Shortened helical antenna cannot provide satisfactory operation above
the third harmonic frequency.
Shortened helical antenna that has not harmonic frequencies could work
effective only at the resonance frequency. Operation of the antenna
on the ranges below or higher the resonance frequency would be
unsatisfactory.
Capacitive top loading shortened helical antenna could
provide effective operation or (if any presents) at the fundamental resonance
frequency either at the second and third harmonic frequency. The
antenna has poor operation on the ranges between the resonance
frequencies.
In the conclusion I would like add that some ham VHF/UHF transceiver
for 144/430 MHz has one helical antenna that work at 144 MHz
and on the 433 MHz band. The 433 MHz is third harmonic frequency
for the antenna with fundamental resonance frequency 144 MHz.

Example
of Calculating a Shortened Helical Antenna for the 40 meter Band
Let's calculate shortened helical antenna for the 40
meter band. Figure
13 shows vertical shortened helical antenna for the
40 meter band. Figure
14 shows dipole shortened helical antenna for the 40
meter band. Lambda for the 7.05 MHz is equal to 41.5 meter. To
make the vertical helical antenna for the 40 meter band it required
length of wire in 21 meter to wound on dielectric form in any
diameter from 10mm to 10cm.
However, practice shows that the length of the wire
should be a bit longer than the required lambda/2. Longer length
allows tune the antenna to the resonance frequency by cutting
the wire by small pieces. Dipole helical antenna for the 40 meter
band will contain two such helixes in the length.
