deriving freq-to-midi from harmonic series?

[MatthewA]

Forgive the ramble but I don't have to problem pinned down completely:

I'm curious about the mathematical relationship between the harmonic series and equal temperament. I'd like to write a program that quantizes glissandos into the harmonic series similar to birdsong but, of course, by definition, you can't transition between equal tempered intervals with the harmonic series.

Every account of 12-TET I read says it's a pragmatic (non mathematical) approach to fixing just intonation. I guess I'm wondering if there's attempts to derive the former from the ladder... if that makes any sense.

I don't know, it's a creative project that just desires to use a keyboard but the tonality take on that super cool harmonic texture of birdsong.

[g***@u.washington.edu]

I haven't followed this so closely, but I do remember a story about
an alternative to equal tempered that has 53 notes/octave.
(Well, it probably wouldn't have been called an octave.)

Equal tempered has the convenience that you can change key without
retuning all the keys. It occurs to me, though, that in the case
of electronic pianos, it would be very easy to switch to just
tempered for each key.

[Les Cargill]

Post by g***@u.washington.edu
I haven't followed this so closely, but I do remember a story about
an alternative to equal tempered that has 53 notes/octave.

It's called 53 T(one)E(qual)T(emperament). Whoda thunk :)?

Post by g***@u.washington.edu
(Well, it probably wouldn't have been called an octave.)

Yes, it still would be - an octave is the most fundamental thing about
any temperament. Er, I don't know of any that don't .. conform to them.

Post by g***@u.washington.edu
Equal tempered has the convenience that you can change key without
retuning all the keys. It occurs to me, though, that in the case
of electronic pianos, it would be very easy to switch to just
tempered for each key.

There are plugins and boxes which enable all manner of temperaments.
It's just a matter of sending a calibrated MIDI pitch bend with each note.

One thing about really old pipe organs is that they may not be in ET
for mechanical and historical reasons. That's part of the sound.

And at the risk of being boring, here's the pedal steel player Buddy
Emmons set of offsets he tuned his steel to. Steel players stare
temperament dead in the face every time they sit down to one.

The up-down is strings ( on two necks ); left-right is a knee
lever or foot pedal.

http://www.buddyemmons.com/TTChart.htm

[g***@u.washington.edu]

On Sunday, July 5, 2020 at 10:22:24 PM UTC-7, Les Cargill wrote:

(I wrote)

Post by Les Cargill

Post by g***@u.washington.edu
I haven't followed this so closely, but I do remember a story about
an alternative to equal tempered that has 53 notes/octave.

It's called 53 T(one)E(qual)T(emperament). Whoda thunk :)?

Post by g***@u.washington.edu
(Well, it probably wouldn't have been called an octave.)

Yes, it still would be - an octave is the most fundamental thing about
any temperament. Er, I don't know of any that don't .. conform to them.

But why is it called octave? What are there eight of?

[Tauno Voipio]

Post by g***@u.washington.edu
(I wrote)

Post by Les Cargill

Post by g***@u.washington.edu
I haven't followed this so closely, but I do remember a story about
an alternative to equal tempered that has 53 notes/octave.

It's called 53 T(one)E(qual)T(emperament). Whoda thunk :)?

Post by g***@u.washington.edu
(Well, it probably wouldn't have been called an octave.)

Yes, it still would be - an octave is the most fundamental thing about
any temperament. Er, I don't know of any that don't .. conform to them.

But why is it called octave? What are there eight of?

Full tone steps, see the white keys on a piano.

[g***@u.washington.edu]

On Monday, July 6, 2020 at 12:37:51 PM UTC-7, Tauno Voipio wrote:

(snip, I wrote)

Post by Tauno Voipio

Post by g***@u.washington.edu
But why is it called octave? What are there eight of?

Full tone steps, see the white keys on a piano.

But that is for the system we have now.

If we had the 53 note system instead, how many keys would be
on a piano for each frequency doubling? (Ignoring the complications
of building and/or playing one.)

Seven white keys and 46 black keys?

Seven white keys and 46 keys of a variety of other colors?

More than seven white keys, along with other colors of keys?

(That is, assume that no pianos like we now know were ever made,
that building piano keys is easy and cheap, and that the problem
of how to play it can be overcome.)

[Les Cargill]

Post by g***@u.washington.edu
(I wrote)

Post by Les Cargill

Post by g***@u.washington.edu
I haven't followed this so closely, but I do remember a story about
an alternative to equal tempered that has 53 notes/octave.

It's called 53 T(one)E(qual)T(emperament). Whoda thunk :)?

Post by g***@u.washington.edu
(Well, it probably wouldn't have been called an octave.)

Yes, it still would be - an octave is the most fundamental thing about
any temperament. Er, I don't know of any that don't .. conform to them.

But why is it called octave? What are there eight of?

Notes in most scales. Some scales drop notes, thwe most common being the
pentatonic ( five note ).

1 2 3 4 5 6 7 8
C,D,E,F,G,A,B - C

[Les Cargill]

Post by MatthewA
Forgive the ramble but I don't have to problem pinned down
I'm curious about the mathematical relationship between the harmonic
series and equal temperament. I'd like to write a program that
quantizes glissandos into the harmonic series similar to birdsong
but, of course, by definition, you can't transition between equal
tempered intervals with the harmonic series.
Every account of 12-TET I read says it's a pragmatic (non
mathematical) approach to fixing just intonation. I guess I'm
wondering if there's attempts to derive the former from the ladder...
if that makes any sense.
I don't know, it's a creative project that just desires to use a
keyboard but the tonality take on that super cool harmonic texture of
birdsong.

Look up "Musical temperament" on Wikipedia for starters.

There are a lot of different temperament systems - Pythagorean.
many forms of Just and Equal. There's Meantone but I know nothing of it.
Pythagorean is nice ratios but you get "wolf tones". The various forms
of Just "fix" these but there are still wolf tones.

ET came about because of the pianoforte - one could play a piece in any
key, so the need arose to distribute the "wolf" equally. It's very
mathematical - each pitch ascending is Fprev*(pow(2,1/12)) .

There is A 440.0000 . T he next note ( A#/Bb ) is 466.1638.

The ratio is 1.05946318... , or "the twelfth root of 2" ( pow(2,1/12) )
which comes down to 1.0594630943592953098431053149397485 given enough :)
digits. And pow(pow(2,1/12),12) is, unsurprisingly a ratio of 2 - an
octave.