Message: 1
Date: Mon, 6 Dec 1999 03:50:00 -0500
From: Joe Monzo
Subject: Re: Ives's tuning: a 48-tET-subset scale
I've been immersed in Aristoxenus for a couple of weeks,
but have been enjoying the discussion of Ives's tuning
very much, and found something at the library that
I thought you all might be interested in.
I've made a webpage with a MIDI-file of this scale,
http://www.ixpres.com/interval/monzo/ives/48-tet.htm
and reproduce the text here for those without web access:
(view in a fixed-width font, such as Courier)
=========================================
A 48-TET-SUBSET SCALE OF CHARLES IVES
-------------------------------------
For the record, I'm all for Johnny Reinhard's work on Ives's
tuning. His Pythagorean version of 'Unanswered Question' had
the most awesomely beautiful sound in the string section that
I've ever heard in any performance of that piece. At the same
time, I agree with Dan Stearns that Ives's conception of tuning
was far more complex than simply having one personal favorite
tuning.
I looked up some stuff on Ives at the library back when
this thread first started (wasn't that a couple of months
ago?), thinking I might find some evidence for Pythagorean.
I copied one double-page from something called 'Scrapbook'
(I think it was part of the 'Memos' but at this point
I don't remember), without having read it, because it had
a scale diagram that caught my eye.
I thought at first that it was some kind of overtone graph,
but upon finally reading it the other day, found that it was
an interesting scale Ives had come up with, based on two pianos
in his church that were about 1/4-tone out of tune and
some 'glasses' his father had made that played 1/8-tones.
Here's the quote - if someone could provide details for
the citation, I'd appreciate it. (I think it's from 'Memos'.)
The editing and footnotes by me have been identified as such;
the unidentified ones are by the editor of the book. I've done
my best to make an ASCII rendering of the musical example.
I'm not sure what the numbers I put in brackets mean - they
appear in the margins of the book, and apparently refer to
the cataloging of the original Ives manuscripts.
> -------Charles Ives, from 'Scrapbook', p 108-110 --------
>
>
> [51m]
>
> ... as a boy I had heard some quarter-tone experiments of
> Father, and this division or other divisions of the tone were
> not entirely unfamiliar to me. In the Sunday-School room of
> the Central Presbyterian Chruch [*5], New York, there were,
> for a while, two pianos which happened to be just about a
> quarter tone apart, and I tried out a few chords then.
>
>
> [m51v]
>
> In this connection, and also referring to Father's glasses
> tuned in different intervals larger and less than quarter tones,
> after hearing the two pianos out of tune in Central Church (but
> asnear as I could tell by listening and with tuning forks, [they]
> were about a quarter tone apart) - a scale (to knock the octaves
> and fifths out by wider intervals, stretching [the] whole and
> half tones a little, but keeping the proportions of the scale)
> - it was started or suggested by these two pianos, and glasses
> between [the quarter tones]. But one piano was moved before I
> could get it well grasped [*6] in my ears. It was mostly worked
> out on paper, which I have in part (see back of _The Indians_
> score)[*7] - taking C as basis, 5 quarter tones up = whole
> interval, and divided in [the] middle by [a] glass = 2&1/2
> [quarter] tones - that is:
[monz note: The diagram was presented horizontally by Ives;
I present it vertically, and add mathematical notation from
two different perspectives to specify the tuning.]
48-tET Ives larger scale Semitones
>
> Notes in
> old scale
>
> Eb 31 -+---- 8 Doh 2^(60/48) = 2^(5/4) 15.00
> |
> 30 -+
> |
> D 29 -+
> |---- 7 Te 2^(55/48) = (2^(5/4))^(11/12) 13.75
> 28 -+
> |
> C# 27 -+
> |
> 26 -+
> |
> 8 C 25 -+
> |
> 24 -+
> |---- 6 Lah 2^(45/48) = (2^(5/4))^(9/12) 11.25
> 7 B 23 -+
> |
> 22 -+
> |
> A# 21 -+
> |
> 20 -+
> |
> 6 A 19 -+
> |---- 5 Soh 2^(35/48) = (2^(5/4))^(7/12) 8.75
> 18 -+
> |
> G# 17 -+
> |
> 16 -+
> |
> 5 G 15 -+
> |
> 14 -+
> |---- 4 Fah 2^(25/48) = (2^(5/4))^(5/12) 6.25
> F# 13 -+
> |
> 12 -+
> |
> 4 F 11 -+---- 3 Me 2^(20/48) = (2^(5/4))^(4/12) 5.00
> |
> 10 -+
> |
> 3 E 9 -+
> |-- Re
> 8 -+
> |
> D# 7 -+
> |
> 6 -+---- 2 Ray 2^(10/48) = (2^(5/4))^(2/12) 2.50
> |
> 2 D 5 -+
> |
> 4 -+
> |-- De
> C# 3 -+
> |
> 2 -+
> |
> 1 C 1 -+---- 1 Doh 2^(0/48) = (2^(5/4))^(0/12) 0.00
>
>
>
>
> [m52v]
>
> - (playing larger scale and then regular one alternately several
> times - and it is quite an interesting sound difference and makes
> a kind of musical sense).
>
> New octaves, that is:
>
>
>
> _________
> / |
> /
> / O
> / ---
> / cycle
> / ---
> /
> / ---
> /
> / --- -O-
> |
> /\ -O- --- \ ---
> | | \
> --|-|------------------------------------\--------------------
> |/ \
> --|----------------------------------------\------------------
> /| \
> |- \-----------------------------------------\----------------
> | / | \ \
> |-|-|-|-----------------------------------#O---\--------------
> \___|/ #O \ \
> ----|-----------------------------------\--------\------------
> \__/ \ 4
> \
> \
> --------------------------------------------\-----------------
> \
> -_____----------------------------------------\---------------
> / \ . O \
> \------|-----#O---------------------------------\-------------
> / . \
> -----/--------------------------------------------\-----------
> / \ \ 3
> ---/--------------------------------------\-------------------
> --- --- \
> -O- -O- \
> \ 2
>
>
[monz note: I don't understand what the notes and numbers
on the right of this example signify - perhaps someone else
out there has a clue. The scale under discussion has cyclic
properties based on 'minor 10ths', the notes on the right
seem to be a cycle based on 'major 10ths'.]
>
> = no octaves nor 5ths during each four octaves, or no
> octaves nor 5ths for 48 half-tones, and [the] only interval
> in common is [the] lower 4th. But [the] trouble is: - [the]
> augmented 9th, taken as a scale length, may be confused with
> [the] minor 3rd. I had some other division, where the scale
> ended on a quarter-tone - can't find it. In this larger scale
> [monz: that is, the one presented here], there are but three
> intervals of even-ratio (so called): {1} the 4th [of the old
> scale] = [the] 3rd [of the larger scale]; or (2) from [the old]
> 4th to the top [of the larger scale] = minor 7th [monz: of
> the old scale] = [augmented]* 6th [monz: of the new scale];
> and [3] the sum of (1) + (2) = from C to Eb top = minor 10th
> [monz: of the old scale = 'octave' of the larger scale].
>
*[monz note: The editor is wrong here; '6th' refers not to
the equivalence of the 12-tET or meantone 'augmented 6th'
= 'minor 7th', but rather indicates that this is the large
scale's analog of the 'major 6th'.]
>
> The other intervals are uneven - some way out from a simple
> ratio [as] 2/1 - for instance 261/712 etc. This, at first,
> seemed very disturbing, - but when the ears have heard more and
> more (and year after year) of uneven ratios, one begins to feel
> that the use, recognition, and meaning (as musical expression)
> of intervals have just begun to be heard and understood. The
> even ratios have been pronounced the true basis of music,
> because man limits his ear, and not because nature does. The
> even ratios have one thing that got them and has kept them in
> the limelight of humanity - and one thing that has kept the
> progress to wider and more uneven ratios very slow - (it is said
> [that] for the power of man's ear to stand up against the
> comparatively uneven 3rds, [when used] to the very even octaves
> and 5ths, was a matter of centuries) - in other words, consonance
> has had a monopolistic tyranny, for this one principal reason:
> - it is *easy* for the ear and mind to use and know them - and
> the more uneven the ratio, the harder it is. The old fight of
> evolution - the one-syllable, soft-eared boys are still on too
> many boards, chairs, newspapers, and concert stages!
>
> --------------------
> Editor's notes:
>
> [*5] Ives was organist there from April 1900 through 1 June 1902.
>
> [*6] This word is hard to make out, but is quite possibly
> "grasped".
>
> [*7] There are three copies of the following diagram differing
> only in minor details: (1) on a rejected title page of "last
> Chorus" of _The Celestial Country_ (Q1718), back of which is
> p. 1 of the score-sketch of _The Indians_ (Q2838) - (2) in m51v
> - and (3) in m52v.
>
------------------- end quote ------------------------------
Ives's 'larger scale' is based on an approximate 48-tET
division of the 'octave'. Its 'whole-step' = 2^(10/48)
= 2^(5/24) = 2.50 Semitones, and 'half-step' = 2^(5/48)
= 1.25 Semitones; in other words, each 'half-step' is
stretched, so that it is 1/8-tone = 2^(1/48) larger than
the usual 12-tET 'half-step' or semitone.
The 12-tET 'minor 10th' = 2^(5/4) is used in Ives's 'larger
scale' as the cyclic equivalent of an 'octave', so that each
'step' in his scale may be represented more clearly as
(2^(5/4))^(x/12), where x = the equivalent 'step' in 12-tET.
Listening to the MIDI file of this scale
http://www.ixpres.com/interval/monzo/ives/lg-qtone.mid
makes it clear to me that it sounds nothing like any 'usual'
scale, but that it has an obvious tetrachordal structure
which gives it musical qualities that are easily recognizable.
Given all the talk in this forum about how Ives considered
sharps to be higher in pitch than flats, I find it very
interesting that his notation here of 'notes in the old scale'
uses sharps for all of the chromatic notes except for the flat
which marks the highest note, the 'octave' of the 'larger scale'.
I don't know if there's any significance to that, but it
seems noteworthy.
-monz
Joseph L. Monzo Philadelphia monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
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Back to my paper on Ives's 48-tET scale