G♭<br>F♯↑

| style="border-left: 5px solid black;" | D

| B♭

| F♭<br>E↑

| B♭

| G♯<br>A♭↓

Two dieses or two kleismas cannot be stacked to produce a chromatic semitone. 11–limit and 13–limit notation can vary (see [[meantone vs meanpop]]).

== True half-sharps and half-flats ==

If true half-sharps and true half-flats are desired, which exactly bisect the chromatic semitone, the meantone fifth is split in half. This creates a 2D new tuning system which is generated by a chain of neutral thirds, with meantone existing as every other note in the generator chain. This adds true half-sharps and half-flats, and creates a "neutral" version of each interval class.

Real-world Arabic and Persian music often involve many fine microtonal details (such as the use of multiple unequal neutral intervals) and exhibit significant regional variations, and as a result they are very difficult to notate exactly. However, they are commonly notated using half-sharps and-half flats. If we take these to be exactly equal to half of a chromatic semitone, then mathematically, this notation system results in a two-dimensional lattice that is generated by a neutral third and an octave. If adjacent sharps and flats, such as C♯ and D♭, are made enharmonically equivalent, this lattice degenerates further into [[24edo]], which is often suggested as a simplified framework and tuning system for notating and playing Arabic and Persian music. But the usual written notation typically lets musicians and composers treat adjacent sharps and flats as two distinct entities if it is decided that they should be different.