I’m trying to replicate in playback the pitches of a natural horn, where the 11th partial is -49 cents from equal temperament and the 14th is -31 cents. Using quarter-tones and three-quarter-tones sharp/flat accidentals sounds quite close, but still not exactly right. Is there a way to fine-tune playback pitch by cents?

Yes – What you do is create a new tonality system with accidentals for each pitch alteration you need.

For example, I am fluent in Ben Johnston’s notation, which starts by tuning the 7 naturals to just intonation in C. The ♯ and ♭ are defined as 25:24 up and down (71 cents), and there are other pairs of symbols for the other commas needed to render extended just intonation.

Once such a system is set up, rendering a harmonic series is easy, but the setup is rather laborious. I am going out in a few minutes but I can be more help when I return tomorrow.

Thanks, Mark. I had got as far as the page where one can change the division of the octave, but I couldn’t work out how to go from there. I’m grateful to you for offering further help when you can. Gareth

Now back. When dealing with different tunings it’s helpful to choose first how you want to notate things. Is this just a harmonic series by itself, or is it happening within a piece that’s otherwise equal-tempered?

If alone, you just need to tune the white notes to the key of the Horn and add accidentals for the 7th, 11th and 13th. The number of divisions you use depends on how precise you want to be. For precision of 1 cent, the number of divisions will add up to 1200. For just intonation there will be rounding errors.

Horn Series.dorico (447.2 KB)

I’ve mocked up a simple file for Horn in F, in Johnston notation with a precision of 1/10 cent (total 12000). A couple of caveats:

- Because this system is based on just intonation in C, in the key of F there are comma issues (which is why 2 of these accidentals need minus signs on them)
- Notation for a transposing instrument is a problem I haven’t solved. We can’t just switch to transposed score (as the player would read) because Dorico transposes only by E.T. intervals – so the accidentals get messed up.

Tell me more about the intended context for your Horn series and I can be more help.

Hi Mark -

I’m so grateful to you for your invaluable advice and hard work on this for me. This is just to acknowledge your reply: I’ll be in a position to look at your file and to provide further details in a few hours’ time.

But - in short - the score I’m working on is for playback only; I decided to keep it separate from the printed score (i.e. to work on a renamed copy of the original file) for various reasons. This means that accidentals/Johnston notation won’t be necessary, a) due to the playback-only nature of the file, and b) because in any case the sections involved are written for natural horn (cf intro and outro of Britten’s Serenade).

The sections are enclosed within a key-signature change involving one of the shipped tonality systems (the rest if the piece using equal temperament), and playback using the provided quarter- and three-quarter-tone sharps/flats works fine except that (to my ears) the resulting pitches don’t match well enough the ones I’m hoping to create.

I’ll have more for you when I’m able to get to my PC in due course.

Thanks again,

Gareth

Hi again, Mark.

The file you sent is exactly what I needed! The pitches remain intact when cut and paste, and – although the notations don’t paste (which I don’t need anyway – see previous email) – the important thing is that the pitch frequencies are accurate. Might I ask how you achieved the adjustment? I had hoped to find a method myself, but I never got any further than being able to use the supplied microtonal accidentals.

Best wishes and many thanks,

Gareth

Hi Gareth – Here’s what I did in brief:

- Sizes of intervals between the white notes in Johnston notation:

(Cents values are always approximate in J.I.)

Notes | Interval | Cents |
---|---|---|

C-D, F-G and A-B | 9 : 8 | 203.9 |

D-E and G-A | 10 : 9 | 182.4 |

E-F and B-C | 16 : 15 | 111.7 |

Multiply the cents by 10 for 12,000 EDO. Unfortunately 3 × 2039 + 2 × 1824 + 2 × 1117 adds up to only 11,999. So we’ll have to fudge one of the intervals. I chose to increase B-C by 1, making it 1118.

- Other intervals needed (accidentals):

Symbols | Ratio | Cents | Description |
---|---|---|---|

♯ / ♭ | 25 : 24 | 70.7 | Chromatic semitone |

+ / – | 81 : 80 | 21.5 | Syntonic comma |

L / 7 | 36 : 35 | 48.7 | Septimal comma |

↑ / ↓ | 33 : 32 | 53.3 | 11th harmonic |

13 / Ɛ❘ | 65 : 64 | 26.9 | 13th harmonic |

Of course there are any number of other ways to notate this! Browsing through the Bravura spec, I find several other systems suitable for J.I. notation. This is just the system I’m most familiar with. In terms of symbols it is more cumbersome than, e.g., Sagittal, but it is unambiguous. (Sagittal is designed to accommodate *any* microtonal scheme, not just J.I.)

In Dorico when you copy notes with custom accidentals to another tonality system, if there are matching pitches, they will display the correct note and accidental. For example, in my file F = 4980/12000 over C; E7♭ = 2669/12000 above C. Otherwise, Dorico displays the closest available note and no accidental – but the exact pitch definition is retained! This is how it’s possible to have your Horn still play back correctly when the symbols are missing.

Any other questions, ask away!

That’s highly impressive, Mark, and thank you enormously again.

I’m still curious though to know how exactly this is done. Are the pitch adjustments possible from within Dorico, and - if so - where can this be accomplished? Dorico’s documentation seems to have little to say on the subject, but that may be because I’m not looking in the right place!

Regards,

Gareth

Actually, Mark, armed with knowledge gained from reading your replies, my search yielded this; How to Set Up Microtonal Tunings | Discover Dorico - YouTube - which I shall look at in detail.

A fascinating topic!

Best wishes,

Gareth