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Similar to this question: How were tuning/temperaments indicated on scores (if they ever were)?

In modern musical notation how should the use of something other than equal temperament be notated?

See also: Are there pieces that require retuning an instrument mid-performance?

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  • I wonder whether strings or voices in particular would stray into anything other that 12tet when used without 12tet instruments anyway.
    – Tim
    Apr 6, 2021 at 8:13
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    Adding to Aaron's answer, this seems like a good article for a longer description, including a section on notation marsbat.space/pdfs/JI.pdf
    – Michael Curtis
    Apr 7, 2021 at 15:42
  • @Tim String players don't stick to12tet. They play a mixture of Pythagorean intonation, just-intonation, expressive intonation and Equal Temperament depending on the circumstances. Also wind instrument players can intonate depending on the circumstances. Here are some short videos on different intonation possibilities on violin: violinmasterclass.com/en/masterclasses/intonation
    – Lars Peter Schultz
    Sep 11, 2022 at 22:00

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The question boils down to how can/should microtonal music be notated. "Microtonality" has come to broadly refer to any music that doesn't conform to 12-tone equal temperament. Microtonality includes a variety of just and meantone temperaments as well as equal temperaments with other than 12 pitches.

Allowing one is going to score for a standard musical staff, then there are essentially two necessary elements:

  1. A set of "accidentals" to indicate the various gradations of a particular pitch. ("Accidental" in this case might be a numerical specification of, say, adjustment in cents.)
  2. A legend explaining the tuning for each pitch.

The specifics of each of these frequently vary from composer to composer, piece to piece, and temperament to temperament, but the set of symbols linked in Bruce Adams's answer are fairly standard for quarter-tone music and have been adapted to other uses as well.

quarter-tone sharp and flat symbols

As a specific example of the two points listed above, here is the "legend" for Ben Johnston's Suite for Microtonal Piano (1978).

"Legend" from Johnston's "Suite for Microtonal Piano"

The numbers indicate positions in the harmonic series. The arcs denote pairs of (just) perfect fifths.

(SOURCE: Wikipedia > Suite for Microtonal Piano)

For chords, one way to notate them is demonstrated by the Sagittal System (see "Sagittal: A Microtonal Notation System" by George D. Secor and David C. Keenan, also the source for the below image). It shows the "core" relationship between pitches combined with symbols for adjustments to those pitches.

Chords in Sagittal notation

Another option that applies to both single notes and chords is Ben Johnston's specification of cents away from C. The example below comes from Kyle Gann's article "How the 13th Harmonic Saved My Music"

Ben Johnston notation example

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    '"Microtonality" has come to broadly refer to any music that doesn't conform to 12-tone equal temperament': I've never heard the term used to refer to early music, nor blues, nor non-European traditional music.
    – phoog
    Apr 6, 2021 at 7:22
  • Quoting from the wikipedia page "just perfect fifths marked with slurs". This may be 'correct' but I find it hard to read. Perhaps it is more 'acceptable' for piano where [slurs](en.wikipedia.org/wiki/Slur_(music) ) mean something different but to me a slur implies a legato technique such as on the guitar or violin: " For bowed string instruments, the notes should be played in one bow stroke. For plucked string instruments, such as guitars, the notes should be played without plucking the individual strings (hammer-ons and pull-offs)."
    – Bruce Adams
    Apr 6, 2021 at 7:26
  • @BruceAdams Is it clear that the posted image is not intended to be played? It's just the composers way of showing how the instrument should be tuned. It's just a set of tuning instructions, not an excerpt from the score proper.
    – Aaron
    Apr 6, 2021 at 11:59
  • @phoog Yes, I realize it's over broad -- not sure how to word it. Happy for a suggestion.
    – Aaron
    Apr 6, 2021 at 12:00
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    You're right, I just tried to get the front page but it was hard to get back. Here maybe w3c.github.io/smufl/latest/index.html
    – Emil
    Sep 14, 2022 at 6:56
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I think what you are describing is called scordatura

I've found three examples of scordatura using different notation:

It might be correct to simply say:

scordatura: meantone

Above the first bar if you were doing something like an arrangement of Fantasia Chromatica by Jan Pieterszn Sweelinck

You could also put something like:

"A = 452 Hz"

To indicate use of a different base pitch. However I cannot find a score with an example of this.

Sibelius and musescore can add an extra staff for scordatura.

For individual notes we have notation for quarter tones which is pretty much standard (and suggestions like saggital for smaller intervals)

caveat: I have no musicological qualifications. This is based on internet research. Others may be able to improve on this answer.

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    Scordatura is not at all the topic of this question. Scordatura is tuning one or more strings to a different note from the standard, for example tuning a violin's E string down to D. The question is asking about using different systems for determining the frequency of each note, such that, given A4 equals 440 Hz, E5 might be 660 Hz or 655 Hz or something else.
    – phoog
    Apr 6, 2021 at 7:13
  • But this is exactly what you are doing if you play in a different temperament. Scordatura was described as "(strings tuned in different intervals)." here - maestronet.com/forum/index.php?/topic/341115-scordatura . Could you not denote different pitches being used using scordata?
    – Bruce Adams
    Apr 6, 2021 at 7:40
  • Scordatura means that the open string is assigned to a different note name, for example D instead of E. Using a different temperament means that the frequency of the note is changed. These changes are far smaller than a half step (the distance between any two adjacent keys on the keyboard). For example, changing the pitch of the E string to D means reducing its frequency by a factor of somewhere between 0.889 and 0.9, depending on the temperament. The ratio between the frequency of an equal-tempered major third and a just major third is a factor of 0.992.
    – phoog
    Apr 6, 2021 at 14:25
  • Furthermore, since a violin, viola, or cello only has four tuned pitches, you cannot specify a temperament by specifying how to tune the four strings. When someone playing a fretless string instrument wants to use a different temperament, it's necessary to place the fingers differently on the fingerboard. For a fretted instrument, you have to move all the frets, and to do so differently on different strings. It's a completely different concept from scordatura. How do you specify that the desired frequency of C sharp is 550 Hz or 551 Hz instead of 554.37 Hz using the concept of scordatura?
    – phoog
    Apr 6, 2021 at 14:30

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