Re: Just Intonation

Hello, Adrian

I'm also on the tuning list and live in Sydney Australia also ! I am
currently organising a forum for people in Australia [mostly Sydney]
focusing on Just Intonation and microtonal music. They will be monthly
meetings where people interested in non 12tet tunings can discuss ideas
share music, make instruments & software, perform pieces and formulate and
better understand theoretical ideas on tuning.

As for your thesis Just intonation and posssibly some form of adaptive JI
would be the most pure sounding for instruments with harmonic overtones
[violins, winds, brass]. The piano has overtones that are slightly
inharmonic [or stretched away from harmonic ideals] and for that reason
some people [namely Sethares] has suggested that the best way to tune such
an instrument is by matching the inharmonic partials. Here also adaptive
tuning could work as although the partials are not harmonic they are not
evenly spaced like 12tet tuning. Another option is to change the timbre to
match the tuning so with an electronic tone generator you coulg create a
timbre that had evenly spaced partials so that 12tet would sound pure [i.e
not beat]

Justin White

DAVID JONES LIMITED ACN 000 074 573

Hello and Hau,
My name is James Starkey. Until recently I have been primarily
focused upon painting, drawing and sculpture as a full-time artist.
However, 3 weeks ago I began learning to build 5-hole, pentatonic
Lakota style Native American flutes. I am now taking an unforseen
crash course in tuning. Native flutes have a couple different common
means of tuning: "Traditional TUning" which is tuning where the holes
happen to fall, and "Concert Tuning" which is in keeping with modes
and cents and all that stuff I am now newly delving into.

I recently ran across the "just intonation" way of tuning. I am
trying to find out how I would tune my E flutes using this theory.
Right now I would tune using my newly purchsed chromatic tuner
thusly: E, G, A, B, D, E. Now, I believe I would still tune this way
but to do so using just intonation some notes would be so many cents
sharp and others so many cents flat?

Please, can somebody break it down in layman's terms how I can tune
my flutes using this more natural and healthy way?
Pilamiyayelo,
James Starkey

To write this note, first I had to check 294 messages I received during the last two days. It took me about two hours to check many repeated posts sent by tuning members so the suggestion given by Ben Miller / Herman Miller would be the right solution of the problem.

Months ago some one was interested on getting a 12 tone scale where E should be 5/4 (1.25). In those days I had proposed the three Piagui scales and finally know that they were already presented in Germany some centuries in the past. Recently, Claudio Di Veroli told me that at the present time it is almost impossible to build a new scale since
almost all the scale modes were already deduced and published. I guess that this information does not include the Just Intonation modes.

Since the development of a JI scale of twelve tones per octave is something new for me, I thought that one of the first numbers, in this case number 16, could give the solution if this number is expanded like the Fourier series, which implies a process of sum. The following details might have been exposed by some tuning members in the past, despite that, I will explain the way I got a self constructing JI scale.

Guessing that numbers 2, 4, 8, 16.....are wavelenghts (m) then their inversions (1/N) are frequencies (Hz). Please disregard the fact that actually the light velocity is the constant that controls the product: wave length by frequency so the extremely low frequencies used in the series are not real values. From frequency (0.25) the constant ( 0.0625) was omited. By taking the theoretical frequency of (1/16) = 0.0625 Hz as the constant addend to expand the series, we get:

0.0625 + [(0.0625)] = (0.125) + [(0.0625)] = (0.1875) + [(0.0625] = (0.25) + 0.3125 +

0.375 + 0.4375 + 0.5 + 0.5625 + 0.625 + 0.6875 + 0.75 + 0.8125 + 0.875 + 0.9375 + 1

+ 1.0625 + 1.125 + 1.1875 + 1.25 + 1.3125 + 1.375 + 1.4375 + 1.5 + 1.5625 + 1.625 +

1.6875 + 1.75 + 1.8125 + 1.875 + 1.9375 + 2 + 2.0625 + 2.125 + 2.1875 .........

The red numbers are the heptatonic Just Intonation frequencies while 1.6875, a provisional A, is the pure fifth of 1.125. The middle of 1.375 and 1.4375 gives 1.40625 that could work as a provisional F#. Similarly, 1.78125 could also work as Bb since 1.78125/1.5 gives 1.1875; besides that, the middle of 1.75 and 1.8125 produces the same frequency: 1.78125 Hz while the middle of 1.5625 and 1.625 gives 1.59375 to work as a provisional Ab; since this value is the pure fifth of 1.0625, this frequency could work as a provisional C#.

The provisional tone frequencies will be modified to get narrow fifths with the proper number of cents. Since these calculations are not finished, next message will contain the full data of the 12 tone Just Intonation scale. The aim of this message is to remark the series process used to determine the first group of JI scale tones.

Thanks

Mario Pizarro

piagui@...

Lima, February 17, 2009

I allready know that what I'm about to write will be at first unacceptable
to most people, but I just have to share this.

As many of you will know I've been on a long quest to find how music works
in Just Intonation.
Tried everything I could think of (which has been a lot), and many times
thought I had found it.
Well.. this is one of those times again :)
I think I've found it :)
(or atleast the start of it)

I think (common practice) music works inside 2 chains of pure 3/2 fifths
which run parallel at a 5/4 interval.

Like this: (subset)
0: 1/1 0.000 unison, perfect prime
1: 135/128 92.179 major chroma, major limma
2: 2187/2048 113.685 apotome
3: 10/9 182.404 minor whole tone
4: 9/8 203.910 major whole tone
5: 32/27 294.135 Pythagorean minor third
6: 1215/1024 296.089 wide augmented second
7: 5/4 386.314 major third
8: 81/64 407.820 Pythagorean major third
9: 4/3 498.045 perfect fourth
10: 10935/8192 499.999 fourth + schisma, 5-limit
approximation to ET fourth
11: 45/32 590.224 diatonic tritone
12: 729/512 611.730 Pythagorean tritone
13: 40/27 680.449 grave fifth
14: 3/2 701.955 perfect fifth
15: 405/256 794.134 wide augmented fifth
16: 5/3 884.359 major sixth, BP sixth
17: 27/16 905.865 Pythagorean major sixth
18: 16/9 996.090 Pythagorean minor seventh
19: 3645/2048 998.044
20: 15/8 1088.269 classic major seventh
21: 243/128 1109.775 Pythagorean major seventh
22: 2/1 1200.000 octave

Where the "consonant" major triad at root position is: 1/1 5/4 3/2
And the "consonant" minor triad at root position is: 1/1 1215/1024 3/2

There are several other major and minor triads possible (and used), but
these either don't have root position at the base of the fifth, or they are
less consonant.

Consonance is not only a function of simplicity of ratios, but in musical
context consonance is mainly a lack of tension to resolve.
A dissonance is a ratio that is not "above" the root (for instance 4/3 is a
dissonance if it is above a 1/1 root)

For instance with the dominant 7th chord, it is 1/1 5/4 3/2 16/9. The 16/9
is the dissonance above the 1/1 root which makes it want to resolve.
The high 1215/1024 minor third is not a dissonance even though its ratio
looks far more complex than the 16/9.
It can also occur in the 7th position like this: 1/1 5/4 3/2 3645/2048 here
it is not a dissonance wishing to resolve, it can live happily above the
root.
The music movement / movement of the "fundamental" bass indicates the
structure which indicates wether a major 7th chord is a dominant chord
wishing to resolve, or if it is the 3645/2048 7th which does not need to
resolve (even though it is fairly "discordant" if that's the right word).

I will give musical examples and post mp3's soon.
But I can allready tell it works like magic. And sounds the best I've ever
heard any tuning sound with common practice music.

Marcel

Sorry the below part wasn't very clear.

Consonance is not only a function of simplicity of ratios, but in musical
> context consonance is mainly a lack of tension to resolve.
> A dissonance is a ratio that is not "above" the root (for instance 4/3 is a
> dissonance if it is above a 1/1 root)
>

What I ment to say was that division by anything other than 1 or 2^x is
dissonant.
x/1, x/2, x/4, x/8, x/16.... x/1024 etc above the root position is not
"fighting" the root, it is "consonant" relevant to the root.
x/3, x/5, x/6, x/9 etc are allways "fighting" the root and are "dissonant"
relevant to the root.

Marcel

Here the first musical proof of my JI system:

Beethoven's Drei Equale no2 (no not no1) in Just Intonation:
http://sites.google.com/site/develdenet/mp3/Drei_Equale_No2_%28M-JI_07-July-2010%29.mid

Here the same in 12EDO for comparison:
http://sites.google.com/site/develdenet/mp3/Drei_Equale_No2_%2812edo%29.mid

Visit www.develde.net for the JI transcription (ratios next to notes in an
image, very readable)
And Scala sequence files.
Will also put an mp3 up tomorrow, and other retuned songs soon.

Now if anybody thinks this does not sound absolutely PERFECTLY IN TUNE, then
you should get your ears checked :)

Marcel