more on Ives stretched 'larger scales'

Here are some further comments and explorations I've
added to my 'Ives's larger scales' webpage.
http://www.ixpres.com/interval/monzo/ives/48-tet.htm

=====================================

> [Ives]
> 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'.]
<< more on this below! >>

>
> = 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.

[monz note: this 'other division, where the scale ended on a
quarter-tone', might be similar to the one illustrated on the
right in the above musical example. It evidently ended on some
quarter-tone pitch which would have been the equivalent of the
'octave' in that scale. The equivalence in the former scale
of the 'augmented 9th' with the 'minor 3rd' - i.e., 'minor 10th'
- apparently bothered Ives and that's why he came up with the
second scale. Too bad we don't know its structure.]

> In this larger scale [monz: that is, the one presented in the
> diagram], 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 'minor 7th' =
'augmented 6th', but rather indicates that this is the large
scale's analog of the 'major 6th'.

It may help to present Ives's statement unedited and in tabular format:

> 3 intervals of even-ratio (so called):
>
> 12-tET diatonic 48-tET-subset
> 'old scale' 'larger scale'
>
> {1} the 4th = 3rd
>
> (2) from 4th to the top
> = minor 7th = 6th
>
> [3] the sum of (1) + (2)
> = from C to Eb top
> = minor 10th. = 8ve]

------------------- end quote ------------------------------

Here's an interval matrix which displays Ives's '3 intervals
of even-ratio (so called)':

Doh 15.00 12.50 10.00 8.75 6.25 3.75 1.25 0.00
Te 13.75 11.25 8.75 7.50 5.00 2.50 0.00 13.75
Lah 11.25 8.75 6.25 5.00 2.50 0.00 12.50 11.25
Soh 8.75 6.25 3.75 2.50 0.00 12.50 10.00 8.75
Fah 6.25 3.75 1.25 0.00 12.50 10.00 7.50 6.25
Me 5.00 2.50 0.00 13.75 11.25 8.75 6.25 5.00
Ray 2.50 0.00 12.50 11.25 8.75 6.25 3.75 2.50
Doh Ray Me Fah Soh Lah Te Doh

What he meant by that was that there are only 3 intervals in
this 48-tET-subset 'larger scale' which have an exact equivalent
in the regular 12-tET scale. This is made plain by the matrix:
the only intervals possible between any two pitches in the
'larger scale' which are exactly the same size as 12-tET intervals
are those of 0.00 (unison in both scales), 5.00 (old 4th = larger
3rd), 10.00 (old minor 7th = larger 6th), and 15.00 (old minor 10th
= larger 8ve) Semitones.

---------------------

I also thought it would be interesting to explore what that
other scale illustrated in the musical examples might be like.

This scale apparently repeats at the 'major 10th', or 2^(4/3)
= 16.00 Semitones, which is the equivalent of the 'octave'.
Each stretched 'semitone' in this larger scale would thus
be (2^(4/3))^(x/12).

Upon analyzing its ratios, we find that it can be described
as a subset of 18-tET:

18-tET Ives larger scale Semitones

Doh 2^(24/18) 2^(4/3) 16.00
Te 2^(22/18) (2^(4/3))^(11/12) 14.&2/3
Lah 2^(18/18) (2^(4/3))^( 9/12) 12.00
Soh 2^(14/18) (2^(4/3))^( 7/12) 9.&1/3
Fah 2^(10/18) (2^(4/3))^( 5/12) 6.&2/3
Me 2^( 8/18) (2^(4/3))^( 4/12) 5.&1/3
Ray 2^( 4/18) (2^(4/3))^( 2/12) 2.&2/3
Doh 2^( 0/18) (2^(4/3))^( 0/12) 0.00

This scale, too, would have probably bothered Ives
because of the equivalence of its stretched 'octave'
with the 'major 10'. Thus he would have been more
interested in that tantalizingly lost other scale
which 'ended on a quarter-tone'.

-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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