Just Intonation

Notes on Designing a Just Intoned Whistle

What we call modes are scales in their own right, and cannot for JI be derived simply by shifting the starting point, as is often done with ET.

But we can describe JI modal scales with simple JI intervals:
T = 9:8 major tone, ~204 cents
t = 10:9 minor tone, ~182 cents
s = 16:15 semi tone, ~112 cents

In the following I used only the following tetrachord patterns:
(Tts) for a major tetrachord
(tsT) for a minor tetrachord
(sTt) for a phrygian tetrachord
To my ear this gives good sounding modal scales, and has some other advantages detailed below.

In the modal scales above all tetrachords form a pure Fourth from first to fourth note,
and each scale has a pure Fifth thanks to the major tone T at the centre (pure fourth plus major tone = pure fifth).

But it also makes it clear that only some patterns can be derived by shifting the stating point:
The Dorian pattern is the same than the Ionian starting on the second.
The Mixolydian pattern is the same than the Ionian starting on the fifth.
But we don't get the Phrygian nor the Aeolian by starting on the third or sixth of the Ionian, instead we get a minor tone t at the centre, which means the fifth is not pure.

The conclusions are exactly the same:
For a JI whistle design we need to introduce some compromise somewhere, or optimise the design to play for certain modes, and other modes need intonation shifts by the player. And there is no one answer.
On a D whistle the notes which needs varying are mainly E and B, depending on which JI scheme the design is based on.

If we optimise the design for a D whistle for G major we get these patterns from the whistle root:

TtsTtsT (using Cnat)
TtsTtTs (using C#)
or D E+ F#- G A B- C-/C#- D
+ and - as indicators for sharpness / flatness (though we have precise values for this).

With such a whistle we can play naturally in G Major, D Mixolydian, A Dorian and F# Phrygian.
And by blowing the B sharp we can play in D Major, E Dorian and A Mixolydian.

Tuning differences from ET in cents:

The B needs to be blown sharp 22 cents to +6 ET for Dmaj, Edor, Emin, Amix.
The C is obtained using a thumb hole.

To play E Aeolian on a D flute/whistle tuned like that would require the player to play the B sharp (relative to ET pitch), as well as the Cnat and the D. If you don't raise the Cnat and D you get an Aeolian tuning of a different character, which to me does not quite sound good enough. Still harmonious, but different.